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Prathap Latch

2011-04-08T16:10:27Z

Pl1208: Prathap Latch moved to Sunkiss pl1208 2

#REDIRECT [[Sunkiss pl1208 2]]

Sunkiss pl1208 2

2011-04-08T16:10:27Z

Pl1208: Prathap Latch moved to Sunkiss pl1208 2

=Introduction to Inorganic Computational Chemistry=

Module one allowed the implementation of different computational techniques towards the rationalisation of organic chemical reactions and characterisation of specific organic molecules. The second module is expected to extrapolate the computational skills learnt; towards the more complex inorganic molecules. In organic molecules, the main elements considered were carbon, hydrogen or halides (high up the periodic table). However, in inorganic chemistry, the type of molecules dealt with could involve transition metal complexes and other heavier elements. This would make calculations carried out to be more resource and time consuming.

Optimisations allow the total minimised energy of different conformers to be attained.This allows determination for which of the possible compounds is a more stable conformer. This determination of the more stable conformer and the location of the transition states or activated complex is essential in fully understanding the mechanism and rationalising the outcome of a reaction. But this cannot be done experimentally and only via the aid of modelling softwares such as gaussian.

<u>
The structure of this module comprises of three main segments as follows</u>

* A small inorganic molecule(BH<sub>3</sub>) will be analysed via geometrical optimisations of the molecule.

* A comparison to evaluate the effectiveness and accuracy of the linear combination of atomic orbitals(LCAO) approach will be carried out. The expected molecular orbitals of the molecule are compared against the computationally derived molecular orbitals.

* A vibrational frequency analysis will be carried out to distinguish the minimum energy level from the transition state. The same approach will then be taken with a heavier G(III) element such as thallium to draw parallels and spot differences in calculations brought about by the substitution for a heavier central atom.

* Molecular modelling of a transition metal complex will be carried out. The importance for the use of Pseudo-potentials and the larger basis sets will be learnt. Also the draw backs of the gaussian software towards inorganic complexes will be understood. The influence of ligands at different positions (cis and trans) has on the overall electronics and strcuture will be analysis via bond length analysis and vibrational analysis as well.

* The final segment will involve a mini-project that looks into the influence of different heteroatoms in benzene analogues. A similar structure in optimisation and vibrational analysis will be undertaken to rationalise the electronic and structural differences.

=Optimisation of BH<sub>3</sub> via Density Functional Theory (DFT)=

The first segment aims to improve familiarisation of the Gaussian and Gaussview interfaces via simple optimisation calculations of the borane molecule, BH<sub>3</sub>. The BH<sub>3</sub> molecule was first drawn and the trigonal-planar geometry was selected. The Density Functional Theory (DFT)-B3LYP based optimisation was carried out. Pre-optimisation paraments were set-bond distance between B and H atoms as 1.5 Å.

{| class="wikitable" border="1" align="center"
|+ Table 1 :'''Parameters Implemented for BH<sub>3</sub> optimisation'''
! <CENTER>Parameter </CENTER> !! <CENTER>Specifics </CENTER>
|-
| <CENTER> Job Type</CENTER> || <CENTER>Optimisation </CENTER>
|-
|<CENTER>Method of Optimisation </CENTER> ||<CENTER>Density Functional Theory (DFT) </CENTER>
|-
| <CENTER>Specific DFT </CENTER> ||<CENTER>B3LYP </CENTER>
|-
| <CENTER> Basis Set used </CENTER>|| <CENTER> 3-21G</CENTER>
|}

The aim of optimisation was to determine the optimum displacement of the hydrogen nuclei from the boron centre under the BH<sub>3</sub> trigonal planar electron configuration. The borane molecule is a small and highly symmetrical molecule with the D<sub>3h</sub> point group where the orbitals involved are the 1s,2s and 2p orbitals. Thus, a simple basis set such as the 3-21G is sufficient to computationally model and determine the optimum bond distance for the B-H bond.

== Results from DFT Optimisation ==
[https://wiki.ch.ic.ac.uk/wiki/images/4/4c/BH3_OPT_pl1208.LOG '''Borane Optimisation''']

{| class="wikitable" border="1" align="center"
|+ Table 2 :'''Comparing the pre and post BH<sub>3</sub> optimisation'''
!<CENTER> Pre-optimised Borane Molecule </CENTER>!!<CENTER> Optimised Borane Molecule </CENTER>!! rowspan="4"|[[image:Reaction_summary_pl1208.JPG |thumb|Diagram 1- calculation summary upon optimisation| centre]]
|-
| width=300px | <CENTER><jmol><jmolApplet><title>Pre-optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>Pre-optimised_bh3_pl1208.mol</uploadedFileContents>
</jmolApplet></jmol></CENTER>||width=300px |<CENTER><jmol><jmolApplet><title>Optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>Optimised_bh3_pl1208.mol</uploadedFileContents>
</jmolApplet></jmol></CENTER>
|-
|<CENTER> B-H bond length = 1.50 Å </CENTER> ||<CENTER> B-H bond length = 1.19 Å </CENTER>
|}

The data in table 2 was extracted from the output log file where the intial bond distance of 1.50 Å was reduced to 1.1944 Å upon optimisation. The optimised H-B-H bond angle was calculated to be 120.0 degrees. When this was compared with literature data (1.18 Å,120 deg) they were in exact agreement with the calculated values.<ref>Fourier transform infrared spectroscopy of the BH, v3 band,Kentarou Kawaguchi, J. Chem. Phys., Vol. 96, No. 5,1 March 1992</ref> The bond angle can be rationalised being due to the sp<sup>2</sup> hybridised centre that subtends the hydrogen atoms at 120 degrees.

For the calculation summary (diagram 1), results are given in atomic units (a.u.) where the unit conversion is 1 a.u. = 2625 KJ/mol. When the RMS gradient is smaller than 0.001 a.u., the change in potential energy compared against the change in bond length results in an almost horizontal plot. This is indicative that the molecule has been fully optimised.

[[Image:BH3_optimisation_output_file.bmp|thumb|300x300px|Diagram 2 - Optimisation Output file for BH3|centre ]]

To determine if the reaction had reached completion, the output file was further analysed. For the calculation to have reached completion, derivatives of the forces and displacements must converge. This is true for borane optimisation(outline in blue,diagram 2) which has both force and displacement derivatives having a magnitude lesser than the present threshold value. This is indicative of the convergence; structure is fully optimised.

=== Further Analysis ===

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 3 : Summary of the different optimisation steps against the Bond lengths of the BH bonds'''</u>
! <CENTER> Optimisation Step No.</CENTER> !!<CENTER> Varying B-H bond Diagram </CENTER> !! <CENTER> Varying Total Energy (a.u.)</CENTER> !! !! <CENTER> Varying B-H Bond Length (Å)</CENTER> !! <CENTER>Plot of Energy and RMS Gradient </CENTER>
|-
| width=150px|<CENTER> 1 </CENTER>|| [[image:Optimisation_step_1_pl1208.JPG|centre|140px]] || width=150px|<CENTER>-26.38 </CENTER>|||| width=150px|<CENTER> 1.50 </CENTER>|| rowspan="5"|[[image: Energy_and_gradient_diagram.JPG|centre|300px]]
|-
|<CENTER> 2 </CENTER>|| [[image: Optimisation_step_2_pl1208.JPG|centre|140px]] || width=150px|<CENTER> -26.42 </CENTER>||||<CENTER> 1.41 </CENTER>
|-
|<CENTER> 3 </CENTER>|| [[image: Optimisation_step_3_pl1208.JPG|centre|140px]] || width=150px|<CENTER> -26.46 </CENTER>||||<CENTER> 1.23 </CENTER>
|-
|<CENTER> 4 </CENTER>|| [[image: Optimisation_step_4_pl1208.JPG|centre|140px]] || width=150px|<CENTER> -26.47 </CENTER>||||<CENTER> '''1.19''' </CENTER>
|}

From table 3, the RMS gradient at step 4 is indicative of the completion of the BH<sub>3</sub> optimisation calculation. This is because by step 4, the difference in total energy between the pre-optimised and optimised molecule falls below the pre-set threshold value. Another trend observed is as the bond lengths get shorter down the table, total energy of the system becomes more negative. Thus, optimisation involves total energy minimisation where the lowest energy configuration is adopted by bringing the H atom closer to the Boron centre. However, any closer could result in significant van der waals repulsions that would instead cause the total energy to rise. Thus, bond length does not decrease further than the 1.19 amstrongs.

== Molecular Orbital Analysis of Borane ==

LCAO is used to rationalise the shape and geometry of the BH<sub>3</sub> molecular orbitals. DFT-based calculations were used to determine the individual molecular orbitals of Borane. The LCAO approach could then be aligned against the DFT-calculated orbitals to determine how accurate LCAO is in predicting the molecular orbitals at the varying energy levels.

=== Molecular Orbital Diagram attained via the LCAO approach ===

[[Image:MO_DIAGRAM_PL1208.jpg|thumb|500X500px|Diagram 3 - MO Diagram for Borane molecule|centre ]]

For the formation of the molecular orbitals via the LCAO approach, it was carried out via the linear combination of two fragment orbitals (FO. The two FOs make up 8 atomic orbitals (AO) (Black Lines). As the number of AOs = number of MOs, 8 molecular orbitals are expected from the linear combinations. The extent of splitting in the formation of the molecular orbitals is approximated to the degree of overlap and the difference in energy levels. The smaller the difference in energy levels of the AOs and the greater the degree of overlap, the larger the bonding-antibonding orbitals splitting. The borane molecule has a point group of D<sub>3h</sub>. Thus, with the aid of the character table, the MO diagram is constructed where each MO is given the symmetry label (in blue).

=== Molecular Orbitals attained via DFT calculations {{DOI|10042/to-7458}} ===

[[Image:DFT_MO.JPG|thumb|300X300px|Diagram 4 - DFT calculation for each MO of borane with specific energy levels |centre ]]

The quantum-based DFT calculations are expected to be more accurate as it takes into account electron and orbital interactions. It is expected to give a more accurate representation of the actual molecular orbitals. The geometrically optimised BH<sub>3</sub> output file was ran via SCAN to yield the results in diagram 4. It is clearly seen that the first four (lowest) MO levels are occupied. This is similar to that depicted in the earlier drawn MO diagram via LCAO (diagram 3).

<u>'''Comparing DFT against LCAO'''</u>

{| class="wikitable" border="1" align="center"
|+ '''Table 4 Comparing the DFT generated MO diagrams against the predicted MO diagrams via LCAO'''
! Molecular Orbital(Lowest to highest energy) !! MO predictions via LCAO !! Orbital Symmetry ( LCAO ) !! DFT Calculation Depiction !! Orbital Symmetry ( DFT ) !! Energy relative to H Atom (amu)via DFT
|-
|<CENTER> 1 </CENTER>|| [[image:MO1_PL1208.bmp|centre|150px]] ||<CENTER> 1a<sub>1</sub>' </CENTER>|| stretch | [[image: MO1_DFT_pl1208.JPG|centre|150px]]||<CENTER> 1a<sub>1</sub>' </CENTER> ||<CENTER> -6.73 </CENTER>
|-
|<CENTER> 2 </CENTER>|| [[image:MO2_PL1208.bmp|centre|150px]] ||<CENTER> 2a<sub>1</sub>' </CENTER> || [[image: MO2_DFT_pl1208.JPG|centre|150px]]||<CENTER> 2a<sub>1</sub>' </CENTER>||<CENTER> -0.517 </CENTER>
|-
|<CENTER> 3 </CENTER>|| [[image:MO3_PL1208.bmp|centre|150px]] ||<CENTER> 1e' </CENTER>|| [[image: MO3_DFT_pl1208.JPG|centre|150px]]||<CENTER> 1e' </CENTER>||<CENTER> -0.356 </CENTER>
|-
|<CENTER> 4 </CENTER>|| [[image:MO4_PL1208.bmp|centre|150px]] ||<CENTER> 1e' </CENTER>|| [[image: MO4_DFT_pl1208.JPG|centre|150px]]||<CENTER> 1e' </CENTER>||<CENTER> -0.356 </CENTER>
|-
|<CENTER> 5 </CENTER>|| [[image:MO5_PL1208.bmp|centre|150px]] ||<CENTER> 1a' '<sub>2</sub> </CENTER>|| [[image: MO5_DFT_pl1208.JPG|centre|150px]]||<CENTER> 1a' '<sub>2</sub> </CENTER> ||<CENTER> -0.074 </CENTER>
|-
|<CENTER> 6 </CENTER>|| [[image:MO6_PL1208.bmp|centre|150px]] ||<CENTER> '''3a<sub>1</sub>'''' </CENTER>|| [[image: MO6_DFT_pl1208.JPG|centre|150px]]||<CENTER> '''2e'''' </CENTER>||<CENTER> 0.188 </CENTER>
|-
|<CENTER> 7 </CENTER>|| [[image:MO8_PL1208.bmp|centre|150px]] ||<CENTER> '''2e'''' </CENTER>|| [[image: MO7_DFT_pl1208.JPG|centre|150px]]||<CENTER> '''2e'''' </CENTER>||<CENTER> 0.188 </CENTER>
|-
|<CENTER> 8 </CENTER>|| [[image:MO7_PL1208.bmp|centre|150px]] ||<CENTER> '''2e'''' </CENTER>|| [[image: MO8_DFT_pl1208.JPG|centre|150px]]||<CENTER>'''3a<sub>1</sub>'''' </CENTER>||<CENTER> 0.192 </CENTER>
|}

The MO diagram attained via LCAO is in close agreement with the individual molecular orbitals calculated via the Gaussian energy function. When the order of the calculated MOs are compared with the predicted MOs against increasing energy levels, the following conclusions were made.

Both methods have very similar orbital representations from MO-1 to MO-8. They are very similar but not exact due to the difference in arrangement of the 2 highest anti-bonding orbitals (MO-7 and MO-8) which have an energy difference of 0.004 Hartrees. This shows the first limitation of the LCAO. By having such a small energy difference, the two anti-bonding orbitals are able to interchange positions bringing about the slight deviation for the order of LCAO molecular orbitals against the actual energy-based ordering of the DFT calculated orbitals.

The limitation of LCAO is that it is not able to derive the specific energy levels and is based on the assumption that the higher the degree of overlap, the greater the orbital splitting. It was expected that the a<sub>l</sub><sup>'</sup> fragment orbitals have a strong splitting due to the high degree of S-S orbital overlap. However, it could not be quantitatively determined if the splitting would have the 3a<sub>l</sub><sup>'</sup> MO to be higher than the 2e<sup>'</sup> molecular orbitals. These are degenerate molecular orbitals formed via side-side p-orbital overlaps which were less effective and resulted in a lesser extent of splitting. In LCAO, it was assumed that the P-P splitting was not large enough for the antibonding 2e<sup>'</sup> orbital to be of a higher energy the 3a<sub>l</sub><sup>'</sup> antibonding MO. But by DFT, it was proven otherwise as seen in the table above.

Pictorially comparing LCAO MOs against DFT MOs, it is concluded that LCAO gives a very good approximation to the shape and size of the orbital lobes where the larger the lobe the greater the electron density about that atom. Comparing the filled molecular orbitals (MO-1 to MO-4), LCAO is very accurate in terms of the orbital phases and relative size of the lobes for the electron occupied molecular orbitals.

However, for the unoccupied MOs, there is a more significant deviation of the predicted LCAO orbital's shape and lobe size from the actual orbital's shape and size derived via DFT methods. For example, in general the Hydrogen orbitals in MO-5 to MO-8 were much larger than predicted in the LCAO. MO 3a<sub>l</sub><sup>'</sup> due to the much larger antibonding phase was expected to have a large central orbital surrounded by 3 smaller and opposite phases about each of the hydrogen. But instead, DFT reveals that all the 4 atoms have almost equal size in orbitals. This can be reasoned as the unoccupied orbitals being unfilled, have a more diffused character which makes it harder to accurately predict to the molecular orbitals. Thus, they usually result in the calculated orbitals to have a much larger, diffused lobe than expected.

In conclusion, apart from the minor deviations LCAO is able to give a good prediction of the orbitals. But the potential for deviations can be inferred and be expected to become more significant when larger molecules with more orbitals are involved. Since the potential for orbital-mixing occurs; making the small energy differences more likely and harder to predict via LCAO.

== Natural Bond Order (NB0) Analysis of Borane {{DOI|10042/to-7458}}==

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Mo_analysis_pl1208.out Borane NBO Analysis]

The NBO analysis carried out allows the electronic distribution within the molecule to be quantitatively calculated. This can be done to produce numerical values to complement the general conclusions made about the electron distribution within the molecule.

Considering the orbitals, Boron being tri-substituted has a total of only 6 electrons in the valence shell. It is expected to be electron deficient as it has not achieved the stable octet configuration. Thus, availability of the unoccupied P<sub>z</sub> orbital makes the compound to be lewis acidic. Being tri-substituted, the boron centre is expected to be SP<sup>2</sup> hybridised; the S-orbital to P-orbital contributions to the hybridised orbital should be the 1:2 ratio.

[[Image:NBO_diagram_pl1208.JPG|thumb|400x300px|Diagram 5-Contoured Colour Depiction of Borane|centre ]]

Using the colour depiction, regions coloured red are indicative of high electron density localised at that specific position (atom) while green is indicative of a region of low electron density which is indicative of electron deficiency. In the diagram, all 3 hydrogens with their valence S-orbital fully filled are electron rich as compared to the borane centre which is colour green due to only 6 of the 8 needed for full valence shell. Thus, as predicted earlier the compound is electron deficient about the centre and is an ideal site for an nucleophilic attack. Similarly, the charge numbers are indicative of the electron density about the atom. -0.101 abotu the H atoms denotes the electron rich region while a the 0.322 about the boron denotes an electron deficient site.

[[Image:data_nbo_pl1208.jpg|thumb|400x400px| Diagram 6- Log file Data|centre]]

The gaussview provided the pictorial interface but no specific NBO data was given. Thus, analysis of the log file allowed the above section to be isolated and the following conclusions to be made. For each boron-hydrogen bond, the Boron centre contributes 44.48% of the electron density while the hydrogen contributes 55.52%. There is a greater contribution by the Hydrogen-S orbitals than the Boron orbitals. The second red box shows that the boron based orbitals have 33% s character and 67% p character. This will be in the 1:2 ratio indicative of the sp<sup>2</sup> hybridised centre. The contribution to the hydrogen is 100% s character. The sp2 centre is as expected and explains the trigonal planar geometry of the molecule. The 100% s-character of the H atom is due to the s-orbitals being the only energetically available orbitals. Thus, NBO analysis can be also used towards justifying the D<sub>3h</sub> point group since it aids in rationalising the trigonal planar geometry adopted.

== Vibrational Analysis of Borane ==

===Introduction & Purpose of Vibrational Analysis <ref>http://www.huntresearchgroup.org.uk/teaching/teaching_comp_lab_year3/6_freq_analysis.html</ref>===

{| align=center
| [[Image:Intro_image_1.JPG|thumb|200x200px|Diagram 7- Distinguishing between a transition state and an minimum <ref>http://www.huntresearchgroup.org.uk/teaching/teaching_comp_lab_year3/6_freq_analysis.html</ref>|centre]]
| [[Image:Intro_image_2.JPG|thumb|200x200px| Diagram 8- Differences in the second derivative <ref>http://www.huntresearchgroup.org.uk/teaching/teaching_comp_lab_year3/6_freq_analysis.html</ref>|centre]]
|}

Vibrational analysis determines if the optimisation of the molecule has gone to completion or failed. Complete optimisation of a molecule should result in the minimum potential energy which is defined as the ''ground state'' (diagram 7). This is mathematically determined by calculating the gradient of the potential energy surface (PES). When the gradient is zero, it is indicative of a convergence. But as seen in the diagram 7, both the transition state and ground state have the first derivative (gradient) to be equal to zero. Thus, the first order differential is inaccurate in determining if the optimisation is complete as it could be zero but at the transition state.

The nature of the turning point (second derivative is < 0 ), is a maximum turning point indicative of the transition state (diagram 8). This indicates that the optimisation is not complete. But when second derivative is > 0, it is at the minimum point and indicates that optimisation is complete (diagram 8). Thus, when the curvature is negative it is represented by a negative vibrational frequency. There is no negative vibrational frequency in real life but when given as a negative value it is due to the nature of the turning point (maximum). Thus, when there are negative vibrational frequencies, optimisation can be concluded to have been carried out towards transition state instead of the ground state. This would mean the optimisation effort is not completed or has failed. This analysis will be applied to the BH<sub>3</sub> molecule optimisation efforts.

=== Vibrational Analysis ===

[https://wiki.ch.ic.ac.uk/wiki/images/a/ad/PRATHAP_BH3_FREQ.LOG Borane Vibrational Analysis]

Upon completion of the Gaussian frequency calculation, the completed file had to be checked to see if the calculation was successful. This was done by comparing the E(RB+HF-LYP) energy against the energy attained via the BH<sub>3</sub> optimisation initially carried out. The results are expressed in the table below. The highlighted segments from the table below show that both have exactly the same energy which was indicative that the frequency optimisation was successful.

{| class="wikitable" border="1" align="center"
|+ '''Table 5 - Comparing the total minimised energy via both Gaussian calculations'''
! BH3 Optimisation !! BH3 Frequency calculation
|-
| [[Image:Energy_comparison_2_pl1208.bmp|thumb|250x250px|Diagram 9-Highlighted total energy|centre]] || [[Image:Energy_comparison_pl1208.bmp|thumb|250x250px|Diagram 10-Highlighted total energy|centre]]
|}

[[Image:Essential_Data_pl1208.JPG|thumb|300x300px|Diagram 11- Isolated Essential information from LOG file|centre]]

The log file is further analysed via specific analysis of the low frequencies segment as shown in diagram 11. In the line that says 'low frequencies' there are a total of 6 different frequencies that correspond to the '-6' in the formula 3N-6 vibrational frequencies. The 6 vibrational frequencies are representative of the motions of the centre of mass of the molecule. They are much smaller than the first vibration listed.

There is a change in the point group for BH<sub>3</sub> molecule from D<sub>3h</sub> to C<sub>3h</sub> in the frequency output. However, the point group should still be the D<sub>3h</sub> and not the C<sub>3h</sub> as the molecule still being trigonal planar has the 3 planar reflections perpendicular to the principal axis of rotation. A summary for the justification of D<sub>3h</sub> is provided below and will be the point group with associcated symmetry elements used instead of the C<sub>3h</sub> point group.

[[Image:Point_group_PL1208.bmp|thumb|300X300px|Diagram 12- Justifying the D3H against C3H |centre]]

As the D<sub>3h</sub> point group has been justified, the expected molecular vibrations are to be aligned against the character table for the D<sub>3h</sub> point group.

[[Image:Character_table_pl1208.JPG|thumb|300x300px|Diagram 13- D<sub>3h</sub> character table <ref>http://www.webqc.org/symmetrypointgroup-d3h.html</ref>|centre ]]

{{DOI|10042/to-7460}}(Checkpoint file for the following vibrational results)

As it was concluded that the fully optimised structure was obtained, frequency analysis was carried out. The positive frequency values are indicative of a minimum point (which is what we hope to see !) while a negative frequency is indicative of the formation of the transition state. However, more than one negative frequency is indicative of an incorrect end result since the critical point is not derived and is indicative of an incomplete optimisation. The IR spectra shows the vibrations stretches expected.

[[Image:Ir_spectrum_pl1208.JPG|thumb|700x600px|Diagram 14- Calculated IR spectrum for Optimised Borane molecule|centre ]]

{| border="1" cellpadding="2" align="center"
|+'''Table 6- Summarising the different types of vibrations contributing to the Borane molecule'''
|-
! scope="col" | No.
! scope="col" | Type of Vibration
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Literature Frequency/cm<sup>-1</sup> <ref>K. Kawaguchi, J. E. Butler, C. Yamada, S. H. Bauer, T. Minowa, H.
Kanamori, and E. Hirota, J. Chem. Phys. 87,243s ( 1987).</ref><ref>Fourier transform infrared spectroscopy of the BH, v3 band,Kentarou Kawaguchi, J. Chem. Phys., Vol. 96, No. 5,1 March 1992</ref>
! scope="col" | Intensity
! scope="col" | Associated Symmetry<br>
D<sub>3h</sub> Point Group
|-
|align="center"| 1 ||align="center"| '''Out of Plane Wagging''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 3;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"| It is the concerted movement of all 3 hydrogen atoms perpendicular to the σ<sub>h</sub> plane, makkng it out-of plane. The boron central atom moves in the opposite direction by a very small displacement relative to the hydorgen's displacement|| align="center"|1144 || align="center"|1141 || align="center"|92.9 ||align="center"| A<sub>2</sub>'
|-
|align="center"| 2 ||align="center"| '''In-Plane Scissoring''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 4;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"| The two hydrogen atoms moves towards each other in a scissoring which results in the central boron atom to be displaced slights as the hydrogen atoms get closer in the σ<sub>h</sub> plane|| align="center"| 1204 ||align="center"| not reported in literature || align="center"|12.3 || align="center"|E'
|-
|align="center"| 3||align="center"| '''In plane rocking''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 5;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"|It involves the scissoring motion of two hydrogen atoms similar to the above, and the other hydrogen atom wags sideways in the plane. Both 2 and 3 are degenerate forms of each other|| align="center"| 1204 ||align="center"| not reported in literature || align="center"|12.3 || align="center"| E'
|-
|align="center"| 4 ||align="center"| '''symmetric stretch''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 6;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"|All 3 hydrogen atoms are displaced in the σ<sub>h</sub> plane where they move in and out together whilest the central boron atom remains stationary|| align="center"| 2598 ||align="center"| 2860|| align="center"| 0 || align="center"| A<sub>1</sub>'
|-
|align="center"|5 ||align="center"| '''Asymmetric stretch''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 7;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"|One of the hydrogen atom remains in a fixed position while the other 2 move in opposite displacements where by when one moves away from the boron centre the other moves closer to the boron centre. The boron centre will very a very small displacement relative to that of the hyodrogen atom's|| align="center"| 2737 ||align="center"| 2720|| align="center"|103.7 || align="center"|E'
|-
|align="center"| 6 ||align="center"| '''Asymmetric stretch''' ||<jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 8;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>PRATHAP_BH3_FREQ_vibrational table.LOG</uploadedFileContents> </jmolApplet></jmol>
|align="center"|All 3 hydrogen atoms are displaced in the σ<sub>h</sub> where by two of the hydrogen atoms are in sync where by when both move towards the boron centre the other hydrogen atom moves away from the central boron where there is minimal displacement by the boron centre.|| align="center"|2737 ||align="center"| 2720|| align="center"|103.7 || align="center"|E'
|}

In view of giving the specific numerical data provided by Gaussian, the frequencies of vibrations were corrected to the nearest integer while the intensities were rounded off to the 1st decimal place. This was done so as to allow the inaccuracies and errors with the harmonic approximation in the calculation to be accounted for. There are errors expected since these anharmonic vibrations have been modelled using the harmonic approximations.

All the values presented above are of a positive magnitude indicative of the minimum turning point. Thus, proving it has been optimised to the ground state and not the transition state; optimisation has been successful. The values are within range of the literature values for the frequency vibrations.

It is noticed that the calculations yields 6 different vibrational frequencies. But when compared with the IR spectra, only 3 peaks were seen. This is because, vibration '''NO. 2 AND 3''' are degenerate of each other and will only be displayed as a single peak. The same is expected of vibrations '''5 AND 6''' which are also degenerate. Thus, these 2 pairs of degenerate frequencies yield only 2 individual peaks. Vibration 4 has zero intensity since there is no change in the dipole moment resulting in a 'Non-IR active stretch' which explains it's zero intensity. Therefore, only 3 peaks seen '''( 1, 2&3 and 5&6)'''which agrees with literature IR spectras which also only show 3 different vibrational frequencies.

=Psuedo-Potentials implemented towards DFT Optimisation of TiBr<sub>3</sub> tri-halide=

For a simple molecule like borane which is made up of elements from the first two periods, the total number of electrons is small allowing the smaller basis set to be implemented towards the vibrational and NBO type analysis. But for Thallium tri-bromide type molecules which form a 186-electron system require a much larger basis set and greater computational resources. This brings about the implementation of the Psuedo-Potentials (PP) where they will be able to substitute the non-valence electrons in the system by which implementing an effective potential. This effective potential being the Pseudo-potential of the many-electron system replaces the non-valence electrons with the (PP) to bring about faster computational processing.

This makes TiBr<sub>3</sub> which has a total of <u>186 electrons</u> an ideal candidate to test the (PP) approach since a non-(PP) approach via the quantum based DFT calculation will be very time consuming and not worth the time and resource allocated. Comparing the basis set used for the BH<sub>3</sub> with TiBr<sub>3</sub>, the tri-halide needs a much larger basis set as there are a greater number of electrons so a greater range of orbitals will be expected to be occupied. Thus, the basis set used accommodates a larger range orbitals and fully describe them to realistic details. The larger basis set, LanL2DZ set will be implemented to fully describe all the available orbitals of the tri-halide. LanL2DZ basis set is a combination of two basis set (D95V) which is ideal for the first row element and to accurately map the much heavier element, it has the Los Alamos ECP subset. <ref>http://pubs.acs.org/doi/abs/10.1021/jp011945l</ref>

== Results from Optimisation ==
[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:TIBR3_LOG_FILE_pl1208.out '''TiBr<sub>3</sub> Optimisation Log File'''] {{DOI|10042/to-7461}}

{| class="wikitable" border="1" align="center"
|+'''Table 7- Comparing the Pre and Post optimisation of TiBr<sub>3</sub> bond angle and bond lengths
!<CENTER> Pre-optimised TiBr<sub>3</sub> Molecule </CENTER>!!<CENTER> Optimised TiBr<sub>3</sub> Molecule </CENTER>!! rowspan="4"|[[image:Optimisation_summary_pl1208_tibr3.JPG |thumb|Diagram 15 - calculation summary| centre]]
|-
| width=300px | <CENTER><jmol><jmolApplet><title>Pre-optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>Pre_optimisation_tibr3_pl1208.mol</uploadedFileContents>
</jmolApplet></jmol></CENTER>||width=300px |<CENTER><jmol><jmolApplet><title>Optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>Post_optimisation_pl1208_tibr3.mol</uploadedFileContents>
</jmolApplet></jmol></CENTER>
|-
|<CENTER> Ti-Br bond length = 2.69 Å </CENTER> ||<CENTER> Ti-Br bond length = 2.65 Å </CENTER>
|}

Similar to the BH<sub>3</sub> optimisation steps undertaken, the TiBr<sub>3</sub> molecule was analysed before and after the optimisation by paying specific attention to the bond length and the angle it was subtended at. Both the molecules had the Br-Ti-Br bond to be at 120.0 degrees. Comparison of the 2 bond lengths shows the slight reduction of the bond towards the optimisation. The gradient showed that the RMS graident has a value lesser than 0.001 a.u. ; indicative that the optimisation is complete. This was a similar analysis to that of the BH<sub>3</sub> optimisation carried out earlier. The only difference is that for the TiBr<sub>3</sub> which has very larger halides as well as the central atom Ti, has the possibility of adopting the pyramidal form to be more sterically stable. Thus, the D<sub>3h</sub> point group was fixed to ensure that all the structures had the same point group. This ensured that the molecule stayed in the trigonal planar orientation even after optimisation.

[[Image:Essential_data_pl1208_tibr3.bmp|thumb|300x300px| Diagram 16 - Essential Date from log file to show convergence |centre]]

The derivatives of the force and displacement have a value lesser than the present threshold value (red box). This shows a convergence and is indicative of a complete optimisation of the TiBr<sub>3</sub> molecule. Similar to the BH<sub>3</sub> analysis, the output file was then used to carry out further analysis.

=== Further Post-Optimisation Analysis===

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 8 : Summary of the different optimisation steps against the Bond lengths of the Ti-Br bonds'''</u>
! <CENTER> Optimisation Step No.</CENTER> !!<CENTER> Varying Ti-Br bond Diagram </CENTER> !! <CENTER> Varying Total Energy (a.u.)</CENTER> !! !! <CENTER> Varying Ti-Br Bond Length (Å)</CENTER> !! <CENTER>Plot of Energy and RMS Gradient </CENTER>
|-
| width=150px|<CENTER> 1 </CENTER>|| [[image:Optimisation_step1_pl1208_tibr3.JPG|centre|140px]] || width=150px|<CENTER>-91.2175 </CENTER>|||| width=150px|<CENTER> 2.69 </CENTER>|| rowspan="5"|[[image: Energy_graph_pl1208_tibr3.JPG|centre|800px]]
|-
|<CENTER> 2 </CENTER>|| [[image: Optimisation_step2_pl1208_tibr3.JPG|centre|140px]] || width=150px|<CENTER> -91.2180 </CENTER>||||<CENTER> 2.65 </CENTER>
|-
|<CENTER> 3 </CENTER>|| [[image: Optimisation_step3_pl1208_tibr3.JPG|centre|140px]] || width=150px|<CENTER> -91.2181 </CENTER>||||<CENTER> 2.65 </CENTER>
|}

For the larger and more electron containing TiBr<sub>3</sub> molecule, the (PP) only considered the valence electrons. Comparing the graphical data provided in the table 8, it is seen that at optimisation step 3, the gradient value is almost at zero. Furthermore, the derivatives are lesser than the present threshold. Both are indicative that the optimisation of the TiBr<sub>3</sub> molecule was complete. Initially the Ti-Br bond length was given as 2.69 Å (optimisation step 1) and upon the successive two optimisation steps the Br atoms were brought closer to the Ti centre. This resulted by step 3 for the minimised bond distance to be 2.65 Å. However, according to literature the bond length was given as 2.52 Å. <ref name="TiBr3 Literature bond length">J.Blixt, J.Glaser, J.Mink, I.Perrson, P.Perrson and M. Sandstrom, J. Am. Chem. Soc., 1995, 117, pp 5089-5104:{{DOI|10.1021/ja00123a011}}</ref> This accounts for the limitation in using the (PP) as it does not consider all the electrons in the molecule and only the valence electrons. Thus some deviation from the experimental was expected. A larger more accurate basis could be considered as it would be describe the orbitals more accurately and give a value closer to that of literature.

=== Vibrational Analysis of TiBr<sub>3</sub> ===

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Tibr3_Freq_vibr_PL1208.out Thallium Tribromide Vibrational Analysis]

{{DOI|10042/to-7462}}

Upon completion of the Gaussian frequency calculation, the completed file had to be checked to see if the calculation was successful. This was done by comparing the E(RB+HF-LYP) energy against the energy attained via the TiBr<sub>3</sub> optimisation initially carried out. The results are expressed in table 9. The highlighted segments from the table show that both have exactly the same energy which was indicative that the frequency optimisation was successful.

{| class="wikitable" border="1" align="center"
|+ '''Table 9-Comparing the total minimised energy via both Gaussian calculations'''
! TiBr<sub>3</sub> Optimisation !! TiBr<sub>3</sub> Frequency calculation
|-
| [[Image:Optimisation_summary_vibfreq_pl1208.bmp|thumb|250x250px|Diagram Highlighted total energy|centre]] || [[Image:Essential_data_vibfreq_pl1208_tibr3.bmp|thumb|250x250px|Diagram Highlighted total energy|centre]]
|}

[[Image:Essential_data_vibfreq2_pl1208_tibr3.JPG|thumb|300x300px|Diagram 17- Isolated Essential information from LOG file|centre]]

The log file is further analysed via specific analysis of the low frequencies as shown in diagram x. In the line that says 'low frequencies' there are a total of 6 different frequencies that correspond to the '-6' in the formula 3N-6 vibrational frequencies. The 6 vibrational frequencies are representative of the motions of the centre of mass of the molecule. Thus are much smaller than the first vibration listed. Further more, as the highest frequency in the 'low frequency' row is row is of 1 order of magnitude smaller than the lowest frequency in the normal vibration modes (diagram 17). It can be concluded that the calculation result are valid and further analysis can be undertaken.

A summary for the justification of D<sub>3h</sub> is provided below and will be the point group with associated symmetry elements used.

[[Image:Point_group_pl1208_tibr3.bmp|thumb|400X400px|Diagram 18- Justifying the D<sub>3h</sub> point group |centre]]

As the D<sub>3h</sub> point group has been justified, the expected molecular vibrations are expected to aligned against the character table for the D<sub>3h</sub> point group.

[[Image:Character_table_pl1208.JPG|thumb|300x300px|Diagram 19-D<sub>3h</sub> character table <ref>http://www.webqc.org/symmetrypointgroup-d3h.html</ref>|centre ]]

The positive frequency values are indicative of a minimum point (which is what we hope to see !) while a negative frequency is indicative of the formation of the transition state. However more than 1 negative frequency is indicative of an incorrect end result since the critical point is not derived; indicative of an incomplete optimisation.

[[Image:IR_spectra_tibr3_pl1208.JPG|thumb|800x600px|Diagram 20- Calculated IR spectrum for Optimised TiBr<sub>3</sub> molecule|centre ]]

{| border="1" cellpadding="2" align="center"
|+'''Table 10-Summarising the different types of vibrations contributing to the TiBr<sub>3</sub> molecule'''
|-
! scope="col" | No.
! scope="col" | Type of Vibration
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Intensity
! scope="col" | Associated Symmetry<br>
D<sub>3h</sub> Point Group
|-
|align="center"| 1 ||align="center"| '''In plane-scissoring like motion''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 3;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| The two bromine atoms moves towards each other in a scissoring which results in the central thallium atom to be displaced slightly as the two bromine atoms get closer in the σ<sub>h</sub> plane. The other bromine atom has a much smaller displacement in the opposite direct to the two bromine atoms|| align="center"|46 || align="center"|3.7 ||align="center"| E'
|-
|align="center"| 2 ||align="center"| '''In plane rocking''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 4;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| When the two bromines are seen to be moving in one direction the other bromine is moving in the opposite direction 'rocking' relative to the other two. The thallium centre is seen to have minimal displacement. All the displacements are in the σ<sub>h</sub> plane || align="center"| 46 || align="center"|3.7 || align="center"|E'
|-
|align="center"| 3||align="center"| '''Out of plane wagging''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 5;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|It involves the concerted movement of all three bromine atoms where they have displacement perpendicular to the σ<sub>h</sub> plane. The thallium also has distince displacement where it has the opposite direction of displacement relative to the bromines displacement || align="center"| 52 || align="center"|5.8 || align="center"| A<sub>2</sub>
|-
|align="center"| 4 ||align="center"| '''symmetric stretch''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 6;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|All 3 bromine atoms are displaced in the σ<sub>h</sub> plane where they move in and out together whilest the central boron atom remains stationary|| align="center"| 165 ||align="center"| 0 || align="center"| A<sub>1</sub>'
|-
|align="center"|5 ||align="center"| '''Asymmetric stretch''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 7;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|One of the bromine atom remains in a fixed position while the other 2 move in opposite displacements where by when one moves away from the thallium centre the other moves closer to the thallium centre. The thallium centre will have a side to side displacement in the plane|| align="center"| 211 || align="center"|25.5 || align="center"|E'
|-
|align="center"| 6 ||align="center"| '''Asymmetric stretch''' ||<jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 8;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>pl1208_Vibrations_TIBR3.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|All 3 bromine atoms are displaced in the σ<sub>h</sub> where by two of the bromine atoms are in sync where by when both move towards the thallium centre the other bromine atom moves away from the central thallium where there is minimal displacement by the thallium centre.|| align="center"|211 || align="center"|25.5 || align="center"|E'
|}

Analysis of the IR spectra shows that only 3 distinct peaks are observed. However, 6 different vibrational modes have been computationally calculated. But of the 6 modes, '''NO 5 AND 6''' are a degenerate pair and only contribute to 1 peak. Similarly '''NO. 1 AND 2''' are also a degenerate pair and contribute to only 1 peak. The final peak is accounted for by NO.3 vibrational mode (Table 10). The IR spectra corresponds to the literature frequencies. <ref>pubs.acs.org/doi/abs/10.1021/ic50122a045</ref> NO.4 vibrational mode being the symmetric stretch would have a zero dipole moment making it a non-IR active vibrational mode which explains why it has an intensity of zero and is not seen in the IR spectra nor in literature at that vibrational frequency of 165 cm<sup>-1</sup>.

==== Compare and Contrasting the TiBr<sub>3</sub> against BH<sub>3</sub> vibrations ====

{| border="1" cellpadding="2" align="center"
|+'''Table 11-Comparing the order of the stretches with respect to increasing frequency'''
|-
! No. !! Order of stretches in the BH<sub>3</sub> molecule!! Order of stretches in the TiBr<sub>3</sub> molecule
|-
| align="center"| 1 || align="center"| '''Out of plane wagging''' || align="center"| '''In plane-scissoring like motion'''
|-
| align="center"| 2 || align="center"| '''In plane scissoring'''|| align="center"| '''In plane rocking'''
|-
| align="center"| 3 || align="center"| '''In plane rocking''' || align="center"| '''Out of plane wagging'''
|-
| align="center"| 4 || align="center"| '''symmetric stretch'''|| align="center"| '''symmetric stretch'''
|-
| align="center"| 5 || align="center"| '''Asymmetric stretch'''|| align="center"| '''Asymmetric stretch'''
|-
| align="center"| 6 || align="center"| '''Asymmetric stretch'''|| align="center"| '''Asymmetric stretch'''
|}

Table 11 summarises the order of the vibrations for both BH<sub>3</sub> and TiBr<sub>3</sub> in acending energy content. A comparison can be accurately carried out since both have the D<sub>3h</sub> point group symmetry and both have been calculated using the same minimal basis set. Both molecules have same number and same types of vibrational modes (3N-6) where N=4. However, there is a different order of the vibrational modes between the two molecules. One such change would be the '''Out of plane wagging''' which was the lowest vibrational wave number in BH<sub>3</sub> but in TiBr<sub>3</sub> it has increased in vibrational wavenumber to NO.3 from NO.1.

This can be rationalised as being due to the significant change in mass of the central atom from B to Ti and from H to Br. Both the changes result in an increase of the reduced mass. Wagging involves the out-of-plane movement of the atoms, TiBr<sub>3</sub> having a larger reduced mass would have much more energy in incurred in the vibration. It is easier for a less heavy molecule to wag than a more heavy one. Thus the wagging has a higher energy content in TiBr<sub>3</sub>.

==Additional Questions==

1. '''In some structures gaussview does not draw in the bonds where we expect, does this mean there is no bond? Why?'''

This problem was encountered in the optimisation of both the BH<sub>3</sub> and the TiBr<sub>3</sub> molecule. This problem with 'invisible' bonds is due to a software limitation. Gaussview 5 which was used for the optimisation efforts has predetermined bond lengths and when two atoms are within this predefined bond length, the bond is 'visible' in Gaussview. Thus during optimisation, the bond lengths are varied through different optimisation steps until the optimisation with the minimum energy is reached. Thus for the intermediate optimisation steps which has the bond lengths beyond the predefined distance, they will be appearing 'invisible. This is more evidently seen in this module as compared to the previous module as the software has the predefined distances mainly catering for organic molecules and complexes. However, based on MO analysis carried out, it is seen that the software despite having 'invisible' bonds would still recognise there is electron density delocalised between the two atoms of the 'invisible' bond.

2. '''What is a bond?'''

Based on the definition from IUPAC a bond is defined as '''chemical bond between two atoms or groups of atoms in the case that the forces acting between them are such as to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent "molecular species''' .<ref>Glossary of Terms used in Physical Organic Chemistry (IUPAC Recommendations 1994), PAC, 1994, 66, pg 1077:{{DOI|10.1351/pac199466051007}}</ref>

However, this is a very vague representation of a chemical bond as it does not include any description about the delocalised electron density expected between the two atoms forming the bond. The bond being formed is a covalent bond and is held together by electrostatic interactions between the two nuclear centres which have similar electro-negativity.

Another type of the bond is the ionic bond which is formed between two nuclear centres when they have significant electronegative difference result in unequal sharing of the bonds. Thus, forming the ionic bond or dipoles when there is still delocalised electrons between the nuclear centres.

= <u>'''Isomer Analysis of Mo(CO)<sub>4</sub>(PPh<sub>3</sub>)<sub>2</sub>'''</u>=

== '''Introduction''' ==

The transition metal complex Mo(CO)<sub>4</sub>(PPh<sub>3</sub>)<sub>2</sub> displays cis-trans stereoisomerism. In the second year inorganic lab course, the Molybdenum complex was synthesised as part of an experiment where identification of the isomers was carried out via analysis of the number of CO vibrational stretches observed in the individual IR spectras. The cis isomer had <u>four carbonyl absorption bands</u> while the trans isomer had <u>a single absorption band</u>. Computational methods of analysis are expected to faciltiated similar identification and characterisation of the different isomers.

[[Image:Literature_Ir_Spectra.JPG|thumb|400x400px|Diagram showing the IR spectra of the two isomers|centre ]]<ref>http://pubs.acs.org/doi/pdf/10.1021/ed079p1249</ref>

As carried out in the earlier exercise, the respective isomers will be optimised and vibrational analysis will be carried out to determine if the computationally attained results draw parallels with the experimental data attained for the two isomers. However, the triphenyl phosphine ligands used presented some computational problems towards effective optimisation of the metal complex. This was mainly due to the three phenyl rings. The aromatic planar rings resulted in the computational calculation to be carried out via a 360 degree rotation about each phenyl ring. As there were six such rings, this will lead to many localised energy minimas that posed as pseudo-ground states. Furthermore, the rings being sterically hindering, required a demanding amount of computational resources (Diagram 21). Thus, another ligand substituent was used with minimal influence to the expected outcome.

[[Image:STERICALLY_HINDERING.bmp|thumb|200x200px|Diagram 21- Representation of the sterics in PPh<sub>3</sub> ligands|centre ]]

The computationally less demanding chloride ligands were used to approximate the phenyl rings in the PPh<sub>3</sub> ligands. They had similar electron-withdrawing capabilities and similar electronic contribution to the phosphorus centre. They were sterically quite large but favourably imposed a much lesser computational demand (Diagram 22).

[[Image:Approximations_pl1208.bmp|thumb|300x300px|Diagram 22- Representation of the phenyl substitution PPh<sub>3</sub> ligands|centre ]]

=='''Approach taken towards Effective Optimisation of both isomers''' ==

Optimisation of the ground state structures will be carried out in three successive optimisation steps. The first optimisation involves the usage of a low basis set via the B3LYP method. This set defined as LANL2MB serves as a loose optimisation to give a rough approximation of the optimised structure.

The second step towards optimisation involves rearrangement of the dihedral angles of the P-Cl bonds in both isomers to facilitate the successive optimisation. This step allows the re-orientation of the bonds to that expected of the final optimised product. This ensures the next optimisation will be done from the better start point that ensures the true lowest minimum point is achieved and not 'stuck' in a psuedo-ground state energy conformation(Diagram 23).

[[Image:Doubel_minima.JPG|thumb|300x300px|Diagram 23- Explaining the need for successive optimisations <ref>http://www.huntresearchgroup.org.uk/teaching/teaching_comp_lab_year3/10b_MoC4L2_opt.html</ref> |centre]]

The final optimisation effort will use a larger and more accurate basis set defined as LANL2DZ.

=== First Optimisation ===
[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Post_first_optimisation_log_file.out Cis Isomer Initial Optimisation Log File] {{DOI|10042/to-7464}}

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Post_first_optimsation_log_file_pl1208.out Trans Isomer Initial Optimisation Log File] {{DOI|10042/to-7465}}

The optimisation using the DFT-B3LYP method via a low level basis set (LANL2MB) gave a rough optimisation since the 'opt=loose' was entered to ensure the convergence criteria was prevented from being more accurate than the method being used. If it was not prevented then the convergence will happen even before the method is complete causing the optimisation to fail. LAN2MB comprised of more than one basis set; it is specifically made up of the (i) STO-G3 set which is specific for the first row elements (ii) Los Alamos ECP (iii) MBS for Na-La and Hf to Bi (essentially all the other elements). <ref>http://www.gaussian.com/g_tech/g_ur/m_basis_sets.htm</ref> <ref>P. J. Hay and W. R. Wadt, “Ab initio effective core potentials for molecular calculations - potentials for K to Au including the outermost core orbitals,” ''J. Chem. Phys.'', '''82''' (1985) 299-310.</ref>

{| class="wikitable" border="1" align="center"
|+ '''Table 12 - Comparing the Total energy of the Cis and Trans isomers'''
! <CENTER> '''Property Compared''' </CENTER> !!<CENTER>Cis Isomer </CENTER> !! <CENTER>Trans Isomer </CENTER>
|-
| <CENTER>Summary </CENTER> || <CENTER>[[Image:Summary_pl1208_cis_01opt.JPG|200x200px|centre]] </CENTER>|| <CENTER>[[Image:Summary_pl1208_trans_01opt.JPG|200x200px|centre]] </CENTER>
|-
| <CENTER> Diagram </CENTER> || [[image: Post_first_optimisation_pl1208_01optcis.JPG|centre|250px]]
<jmol>
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<text>Cis optimised metal complex Jmol </text>
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<script>spin on</script>
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|| [[image:Post_first_optimisatin_diagram_pl1208.JPG|centre|250px]]
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<text>Trans optimised metal complex Jmol </text>
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Total energy of the Cis isomer upon the first optimisation is -617.5250 a.u. while the trans isomer had an energy of -617.5220 a.u. The difference between the two isomers being 0.0030 a.u. corresponded to ca. 9 KJ mol<sup>-1</sup>(a small energy difference between the two isomers). As the energy difference between the two isomers are almost equal, they would be labile at room temperature. This could be avoided by reconsidering the type of ligands used. The steric influence could be increased by having more bulky ligands such as PPh<sub>3</sub> that makes the cis-isomer much more unstable. This results in the energy difference between the two isomers to become significantly larger; allowing selectivity for one isomer over the other since the steric influence plays a greater role in the cis isomer relative to the influence it brings to the trans isomer.

This was an interesting result as the trans product which had the significantly lesser steric clash was expected to be more readily formed and have a much lower energy level as compared to the cis isomer which has the bulky ligands adjacent to each other. However, it is too soon to rule out the influence of sterics since the reaction under kinetic control has not been looked into. According to literature sources, the cis isomer is the kinetically stable product that isomerises to the thermodynamically stable trans isomer.<ref>G. Hogarth, T. Norman, Inorganica Chemica Acta 254 (1997) 167 - 171.</ref> As this is only the rough optimisation, the validity of this statement is yet to be verified and will be done so upon the successive optimisations.

The RMS gradient for both isomers is less than 0.001 a.u. (indicating convergance). Hence suggesting that the optimisation has reached a completion. Thus, further analysis via the log file can be considered since it is a completed optimisation.

{| class="wikitable" border="1" align="center"
|+ '''Table 13 - Comparing Initial Optimised Trans isomer bond lengths and angles against Literature'''
! Bond length (Å)/ Bond Angle (deg) Compared!!Literature Trans isomer <ref>G. Hogarth, T. Norman, Inorganica Chemica Acta 254 (1997) 167 - 171.</ref>!! Calculated Trans Data
|-
| <CENTER> P-Cl bond </CENTER>|| <CENTER> 1.828 </CENTER>|| <CENTER> 2.404 </CENTER>
|-
| <CENTER> P-Mo bond </CENTER>|| <CENTER> 2.500 </CENTER>|| <CENTER>2.481 </CENTER>
|-
| <CENTER>C-O (top Axial) </CENTER>|| <CENTER> 1.164 </CENTER>|| <CENTER>1.192 </CENTER>
|-
| <CENTER>C-O (Bottom Axial) </CENTER>||<CENTER>1.162 </CENTER>|| <CENTER>1.192 </CENTER>
|-
| <CENTER>Mo-C (Axial) </CENTER>|| <CENTER> 2.005 </CENTER>|| <CENTER>2.109 </CENTER>
|-
| <CENTER> Mo-C (Equatorial) </CENTER>||<CENTER> 2.005 </CENTER>|| <CENTER> 2.109 </CENTER>
|-
| <CENTER>P-Mo-P angle </CENTER>|| <CENTER> 180.0 </CENTER>|| <CENTER>180.0 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER>87.2 </CENTER>|| <CENTER>90.6 </CENTER>
|-
| <CENTER>C-Mo-C (Axial)cis </CENTER>|| <CENTER> 92.1 </CENTER>|| <CENTER>89.946 </CENTER>
|-
| <CENTER> C-Mo-C (Equatorial)trans </CENTER>|| <CENTER> 180.0 </CENTER>|| <CENTER> 180.0 </CENTER>
|}

{| class="wikitable" border="1" align="center"
|+ '''Table 14 - Comparing Initial Optimised Cis isomer bond lengths and angles against Literature (L=PPh3)'''
! Bond length (Å)/ Bond Angle (deg) Compared!!Literature Cis isomer <ref> F.A. Cotton, D.J. Darensbourg, S. Klein and B.W.S. Kolthammer. Inorg. Chem. 21 (1982), p. 294</ref>!! Calculated Cis Data
|-
| <CENTER> P-Cl bond </CENTER>|| <CENTER> - </CENTER>|| <CENTER> 2.405 </CENTER>
|-
| <CENTER> P-Mo bond </CENTER>|| <CENTER> 2.577 </CENTER>|| <CENTER>2.5253 </CENTER>
|-
| <CENTER>C-O (top Axial) </CENTER>|| <CENTER> - </CENTER>|| <CENTER>1.193 </CENTER>
|-
| <CENTER>C-O (Bottom Axial) </CENTER>||<CENTER> - </CENTER>|| <CENTER>1.193 </CENTER>
|-
| <CENTER>Mo-C (Axial) </CENTER>|| <CENTER> 2.022 </CENTER>|| <CENTER>2.109 </CENTER>
|-
| <CENTER> Mo-C (Equatorial) </CENTER>||<CENTER> 1.972 </CENTER>|| <CENTER> 2.061 </CENTER>
|-
| <CENTER>P-Mo-P angle </CENTER>|| <CENTER> 104.62 </CENTER>|| <CENTER>95.511 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER>163.7 </CENTER>|| <CENTER>177.055 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER> 80.6 </CENTER>|| <CENTER>90.646 </CENTER>
|-
| <CENTER> C-Mo-C mutually cis carbonyls </CENTER>|| <CENTER> 83.0 </CENTER>|| <CENTER> 89.401 </CENTER>
|-
| <CENTER> C-Mo-C mutually cis carbonyls </CENTER>|| <CENTER> 174.1 </CENTER>|| <CENTER> 178.238 </CENTER>
|}

A comparison of the key bond lengths against literature was carried out for the trans isomer. However the same was not replicated for the cis isomer as there no literature data available for that isomer. Instead, the crystallographic data for the complex where L=PPh<sub>3</sub> was used. This was a sound comparison since apart from the two ligands, it is a similar transition metal complex which allowed bond comparisons to be made. Comparing the Mo-CO bond in both isomers, the bonds trans to the PR<sub>3</sub> ligands are shorter than the bonds cis to the ligands. Similary the Mo-P bonds in the trans isomer is shorter than those in the cis isomer. This is because, the cis has the bonds slightly elongated to minimise the non-favourable non-bonding interactions between the bulky ligands adjecent to each other.

However, the calculated data varies significantly from the literature data especially for the trans which was compared against the same compound. This was the first indication that the basis set used was not very effective in the optimisation efforts and raises the need for a secondary optimisation to be carried out.

[[Image:Rms_gradient_pl1208_cisopt01.JPG|thumb|600x600px|Diagram 24-Cis Isomer's 11 optimisation steps |centre]]

[[Image:Rms_gradient_pl1208_transopt01.JPG|thumb|600x600px|Diagram 25-Trans Isomer's 8 optimisation steps |centre]]

From both the diagrams, it is rationalised that the larger molecular structure brings about a need for additional optimisation steps as compared to a smaller molecule like BH<sub>3</sub> which only needed 6 steps to be fully optimised. The 'horizontal effect' seen shows that the level of accuracy implemented by the calculation is not very much higher than the convergence criteria and so it has the horizontal stagnation of the total energy towards convergence. This shows that a more accurate basis set should be adopted instead.

=== Second Optimisation ===

This step involves the 'manual' realignment of both Phosphrous based substituents by defining the dihedral angle between the Cl-P-Mo-CO for both the isomers as shown below. The 'manual' adjustment will facilitate the second optimisation to give the lowest energy conformer for both the isomer and not a psuedo-lowest if it got 'stuck' in a intermediate minimum point.

{| class="wikitable" border="3" align="center"
|+ <u>'''Diagram 26 representation of the Re-aligned Cis isomers'''</u>
|-
| [[Image:Rearranged_cis_isomer.bmp|thumb|200x200px|Schematic of the realigned P-Cl bonds ]] || [[Image:Rearranged_cis_isomer.JPG|thumb|200x200px| Diagram of the realigned P-Cl bonds]]
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<script>spin on</script>
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{| class="wikitable" border="3" align="center"
|+ <u>'''Diagram 27- representation of the Re-aligned Trans isomers'''</u>
|-
| [[Image:Realigned_transisomer_pl1208.bmp|thumb|200x200px|Schematic of the realigned P-Cl bonds ]] || [[Image:Rearranged_isomer_pl1208_trans.JPG|thumb|200x200px| Diagram of the realigned P-Cl bonds]]
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=== Third Optimisation ===

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Post2nd_optimisation_log_file_pl1208cis.out Cis Isomer Second Optimisation Log File] {{DOI|10042/to-7469}}

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Post_2nd_optimisation_log_filepl1208trans.out Trans Isomer Second Optimisation Log File] {{DOI|10042/to-7468}}

The final optimisation efforts involved use of the more accurate LANL2DZ pseudo-potential basis set. This resulted in a more accurate optimisation the isomers. The additional keywords "int=ultrafine" and "scf=conver=9" added enabled a tighter convergence criteria to be imposed to ensure the minimum energy level was attained.

{| class="wikitable" border="2" align="center"
|+ '''Table 15 - Comparing the Total energy of the Cis and Trans via secondary optimisation'''
! <CENTER> '''Property Compared''' </CENTER> !!<CENTER>Cis Isomer </CENTER> !! <CENTER>Trans Isomer </CENTER>
|-
| <CENTER>Summary </CENTER> || <CENTER>[[Image:Summary_pl1208_2ndopt_cis.JPG|200x200px|centre]] </CENTER>|| <CENTER>[[Image:Summary_pl1208_2ndopt_transpl1208.JPG|200x200px|centre]] </CENTER>
|-
| <CENTER> Diagram </CENTER> || [[image: Post_2nd_optimisation_diagram_pl1208cis.JPG|centre|250px]]
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<script>spin on</script>
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|| [[image:Post_2nd_optimisationpl1208trans.JPG|centre|250px]]
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Comparing the RMS gradient of the second optimisation(Table 15) against the first optimiasation (Table 12) shows that the gradient via the second optimisation result is much more closer to zero as compared to that in the first optimisation. This could be due to the tigher restrictions imposed on the convergence that bring about a much more significant optimisation of the isomers.

{| class="wikitable" border="2" align="center"
|+ '''Table 16 - Comparing the Total energy from the two optimisation'''
! <CENTER> '''Isomer''' </CENTER> !! <CENTER> '''Total Energy (1st Optimisation) a.u.''' </CENTER>!! <CENTER> '''Total Energy (2nd Optimisation) a.u'''. </CENTER>
|-
| <CENTER> Cis </CENTER> || <CENTER> -617.5250 </CENTER>|| <CENTER> -623.5771 </CENTER>
|-
| <CENTER> Trans </CENTER> || <CENTER> -617.5220 </CENTER>|| <CENTER> -623.5760 </CENTER>
|-
| <CENTER> Trans - Cis </CENTER> || <CENTER> 0.003 </CENTER>|| <CENTER> 0.0011 </CENTER>
|}

In both the first and the second optimisations, the trans-isomer is thermodynamically more stable than the cis-isomer. However, the second optimisation results in the difference between the energy levels to increase to a difference of about 40 KJ/mol from about 9 KJ/mol. Although the energy gap is not large but it shows that the second optimisation results in a more accurate depiction of the thermodynamically more stable trans isomer. This is rationalised since it does not have the un-favourable non-bonding interactions that the cis-isomer is subjected to. As stated before, much larger ligands could be considered if the energy gap is to be more distinct.

{| class="wikitable" border="2" align="center"
|+ '''Table 17 -Comparing Second Optimised Trans Isomer against Literature'''
! Bond length (Å)/ Angle (deg) Compared!! Literature Trans isomer <ref>G. Hogarth, T. Norman, Inorganica Chemica Acta 254 (1997) 167 - 171.</ref>!! Calculated (second optimisation)
|-
| <CENTER> P-Cl bond </CENTER>|| <CENTER> 1.828 </CENTER>|| <CENTER> 2.240 </CENTER>
|-
| <CENTER> P-Mo bond </CENTER>|| <CENTER> 2.500 </CENTER>|| <CENTER>2.445 </CENTER>
|-
| <CENTER>C-O (top Axial) </CENTER>|| <CENTER> 1.164 </CENTER>|| <CENTER>1.172 </CENTER>
|-
| <CENTER>C-O (Bottom Axial) </CENTER>|| <CENTER>1.164 </CENTER>|| <CENTER>1.173 </CENTER>
|-
| <CENTER>Mo-C (Axial) </CENTER>|| <CENTER> 2.005 </CENTER>|| <CENTER>2.0600 </CENTER>
|-
| <CENTER> Mo-C (Equatorial) </CENTER>|| <CENTER> 2.005 </CENTER>|| <CENTER> 2.0604 </CENTER>
|-
| <CENTER>P-Mo-P angle </CENTER>|| <CENTER> 180.0 </CENTER>|| <CENTER>177.4 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER>87.2 </CENTER>|| <CENTER>90.02 </CENTER>
|-
| <CENTER>C-Mo-C (Axial)cis </CENTER>|| <CENTER> 92.1 </CENTER>|| <CENTER>89.503 </CENTER>
|-
| <CENTER> C-Mo-C (Equatorial)trans </CENTER>|| <CENTER> 180 </CENTER>|| <CENTER> 179.001 </CENTER>
|}

{| class="wikitable" border="1" align="center"
|+ '''Table 18 - Comparing second Optimised Cis isomer bond lengths and angles against Literature (L=PPh3)'''
! Bond length (Å)/ Bond Angle (deg) Compared!!Literature Cis isomer <ref> F.A. Cotton, D.J. Darensbourg, S. Klein and B.W.S. Kolthammer. Inorg. Chem. 21 (1982), p. 294</ref>!! Calculated Cis Data
|-
| <CENTER> P-Cl bond </CENTER>|| <CENTER> - </CENTER>|| <CENTER> 2.240 </CENTER>
|-
| <CENTER> P-Mo bond </CENTER>|| <CENTER> 2.577 </CENTER>|| <CENTER>2.512 </CENTER>
|-
| <CENTER>C-O (top Axial) </CENTER>|| <CENTER> - </CENTER>|| <CENTER>1.176</CENTER>
|-
| <CENTER>C-O (Bottom Axial) </CENTER>||<CENTER> - </CENTER>|| <CENTER>1.171 </CENTER>
|-
| <CENTER>Mo-C (Axial) </CENTER>|| <CENTER> 2.022 </CENTER>|| <CENTER>2.012 </CENTER>
|-
| <CENTER> Mo-C (Equatorial) </CENTER>||<CENTER> 1.972 </CENTER>|| <CENTER> 2.058 </CENTER>
|-
| <CENTER>P-Mo-P angle </CENTER>|| <CENTER> 104.6 </CENTER>|| <CENTER>94.2 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER>163.7 </CENTER>|| <CENTER>176.1 </CENTER>
|-
| <CENTER>P-Mo-C angle </CENTER>|| <CENTER> 80.6 </CENTER>|| <CENTER>89.2 </CENTER>
|-
| <CENTER> C-Mo-C mutually cis carbonyls </CENTER>|| <CENTER> 83.0 </CENTER>|| <CENTER> 89.1 </CENTER>
|-
| <CENTER> C-Mo-C mutually cis carbonyls </CENTER>|| <CENTER> 174.1 </CENTER>|| <CENTER> 178.4 </CENTER>
|}

Comparisons made between the literature and calculated values for the trans isomer have little difference in values. The cis was compared with a different ligand based isomer so slight deviations can be anticipated. But overall the deviations were minimal suggesting that the optimisation via the pseudo-potential basis set was more accurate.

Both the optimisations have the Mo-C bonds to be shorter than the Mo-P bonds. This could be due to the steric bulk of the PR<sub>3</sub> ligands and the backbonding of the Metal to the antibonding orbital of the CO pi*. The back-bonding increases the electron density between the Mo and the C of the carbonly and a shorter bond length is representation of greater delocalised electron density between the two atoms. Hence the shorter Mo-C bond since Mo-P does not have such back-bonding interactions. This results in the slightly distorted octahedral (mainly due to steric) that results in the C1 point group instead of C2v.

[[Image:Rms_gradient_cis_pl1208_2opt.JPG|thumb|400x400px|Diagram 28-Cis Isomer's 18 optimisation steps |centre]]

[[Image:Rms_gradient_trans_pl1208.JPG|thumb|400x400px|Diagram 29-Trans Isomer's 9 optimisation steps |centre]]

=== Conclusion towards Optimisation efforts ===

From the analysis of the changes to the bond lengths and bond angle, the second optimisation does bring the computationally modelled complex closer to that of literature. If a larger more accurate basis set was used, the differences against literature data can be expected to be further minimised.

== Vibrational Frequency Analysis ==

The frequency calculations were carried using the same method and basis set as that of the optimisation. This ensures accurate comparisons could be made. As mentioned in the introduction segment (Diagram 21), the different isomers can be distinguished from each other based on the number of CO stretches observed. This will be the key observation followed up via computational vibrational frequency analysis.

=== Analysis of the Cis-Metal Complex ===

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cisvibrational_PL1208.out Vibrational Analysis Cis Isomer Log File] {{DOI|10042/to-7482}}

Upon completion of the Gaussian frequency calculation, the completed file had to be checked to see if the calculation was successful. This was done by comparing the <u>E(RB+HF-LYP)</u> energy against the energy attained via the optimisation initially carried out. The results are expressed in the table below. Table 19 shows that both have exactly the same energy which is indicative that the frequency optimisation was successful.

{| class="wikitable" border="1" align="center"
|+ '''Table 19 - Comparing the total minimised energy via both Gaussian calculations'''
! Cis Isomer Optimisation !! cis Isomer Frequency calculation
|-
| [[Image:Summary_pl1208_2ndopt_cis.JPG|thumb|250x250px|centre]] || [[Image:Energy_summary_pl1208_cis_vibfreq.JPG|thumb|250x250px|centre]]
|}

[[Image:Essential_data_vibfreq_cis.JPG|thumb|300x300px|Diagram 30 - Isolated Essential information from LOG file|centre]]

The log file is further analysed via specific analysis of the low frequencies segment as shown in diagram 30. In the line that says 'low frequencies' there are a total of 6 different frequencies that correspond to the '-6' in the formula 3N-6 vibrational frequencies. Further more, as the highest frequency in the 'low frequency' row it is 1 order of magnitude smaller than the lowest frequency in the normal vibration modes. It can be concluded that the calculation results are valid and further analysis can be undertaken.

[[Image:Point_group_vibrfreq_cis.JPG|thumb|300x300px|Diagram 31-Point group symmetry of the Cis isomer|centre ]]

The Cis-isomer having the C<sub>2V</sub> point group symmetry will have the carbonyl bond stretches assigned based on the symmetry elements of the point group.

[[Image:Ir_spectra_Annotated_pl1208_cisvibanaly.jpg|thumb|800x800px| Diagram 32- Annotated IR spectra for the Cis isomer showing the distince carbonyl peak|centre]]

The IR spectra as expected would show the presence of multiple peaks in the carbonyl region of about 1900 cm<sup>-1</sup>. These correspond with the literature IR spectra (Diagram 21). Table 20 provides more detailed analysis into the bond stretches.

{| border="1" cellpadding="2" align="center"
|+'''Table 20 - Summarising the Carbonyl Stretches in the Cis isomer'''
|-
! scope="col" | No.
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Literature Frequency/cm<sup>-1</sup><ref>ELmer C. Alyea and Shuquan Song, ''Inorg. Chem.'', '''1995''', ''34'', 3864-3873 </ref>
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Intensity
! scope="col" | Associated Symmetry C<sub>2v</sub> Point Group
|-
|align="center"| 1 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 44;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39927_pl1208.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| The Equatorial carbonyl stretches vibrate out of phase with respect to each other. Similarly, the Axial carbonyl stretches also vibrate out of phase with respect to each other. Comparing the extent of the vibrations, the equatorial carbonyls have a greater displacement than the axial carbonyls.|| align="center"|1986 || align="center"|1946 || align="center"|761 || align="center"| B<sub>2</sub>
|-
|align="center"| 2 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 45;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39927_pl1208.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The Equatorial carbonyl stretches vibrate out of phase with respect to each other. Similarly, the Axial carbonyl stretches also vibrate out of phase with respect to each other. Comparing the extent of the vibrations, the axial carbonyls have a greater displacement than the equatorial carbonyls. ||align="center"|1994 || align="center"| 1949 || align="center"|1501 || align="center"| B<sub>1</sub>
|-
|align="center"| 3 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 46;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;;
</script><uploadedFileContents>Log_39927_pl1208.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The Equatorial carbonyl stretches vibrate in phase with respect to each other. Similarly, the Axial carbonyl stretches also vibrate in phase with respect to each other. However, the two set of carbonyl vibrations are out of phase with respect to each other. || align="center"|2004 ||align="center"| 1959 || align="center"|636 || align="center"| A<sub>1</sub>
|-
|align="center"| 4 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 47;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
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|align="center"|The Equatorial carbonyl stretches vibrate in phase with respect to each other. Similarly, the Axial carbonyl stretches also vibrate in phase with respect to each other. The two set of carbonyl vibrations are in phase with respect to each other.||align="center"|2072 || align="center"| 2024 ||align="center"| 595 || align="center"| A<sub>1</sub>
|}

The table shows the 4 different carbonyl bond stretches seen in the cis isomer. All 3 possible asymmetric vibrations and the fully symmetric vibrations by the carbonyls are accounted for. These corresponding to the 4 peaks seen in literature make the computationally calculated results reliable to a large extent. This is theoretically viable as all the stretches bring about a change in the dipole and would make them IR active.

However, comparisons of the attained values against literature is indicative of an approximate ~50 cm<sup>-1</sup> error in all the compounds. This as initially explained is a internal error of the software since the calculations of anharmonic functions are carried out via an harmonic approximation. This brings about a 10% error in the computationally calculated results and thus the deviation from literature is observed.

{| border="1" cellpadding="2" align="center"
|+'''Table 21- Summarising the low frequency Stretches in the Cis isomer'''
|-
! scope="col" | No.
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Literature Frequency/cm<sup>-1</sup><ref>ELmer C. Alyea and Shuquan Song, ''Inorg. Chem.'', '''1995''', ''34'', 3864-3873 </ref>
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Intensity
! scope="col" | Associated Symmetry C<sub>2v</sub> Point Group
|-
|align="center"| 5 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 3;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39927_pl1208.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| The P atoms were stationary with the Cl atoms 'rocking' about them. The displacement of the Cl ligands were out of phase with the Metal-carbonyl (MoCO<sub>4</sub>) displacements|| align="center"|- || align="center"|10.7 || align="center"|0.03 || align="center"| -
|-
|align="center"| 6 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 4;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39927_pl1208.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The Cl ligands rock about the non-fixed P atoms where by the CO ligands rock out of phase with respect to the Cl ligand's displacement. The transition metal had a fixed centre. ||align="center"|- || align="center"| 17.6 || align="center"|0.008 || align="center"| -
|}

The table 21 shows the two low energy frequencies that complemented the fact that the structure were optimised. This is because the frequencies in the table are positive which was indicative that the second derivative was positive and the turning point was a minimum point. This accurately describes that of the ground state minima instead of the transition state maxima.

=== Analysis of the Trans-Metal Complex ===

[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Trans_vibanaly_pl1208_2ndopt.out Vibrational Analysis Trans Isomer Log File] {{DOI|10042/to-7471}}

Upon completion of the Gaussian frequency calculation, the completed file had to be checked to see if the calculation was successful. This was done by comparing the <u>E(RB+HF-LYP)</u> energy against the energy attained via the optimisation initially carried out. The results are expressed in the table below. Table 22 shows that both have exactly the same energy which is indicative that the frequency optimisation was successful.

{| class="wikitable" border="1" align="center"
|+ '''Table 22 - Comparing the total minimised energy via both Gaussian calculations'''
! Trans Isomer Optimisation !! Trans Isomer Frequency calculation
|-
| [[Image:Summary_pl1208_2ndopt_transpl1208.JPG|thumb|250x250px|centre]] || [[Image:Summarised_energy_pl1208transiosmer.JPG|thumb|250x250px|centre]]
|}

[[Image:Essential_data_pl1208trans_isomer.JPG|thumb|500x500px|Diagram 35 - Isolated Essential information from LOG file|centre]]

The log file is further analysed via specific analysis of the low frequencies segment as shown in diagram 30. In the line that says 'low frequencies' there are a total of 6 different frequencies that correspond to the '-6' in the formula 3N-6 vibrational frequencies. Further more, as the highest frequency in the 'low frequency' row it is 1 order of magnitude smaller than the lowest frequency in the normal vibration modes. It can be concluded that the calculation results are valid and further analysis can be undertaken.

[[Image:Point_group_pl1208Trans.JPG|thumb|300x300px|Point group symmetry of the Trans isomer|centre ]]

The Trans-isomer having the D<sub>4h</sub> point group symmetry will have the carbonyl bond stretches assigned based on the symmetry elements of the point group.

[[Image:pl1208transIR.OUT|thumb|800x800px| Diagram 36- Annotated IR spectra for the Trans isomer showing the distance carbonyl peak|centre]]

{| border="1" cellpadding="2" align="center"
|+'''Table 23- Summarising the Carbonyl Stretches in the Trans isomer'''
|-
! scope="col" | No.
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Literature Frequency/cm<sup>-1</sup><ref>ELmer C. Alyea and Shuquan Song, ''Inorg. Chem.'', '''1995''', ''34'', 3864-3873 </ref>
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Intensity
! scope="col" | Associated Symmetry D<sub>4h</sub> Point Group
|-
|align="center"| 1 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 44;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| Only upon close observations will the equatorial carbonyl appear to have very minimal stretches. The axial carbonyls have strong vibrations and would vibrate out of phase relative to each other. By relative comparisons, the equatorial carbonyls are assumed to be stationary.|| align="center"|1896 || align="center"|1950 || align="center"|1475 || align="center"| E<sub>u</sub>
|-
|align="center"| 2 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 45;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The Axial carbonyls bonds do not vibrate. But the Equatorial carbonyl stretches strongly vibration where both the vibrations are out of phase. ||align="center"|1896 || align="center"| 1951 || align="center"|1467 || align="center"| E<sub>u</sub>
|-
|align="center"| 3 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 46;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The axial carbonyl stretches are in phase with each other. The equatorial carbonyls vibrate in-phase as well. But the axial carbonyl stretches have a similar vibrational stretching magnitude to the equatorial carbonyl stretches. But the axial stretches are out of phase with respect to the equatorial stretches. || align="center"|- ||align="center"| 1977 || align="center"|0.61 || align="center"| B<sub>1g</sub>
|-
|align="center"| 4 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 47;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The axial carbonyl stretches are in phase with each other. The equatorial carbonyls vibrate in-phase as well. But the axial carbonyl stretches have a similar vibrational stretching magnitude to the equatorial carbonyl stretches. The axial stretches are in phase with respect to the equatorial stretches.||align="center"|- || align="center"| 2031 ||align="center"| 3.73 || align="center"| A<sub>1g</sub>
|}

The reasons for the deviation from the literature expected stretches would be the same reason as that explained earlier. For the trans isomer, 4 carbonyl stretches were noted (table 23). However, only one peak is seen in the IR spectra. This is rationalised by the following reasons.

* The trans isomer has a very different point group symmetry (D4h) as compared to the cis isomer (C2v). Thus, it is noted that the D4h point group brings about a greater order of symmetry for the trans isomer than for the cis isomer. This higher order of symmetry is expected to influence the bond stretches seen since higher symmetry usually results in less intense peaks seen.

* From the table No.1 AND 2 would have very similar type of carbonyl stretches (just different axis) so the expected change in dipole moment is going to be the same.They are thus theoretically expected to be degenerate to each other and only contribute to a single peak in the spectra. However, computationally they are non-degenerate vibrations having slightly different vibrational frequencies differing only by 1 cm<sup>-1</sup>. But they are so close they overlap and are seen as a single peak.

* No2 and 3 have all 4 carbonyl moving about the highly symmetric molecule that results in the dipoles trying to cancel out each other. This result in the overall dipole moment to be very small and would explain why they are not significant enough to be seen on the IR although computationally accounted for.

{| border="1" cellpadding="2" align="center"
|+'''Table 24- Summarising the low frequency Stretches in the Trans isomer'''
|-
! scope="col" | No.
! scope="col" | Model of Vibration
! scope="col" | Description Of the vibration
! scope="col" | Literature Frequency/cm<sup>-1</sup><ref>ELmer C. Alyea and Shuquan Song, ''Inorg. Chem.'', '''1995''', ''34'', 3864-3873 </ref>
! scope="col" | Calculated Frequency/cm<sup>-1</sup>
! scope="col" | Intensity
! scope="col" | Associated Symmetry D<sub>4h</sub> Point Group
|-
|align="center"| 5 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 3;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"| The phosphrous centres remain fixed while the Cl substituents rock about these fixed centres. The central (MoCO<sub>4</sub>) core has a rocking that is out of phase to the Cl's rocking displacement || align="center"|- || align="center"|5.03 || align="center"|0.10 || align="center"| -
|-
|align="center"| 6 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 4;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_39924_nithya'sbfvibrationaltrans.out</uploadedFileContents> </jmolApplet></jmol>
|align="center"|The axial carbonyls remain in their fixed position and so does the P centres. The Cl ligands rock about the fixed P centres. But the rocking of one PCl<sub>3</sub> unit is out of phase to the other PCl<sub>3</sub> unit. The equatorial carbonyl ligands have a very small rocking effect that is perpendicular to the rocking of the Cl substituents.||align="center"|- || align="center"| 6.09 || align="center"|0.0001 || align="center"| -
|}

==Conclusion==
Although the computationally calculated values correspond to the literature values to differentiate the cis from the trans isomer. This is only done against the PCl<sub>3</sub> references. At the start of the experiment, the PPh3 ligand was replaced with PCl3 as it was defined as a good substituent theoretically. This has yet to be proved experimentally and will be done via table 25 and 26 which compares the calculated Ir Carbonyl bond stretches against that of literature where the ligand is PPh3.

{| class="wikitable" border="1" align="center"
|+ '''<u>Table 25- Cis isomer (L=PCl<sub>3</sub>) against literature Cis isomer (L=PPh3)</u>'''
! Calculated C=O stretches(cm<sup>-1</sup>) !! Literature C=O stretches(cm<sup>-1</sup>)<ref>W.Hieber, J.Peterhans, Z. Naturforsch., 1959, 14b, pg 462</ref>
|-
|<CENTER> 1946 </CENTER>|| <CENTER> 1897 </CENTER>
|-
|<CENTER> 1949 </CENTER>|| <CENTER> 1908 </CENTER>
|-
|<CENTER> 1959 </CENTER>|| <CENTER> 1927 </CENTER>
|-
|<CENTER> 2024 </CENTER>|| <CENTER> 2023 </CENTER>
|}

{| class="wikitable" border="1" align="center"
|+ '''<u>Table 26- Trans isomer (L=PCl<sub>3</sub>) against literature Cis isomer (L=PPh<sub>3</sub>)</u>'''
! Calculated C=O stretches(cm<sup>-1</sup>) !! Literature C=O stretches(cm<sup>-1</sup>)<ref>W.Hieber, J.Peterhans, Z. Naturforsch., 1959, 14b, pg 462</ref>
|-
|<CENTER> 1950 </CENTER>|| rowspan="4" |<CENTER> 1902 </CENTER>
|-
|<CENTER> 1951 </CENTER>
|-
|<CENTER> 1977 </CENTER>
|-
|<CENTER> 2031 </CENTER>
|}

From both the tables, it can be seen that although the PCl3 was justified initially and is indeed computationally less demanding, it is how ever not the ideal replacement for the PPh3 ligand. This can be justified by the difference in the values to be rather significant and beyond that of the software error of 10%. Thus other ligands could be considered and tested for suitability in substituting the PPh3 ligand.

= Mini Project : Detailed Analysis of Boron and Nitrogen Analogues of Benzenes =

[[Image:Summary_MP_pl1208.bmp|thumb|200x200px|Diagram 37 - The 3 different analogues of benzene |centre ]]

The mini project carried out will aim to compare, contrast and rationalise the specific structures, bond stretches and the electronic characters of the three different analogues of benzene (Diagram A).

== Introduction to the three different analogues ==

<u>Pyridinium ion</u>

Pyridinium is the cationic form of pyridine where due to the positive charge being localised on the nitrogen centre it can be readily used as an organic base in chemical reactions. The cation has a pKa of 5.2 which is very close to that of acids such as Acetic acid which has a pKa of 4.8.<ref><font>E. P. Serjeant and B. Dempsey (eds.), ''Ionization Constants of Organic Acids in Solution'', IUPAC Chemical Data Series No. 23, Pergamon Press, Oxford, UK, 1979</font></ref> This is because the heteroatom being from group 5 in the periodic table has an electronegativity of about 3.04 on the pauling scale. This makes it distinctly electronegative and would favourably lose the proton (H-N) to remove the +1 charge. The pyridinium ion having the C2<sub>(x)</sub> and sigma<sub>v</sub> symmetry elements would belong to the C<sub>2v</sub> point group symmetry.

[[Image:Plane_symmetry_mp_pl1208.bmp|thumb|200x200px|Diagram 38 -Symmetry elements of the cation|centre ]]

<u>Benzene</u>

Kekulé was the first to introduce the sensible structure for benzene where all the 6 carbon atoms are arranged in a hexagon. All the carbons in the plane being of equal electronegativity (2.5 on the pauling scale) would have equal electron sharing and would form covalent bonds. The carbon not being as electronegative as nitrogen has an pKa of 43. This highly large value is indicative that it would not be as easily deprotonated as the carbon would not readily hold on to a (-) charge. Benzene has a D6h point group symmetry due to the C6 rotation along the principal axis and the planar reflection along sigma(h)

[[Image:Plane_symmetrymp_2_pl1208.bmp|thumb|300x300px|Diagram 39-Symmetry elements towards D<sub>6h</sub>|centre ]]

<u>Pyridinium ion</u>

Boratabenzene having the boron heteroatom forms a heteroaromatic compound similar to that of pyridinium -both would have the same point group symmetry of C<sub>2v</sub>. However, chemically they are very different since boron has an electronegativity of 2.04 on the pauling scale. The presence of the anionic charge is expected to increase the electron density of the ring. Thus, making it more favourable to reactions such as Diels-alder reactions which result in the loss of aromaticity to form borabarrelenes type compounds.<ref>1-Borabarrelene Derivatives via Diels-Alder Additions to Borabenzenes Thomas K. Wood, Warren E. Piers, Brian A. Keay, and Masood Parvez Org. Lett.; 2006; 8(13) pp 2875 - 2878; (Letter) doi:10.1021/ol061201w</ref>

== Aims of the Mini Project ==

* The influence of the different heteroatoms (B and N) are expected to bring about different influences to the benzene ring which as benzene has equal charge / electron distribution. Hence benzene has equal C-C bond lengths. The utility of an effective basis set to optimise the individual analogues to the lowest configuration. This will then be compared against literature to determine the accuracy of the basis set used and to validate the literature results where applicable.

* The key factor for influences is the difference in electronegativity of the heteroatoms. Thus, an NBO analysis will be carried out to justify the pKa values stated for pyridinium ion and benzene while proving the hypothesis for the pKa of the boratabenzene.

* A vibrational frequency analysis will be carried out to complement that the lowest energy conformation is indeed computationally attained. Also the differences between the bond stretches in the different analogues will be accounted for and rationalised.

*Finally an MO analysis will be carried out to justify the electronic nature of the analogues and facilitate in justifying the pKa values as well as the C-C bond stretches.

== Optimisation of the 3 Analogues ==

The method adopted for the optimisation was the Density Functional Theory (DFT) via the B3LYP (Becke, three-parameter, Lee-Yang-Parr) exchange-correlation functional. Initially the 3-21G basis set was used to attain a 'rough' optimisation for each analogue. Upon the first optimisation, a more accurate basis set was adopted 6-31G(d,p) to be carried out in the post 3-21G optimised analogues. This basis set was deemed to be more accurate since the 3-21G basis set was of 3 primitive GTO for core electrons where 2 were for inner valence orbitals and 1 for outer. This was a poor representation and was deemed only useful for preliminary geometry optimisation but poor for energy minimisation. The 6-31G(d,p) basis set adds polarisation to all atoms and thus better improves the modelling of the core electrons. This allows more accurate energy minimisation. For this project it will be chosen as the basis set as it leads to convergence to present threshold value (to be seen) and is the best compromise of speed and accuracy.

{| class="wikitable" border="1" align="center"
|+ Table 27 :'''Successive Optimisation for Benzene'''
!<CENTER>Properties Compared</CENTER> !!<CENTER> 3-21G Optimised Benzene </CENTER>!!<CENTER> 6-31 G(d,p) Optimised Benzene </CENTER>
|-
| <CENTER>Diagram of Molecules</CENTER> || width=300px | <CENTER><jmol><jmolApplet><title>3-21G optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>3-21g_benzene_pl1208.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>|| width=300px |<CENTER><jmol><jmolApplet><title> 6-31 G(d,p)Optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>6-31gdp_pl1208_benzne.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>
|-
|<CENTER> Summary of Energy and Gradient</CENTER> || [[Image:Summary_pl1208_3-21g_benzene.JPG|220px|centre]] || [[Image:Summary_pl1208_benzene_6-316dp.JPG|220px|centre]]
|-
| <CENTER>Progress of Optimisation</CENTER> || [[Image:Gradient_pl1208_3-21g_benzene.JPG|250X220px|centre]]
|| [[Image:Gradient_pl1208_6-31g_benzene.JPG|220X220px|centre]]
|-
|<CENTER> C<sub>1</sub>-C<sub>2</sub> bond length(Å)</CENTER> ||<CENTER> 1.397 </CENTER>||<CENTER> 1.396 </CENTER>
|-
|<CENTER> C<sub>1</sub>-C<sub>2</sub> bond length(Å)</CENTER> ||<CENTER> 1.397 </CENTER>||<CENTER> 1.396 </CENTER>
|-
|[https://wiki.ch.ic.ac.uk/wiki/images/e/e7/3-21g_benzene_pl1208.LOG 3-21G Benzene Optimisation][https://wiki.ch.ic.ac.uk/wiki/images/a/ae/6-31gdp_pl1208_benzne.LOG 6-31G(d,p) Benzene Optimisation]
|}

{| class="wikitable" border="1" align="center"
|+ Table 28 :'''Successive Optimisation for 1-H-Boratabenzene'''
!<CENTER>Properties Compared</CENTER> !!<CENTER> 3-21G Optimised 1-H-Boratabenzene </CENTER>!!<CENTER> 6-31 G(d,p) Optimised 1-H-Boratabenzene </CENTER>
|-
| <CENTER>Diagram of Molecules</CENTER> || width=300px | <CENTER><jmol><jmolApplet><title>3-21G optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>1ST_OPT_pl1208_borata.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>|| width=300px |<CENTER><jmol><jmolApplet><title> 6-31 G(d,p)Optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>6-31gdp_boratabenzene_pl1208.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>
|-
|<CENTER> Summary of Energy and Gradient</CENTER> || [[Image:Summary_boron_3-21_pl1208.JPG|220px|centre]] || [[Image:Summary_boron_6-31_pl1208.JPG|220px|centre]]
|-
| <CENTER>Progress of Optimisation</CENTER> || [[Image:Gradient_pl1208_3-21g_boron.JPG|250X220px|centre]]
|| [[Image:Gradient_pl1208_6-31g_boron.JPG|220X220px|centre]]
|-
|<CENTER> B<sub>1</sub>-C<sub>2</sub> bond length(Å)</CENTER> ||<CENTER> 1.515 </CENTER>||<CENTER> 1.514 </CENTER>
|-
|<CENTER> C<sub>2</sub>-C<sub>3</sub> bond length(Å)</CENTER> ||<CENTER> 1.402 </CENTER>||<CENTER> 1.398 </CENTER>
|-
|[https://wiki.ch.ic.ac.uk/wiki/images/5/56/1ST_OPT_pl1208_borata.LOG 3-21G 1-H-Boratabenzene Optimisation][https://wiki.ch.ic.ac.uk/wiki/images/8/8a/6-31gdp_boratabenzene_pl1208.LOG 1-H-Boratabenzene Optimisation]
|}

{| class="wikitable" border="1" align="center"
|+ Table 29 :'''Successive Optimisation for Pyridinium'''
!<CENTER>Properties Compared</CENTER> !!<CENTER> 3-21G Optimised Pyridinium </CENTER>!!<CENTER> 6-31 G(d,p) Optimised Pyridinium </CENTER>
|-
| <CENTER>Diagram of Molecules</CENTER> || width=300px | <CENTER><jmol><jmolApplet><title>3-21G optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>pl1208_Pyridinium_3-21gopt.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>|| width=300px |<CENTER><jmol><jmolApplet><title> 6-31 G(d,p)Optimised molecule</title><color>black</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 180 120;spin 2;</script>
<uploadedFileContents>6-31gdp_pyridinium_pl1208.LOG</uploadedFileContents>
</jmolApplet></jmol></CENTER>
|-
|<CENTER> Summary of Energy and Gradient</CENTER> || [[Image:Summary_3-21_pyrinidine_pl1208.JPG|220px|centre]] || [[Image:Summary_6-31_pyrinidine_pl1208.JPG|220px|centre]]
|-
| <CENTER>Progress of Optimisation</CENTER> || [[Image:Gradient_3-21_pyrinidine_pl1208.JPG|250X220px|centre]]
|| [[Image:Gradient_6-31_pyrinidine_pl1208.JPG|220X220px|centre]]
|-
|<CENTER> N<sub>1</sub>-C<sub>2</sub> bond length(Å)</CENTER> ||<CENTER> 1.360 </CENTER>||<CENTER> 1.352 </CENTER>
|-
|<CENTER> C<sub>2</sub>-C<sub>3</sub> bond length(Å)</CENTER> ||<CENTER> 1.384 </CENTER>||<CENTER> 1.383 </CENTER>
|-
|[https://wiki.ch.ic.ac.uk/wiki/images/2/25/Pl1208_Pyridinium_3-21gopt.LOG 3-21G Pyridinium Optimisation][https://wiki.ch.ic.ac.uk/wiki/images/5/5a/6-31gdp_pyridinium_pl1208.LOG 6-31G(d,p) Pyridinium Optimisation]
|}

Overall it is observed that the second more accurate 6-31G basis set brought about a lowering of the total energy of the individual molecules. It also resulting in the RMS gradient to be within the same range of being lesser than 0.001 a.u. which was indicative of an almost horizontal plot as seen from the progress of optimisations above. This was indicative of the derivative forces had converged at the present threshold level and was an indication for the completion of the optimisation. This allowed the optimised molecules to be further analysis via vibrational analysis which upon analysis of the 'low' frequencies will further complement or contradict the claimed successful optimisation. The convergence of the force derivatives against present threshold was further checked with all the respective log files.

There is a difference in the point group symmetries in the table above as compared to that given in the introduction. Analysis of each individual symmetry element was indicative that the programme had assigned the incorrect point group where benzene was not C<sub>1</sub> and instead D6h while boratabenzene anion and pyridinium cations are both not C<sub>1</sub> but instead C<sub>2v</sub>. This would be expected to occur as the benzene molecule in the gaussview is distorted upon optimisation resulting in the deviation from the expected point group. However, for the vibration-based analysis they will be subjected to the symmetry elements of the D<sub>6h</sub> and C<sub>2v</sub> point groups respectively.

=== Bond Analysis ===
[[Image:Carbon_numbering.bmp|thumb|200x200px|Diagram 40-carbon numbering in benzene analogues |right]]

{| class="wikitable" border="1" align = "centre"
|+ '''Table 30- Comparing the Calculated Bond lengths against the Literature values'''
!| !!colspan="7"| Bond Length/Å
|-
! Bond !! Calculated- C<sub>6</sub>H<sub>6</sub> !! Literature - C<sub>6</sub>H<sub>6</sub> !! Calculated-C<sub>5</sub>BH<sub>6</sub><sup>-</sup> !!Literature C<sub>5</sub>BH<sub>6</sub><sup>-</sup> <ref>http://pubs.acs.org/doi/abs/10.1021/ja00137a030</ref> !!Calculated-C<sub>5</sub>NH<sub>6</sub><sup>+</sup>!!Literature C<sub>5</sub>NH<sub>6</sub><sup>+</sup><ref>http://pubs.acs.org/doi/pdfplus/10.1021/jo051354h</ref>
|-
| align="center" |'''X<sub>1</sub>-H''' ||align="center" | 1.09 ||align="center" |1.08 || align="center"|1.22 ||align="center" |1.02 || align="center"|1.017||align="center" |1.052
|-
| align="center"|'''X<sub>1</sub>-C<sub>2</sub>''' || align="center"|1.40 ||align="center" |1.40 ||align="center"|1.541 ||align="center" | 1.453||align="center"|1.352 || align="center" |1.351
|-
| align="center"|'''C<sub>2</sub>-C<sub>3</sub>''' || align="center"|1.40 ||align="center" | 1.40||align="center"|1.399 ||align="center" | 1.389||align="center"|1.383 || align="center" | 1.381
|-
| align="center"|'''C<sub>3</sub>-C<sub>4</sub>''' || align="center"|1.40 ||align="center" | 1.40||align="center"|1.405 ||align="center" | 1.388||align="center"|1.399 || align="center" | 1.396
|-
| align="center"|'''C<sub>4</sub>-C<sub>5</sub>''' || align="center"|1.40 ||align="center" | 1.40||align="center"|1.405 ||align="center" | 1.380||align="center"|1.399 || align="center" | 1.396
|-
| align="center"|'''C<sub>5</sub>-C<sub>6</sub>''' || align="center"|1.40 ||align="center" | 1.40||align="center"|1.399 ||align="center" | 1.410||align="center"|1.383 || align="center" | 1.381
|-
| align="center"|'''C<sub>6</sub>-X<sub>1</sub>''' || align="center"|1.40 ||align="center" | 1.40||align="center"|1.541 ||align="center" | 1.481||align="center"|1.352 || align="center" | 1.351
|}

A comparison of the X(heteroatom)-C bond and C-C bond lengths were carried out. Comparisons made against literature values for all three analogues showed that the calculated values had minimal deviation from the literature values. Thus the geometry and structure of the computationally calculated analogues were highly reliable. Comparing the X-H bond across the table, it is seen that the nitrogen analogue has the smallest N-H bond followed by benzene and lastly the boron. Thus the more electronegative X group used results in the X-H bond to be shorter which can be accounted for due to the electronegative character of the X atom.

Comparing the C-C bond lengths it seen that the use of the N atom results in the ring to contract slightly by 0.01 angstroms. This is a very small distance to be be making any conclusions. It is however anticipated that there should not be much changes in the N analogue benzene apart from inductive effects as the only available p-oribtal lone pair is not localised in the N-H bond. Thus it is unable to have resonanace type interactions with the π* orbitals of the ring. Thus, minimal changes the the C-C bonds is expected. However, being the more electronegative atom it will have inductive electron withdrawing. But for the Boron which has the similar situation of the empty orbital now localised in the B-H bond it is more electropositive in nature and should have more inductive electron donation into the ring and would be expected to slightly increase the C-C bond since donation of electrons into the π* orbital inductively decreases the energy of the π bond which causes a slight lengthening of the bond.

== Population Analysis ==

The population analysis will be carried out as two seperate segments where the first will be an Natural Bond Order and Mullikien charge analysis followed by a Molecular Orbital (MO) analysis. The analysis will be used to justify and compare the electronic distribution within the different benzene analogues.

=== Natural Bond Order & Mullikien Charges Analysis ===

The NBO analysis carried out is expected to help rationalise the change in charge density throughout the molecule. The green regions indicate high positive charges while the red regions indicate high negative charges. The more negative the charge the greater the electron density about the region. The mullikien charge density map is just another way of determining the charge density allocations within the moelcule.

{| align=center
|+ '''Diagram 41 - NBO and MC for benzene'''
| [[Image:NBO_BENZENE_PL1208.JPG|thumb|200px|NBO [https://wiki.ch.ic.ac.uk/wiki/images/9/98/MO_ANALYSIS_PL1208_BENZNEN.LOG NBO and MC log File]]]
| [[Image:MULLIKIN_CHARGE_PL1208_BENZENE.JPG|thumb|200px|mullikin charges]]
|}

The carbon framework of the benzene ring is in red as expected with a numerical value of -0.238 since it accounts for the delocalised pi-electron could expected from an aromatic compound. Having the delocalised centre would explain the electron deficient nature of the protons which are yellow and a positive value of 0.239. However, based on the bond lengths seen earlier, benzene has an equal electron distribution so all the carbons have the same electron density and so do all the hydrogens relative to each other. However it is expected to change for the other two analogues which introduces the heteroatom of a varying electronegativity.

{| align=center
|+ '''Diagram 42 - NBO and MC for Boratabenzene'''
| [[Image:NBO_BORON_PL1208.JPG|thumb|200px|NBO [https://wiki.ch.ic.ac.uk/wiki/images/0/0f/MO_PL1208_BORON.LOG NBO and MC log File]]]
| [[Image:MULLIKIN_CHARGE_PL1208_BORON.JPG|thumb|200px|mullikin charges]]
|}

Looking at the NBO charges, replacement of the C-H with B-H where boron being from G(III) is much more electropositive. This is seen from the difference in charges where a significant change is magnitude relative to benzene from -0.235 to 0.202 is observed. This shows an inversion in the polarity where the X atom in boratabenzene is electron deficient. Boron being less electron withdrawing is seen by comparing the C-H against the N-H where the charge density on the hydrogen decreases from 0.235 in benzene to -0.097 in boratabenzene. This shows that the hydrogen is no longer electron deficient and would less readily dissociate as H+. Thus explaining why borata benzene is less acidic than benzene. The effects of being electropositive are inductive and resonance based. This is seen in the 2 adjecent carbons (position 2 and 6) which has a charge of -0.588 whereas benzene carbons have it as -0.235. The two next furthest (position 3 and 5) have it as -0.250 which is half the magnitude of position 2 and 6. This shows that these 2 carbons being 2 bonds away from the boron would be less electron dense as it experiences less of the 'electon-pushing' effects of the boron. The furthest carbon (position 4) has a value of -0.340. This shows the furthest has the second most electon density in the ring. Thus the un-equal electron density of the system can be rationalised by the scheme below. As electron delocalisation into the ring is not as significant, the C-H bond hydrogen would be less electron deficient of 0.184 (smaller positive value) that the benzene hydrogens.

[[Image:SCHEME_1_boron_pl1208.bmp|thumb|400X400px|Diagram- Resonance rationalising the difference in electron density within the framework|centre]]

{| align=center
|+ '''Diagram 43 - NBO and MC for Pyridinium'''
| [[Image:NBO_NITROGEN_PL1208.JPG|thumb|200px|NBO [https://wiki.ch.ic.ac.uk/wiki/images/d/df/MO_PL1208_NITROGEN.LOG NBO and MC log File]]]
| [[Image:MULLIKIN_CHARGE_PL1208_NITROGEN.JPG|thumb|200px|mullikin charges]]
|}

Again comparing the charges within the pyridinium analogue, the nitrogen(Group V) being more electron withdrawing has a larger negative magnitude of -0.475 as compared to benzene which has -0.239. This fits the logic of nitrogen being more electron withdrawing than carbon. This would however similar to boratabenzene result in unequal electron distribution within the benzene analogue. The inductive effects result in the N-H hydrogen to be strongly electron deficient of 0.483 as compared to 0.239 in benzene. The two adjecent carbons are a positive magnitude of 0.071 as compared to the other two frame works which are negative. This shows that the ortho positions are electron deficient and more deficient that position 4 due to added resonance + inductive effects (Diagram 43). The meta position would not be as electron deficient as it is less influenced by inductive effects and less by resonance so a negative value.

[[Image:SCHEME_1_nitrogen_pl1208.bmp|thumb|400X400px|Diagram 44- Resonance rationalising the difference in electron density within the framework|centre]]

=== Molecular Orbital Analysis ===

[[Image:CLOUD_REPRESENTAATION.bmp|thumb|400X400px|Diagram 45- showing the orbitals of all 3 analogues|centre ]]

The cloud representation shows the different type of analogues formed where the nitrogen analogue being electronegative would be inductively withdrawing and would have an electron deficient nitrogen centre. Inversely, the electropositive boron results in an inductive electron pushing into the carbon framework. Benzene having the same atoms within the framework would have equal electron distribution.

The below molecular diagrams shows the occupancy of the 3 bonding orbitals and to be specific in the analysis, only the Pi molecular oribital are looked at. Thus these specific 6 molecular orbitals will be looked into through all 3 analogues and their respective energy levels will be compared and rationalised

====<u>'''Approach towards B and N analogues'''</u>====

Both the boron and nitrogen analogues of benzene have the same point group (C<sub>2v</sub>) and so would be expected to have the same type of molecular orbitals with varying energy levels. The molecular orbitals would be of varying energy levels as the heteroatoms are of different electro-negativities (shown above in NBO analysis). The Character table will be used to determine the irreducible representations of the p-orbtals in the basis set. Subsequently the rotation and projection operators are used to determine how each other is expected to look like. As the p-orbitals are delocalised to form the Pi-System and there being a total of 21 occupied molecular orbitals, only the p-orbitals will be taken into account to compare the LCAO approach against the computationally derived orbitals. This will be done by using the rotation and projection operators to determine how the 6 p-orbitals 'look' like.

{| align=center
|+ '''Diagram 46 - Molecular Orbital Diagram for Pyridinium ion'''
| [[Image:Mo_diagram_nitrogen_pl1208.jpg|thumb|500px|Molecular Diagram for the Pyridinium cation via LCAO|centre]] || [[Image:C2v_point_group_pl1208.JPG|thumb|300px|C2v Point group character table|centre]]
|}

{| align=left
|+ '''Diagram 47 - Annotated Pi- Molecular Orbitals of Pyridinium ion'''
| [[Image:TABLE_PL1208NITROGEN.bmp|thumb|400x400px|'''Selected Pi-molecular orbitals''']]
|}

{| align=center
|+ '''Diagram 48 - Homo-Lumo Molecular Orbitals of Pyridinium ion'''
| [[Image:Mo1_pl1208_nitrogen.JPG|thumb|200x200px|'''MO 1 (HOMO-4)-0.640 a.u.''']]
| [[Image:Mo2_pl1208_nitrogen1.JPG|thumb|200x200px|'''MO 2 (HOMO-1)-0.508 a.u.''']]
| [[Image:Mo3_pl1208_nitrogen.JPG|thumb|200x200px|'''MO 3(HOMO) -0.478 a.u.''']]
|}
{| align=center
| [[Image:Mo4_pl1208_nitrogen.JPG|thumb|200x200px|'''MO 4 (LUMO)-0.258 a.u.''']]
| [[Image:Mo5_pl1208niitrogen.JPG|thumb|200x200px|'''MO 5 (LUMO+1)-0.220 a.u.''']]
| [[Image:Mo6_pl1208_nitrogen.JPG|thumb|200x200px|'''MO 6 (LUMO+3)-0.073 a.u.''']]
|}

As stated before, only the specific orbitals that correspond similarly to the orbitals via LCAO are considered as they are the Pi-oribtals. Diagram 47 shows that the HOMO-LUMO orbitals have the orbital phases as expected in the LCAO approach. However, Mo1 and MO 6 are not in direct sequence with the other orbitals (highlighted) and is because those in between are possibly due to other orbital mixings which will not be considered.

{| align=left
|+ '''Diagram 49- Annotated Pi- Molecular Orbitals of Boratabenzene ion'''
| [[Image:Table_pl1208_boron.bmp|thumb|400x400px|'''Selected Pi-molecular orbitals''']]
|}

{| align=center
|+ '''Diagram 50 - Homo-Lumo Molecular Orbitals of Boratabenzene ion'''
| [[Image:MO_1_pl1208_boron.JPG|thumb|200x200px|'''MO 1 (HOMO-4)-0.132 a.u.''']]
| [[Image:MO_2_pl1208_boron.JPG|thumb|200x200px|'''MO 2 (HOMO) 0.010 a.u.''']]
| [[Image:MO_3_pl1208_boron.JPG|thumb|200x200px|'''MO 3 (HOMO-1) -0.034 a.u.''']]
|}
{| align=center
| [[Image:MO_4_pl1208_boron.JPG|thumb|200x200px|'''MO 4 (LUMO+1) 0.232 a.u.''']]
| [[Image:MO_5_pl1208_boron.JPG|thumb|200x200px|'''MO 5 (LUMO) 0.214 a.u.''']]
| [[Image:MO_6_pl1208_boron.JPG|thumb|200x200px|'''MO 6 (LUMO+7) 0.370 a.u.''']]
|}

The comparison of the molecular orbitals is similar to that seen in the nitrogen analogue where the distinct difference is with the energy levels of the specific molecular orbitals that will be compared subsequently.

====<u>'''Approach towards Benzene'''</u>====

{| align=center
|+ '''Diagram 51 - Molecular orbital Diagram with nodal planes for benzene'''
| [[Image:Mo_diagram_pl1208_benzene.jpg|thumb|500px|Molecular Diagram for Benzene via LCAO|centre]] || [[Image:Point_group_d6h_pl1208.JPG|thumb|300px|D6h Point group character table|centre]]
|}

{| align=left
|+ '''Diagram 52 - Annotated Pi- Molecular Orbitals of Benzene'''
| [[Image:TABLE_2.bmp|thumb|400x400px|'''Selected Pi-molecular orbitals''']]
|}

The diagram shows the equal splitting of the molecular orbitals which is expected since all the atoms of the 6-membered framework are carbons and of the same electronegativity. It is also seen with greater out of phase interactions there is a greater number of nodal planes and as such there is more destabilising interactions resulting in a higher energy level molecular orbital

{| align=center
|+ '''Diagram 53 - Homo-Lumo Molecular Orbitals of Benzene'''
| [[Image:HOMO-4_pl1208_benzene.JPG|thumb|200x200px|'''MO 1 (HOMO-4)-0.359 a.u.''']]
| [[Image:Homo-1_pl1208_mpbenzene.JPG|thumb|200x200px|'''MO 2 (HOMO-1)-0.246 a.u.''']]
| [[Image:Homo_pl1208_mp_benzene.JPG|thumb|200x200px|'''MO 3(HOMO) -0.246 a.u.''']]
|}
{| align=center
| [[Image:LUMO_pl1208_benzene.JPG|thumb|200x200px|'''MO 4 (LUMO)0.002 a.u.''']]
| [[Image:Lumo+1_pl1208_benzene.JPG|thumb|200x200px|'''MO 5 (LUMO+1)0.002 a.u.''']]
| [[Image:Lumo_+5_pl1208_benzene.JPG|thumb|200x200px|'''MO 6 (LUMO+5)0.161 a.u.''']]
|}

====<u>'''Comparing the influence of the Heterogroup'''</u>====

{| class="wikitable" border="1" align="left"
|+ '''Table 31 - Energies of Molecular Orbitals'''
! Molecular Orbital !! C<sub>6</sub>H<sub>6</sub>(a.u.) !! C<sub>5</sub>BH<sub>6</sub><sup>-</sup>(a.u.) !! C<sub>5</sub>H<sub>6</sub>N<sup>+</sup>(a.u.)
|-
| align="center"|'''MO 1''' ||align="center"| -0.359||align="center"|-0.132||align="center"|-0.640
|-
| align="center"|'''MO 2''' || align="center"|-0.246||align="center"|0.010||align="center"|-0.508
|-
| align="center"|'''MO 3''' || align="center"|-0.246||align="center"|-0.034||align="center"|-0.478
|-
| align="center"|'''MO 4''' || align="center"|0.002||align="center"|0.232||align="center"|-0.258
|-
| align="center"|'''MO 5''' ||align="center"| 0.002||align="center"|0.214||align="center"|-0.220
|-
| align="center"|'''MO 6''' || align="center"|0.161||align="center"|0.370||align="center"|-0.073
|}

[[Image:Graph_of_energies_pl1208_mp.JPG|thumb|700x600px|Graph representing the Molecular orbital energies for all 3 analogues|centre ]]

As seen from the above tables and diagrams, it can be seen that for all there analogues the expected orbital phases are highly similar to the calculated molecular orbitals. However there is a difference in the HOMO-LUMO gap for a three analogues and is due to the difference of the R atom substituted into the ring. The benzene has a difference between MO 3 and MO 4 of 0.248 a.u. while the Boron analogue has a difference between MO 2 and MO 5 of 0.224 a.u. which is a slightly smaller gap and finally the nitrogen analogue has the gap between MO 3 and MO 4 of 0.220 a.u. It would show that the nitrogen analogue as expected is the better pi acceptor due to the electronegative nature of nitrogen.

Comparisons of the plots in the graph would show that the Nitrogen analogue being more electronegative will be lower in energy level and thus having the lowest lumo explains it higher pKa for the deprotonation as compared to the other analogues. Similarly, boron being electro positive will have the higher energy p orbitals that result in higher energy pi-orbitals. This would make the LUMO energetically less accessible and thus would have a much higher pKa and is instead a much better base with the higher more readily available HOMO orbital.

== Vibrational Analysis ==
[https://wiki.ch.ic.ac.uk/wiki/images/d/d1/VIBRATIONAFREQINPUT_pl1208_benzene.LOG Benzene vibrational analysis Log File]
[https://wiki.ch.ic.ac.uk/wiki/images/a/a6/VIBRATIONAANALYSIS_pl1208_boron.LOG Boratabenzene vibrational analysis Log File]
[https://wiki.ch.ic.ac.uk/wiki/images/a/a9/VIBRATIONALFREQ_pl1208_nitrogen.LOG Pyridinium vibrational analysis Log File]

{| align=center
| [[Image:Ir_spectra_pl1208_benzene.JPG|thumb|400px|IR spectra for Benzene ]]
| [[Image:Ir_spectra_pl1208_borane.JPG|thumb|400px|IR spectra for Boratabenzene ]]
| [[Image:Ir_spectra_pl1208_nitrogen.JPG|thumb|400px|IR spectra for Pyridinium ]]
|}

{| class="wikitable" border="1" align="center"
|+ <u>'''Table comparing the Calculated vibrational stretches against literature for nitrogen analogue'''</u>
! Type of Stretch !! Literature value (cm<sup>-1</sup>)<ref>http://pubs.acs.org/doi/pdf/10.1021/jp013080h</ref> !! Calculated Value !! Diagramatic representation
|-
| Ring Deformation || 392 || 392.5 || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 3;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| Ring Deformation || 606 || 620|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 5;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| CH Bend || 661 || 676|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 7;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| Ring Deformation || 859 || 855|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 9;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| CH Bend || 969,990 || 992|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 10;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| CH Bend + Ring Deformation || 1048 || 1023, 1048|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 14;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| CH-IP bend || 1165 || 1199|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 18;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| CH bend || 1254 || 1229|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 19;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| Ring Stetch || 1321,1375 || 1374|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 21;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| Ring Deformation || 1474,1530 || 1416,1524|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 23;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| C-H stretch || 3112,3129, 3139,3141 || 3243, 3256|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 30;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|-
| N-H stretch|| 3451 || 3569|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>200</size>
<script>zoom 100;frame 32;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>VIBRATIONALFREQ_pl1208_ni.LOG</uploadedFileContents> </jmolApplet></jmol>
|}

From the table above it can be seen that the calculated results are in good agreement with the literature values and analysis of all 3 log files show that the optimisation did lead to a positive turning point as explained in earlier sections. Thus the vibrational analysis complemented on the successive optimisation carried out for all three analogues. But do to lack of time vibrational analysis was only carried out for the nitrogen analogue. It was also not the 'focused' on method of analysis as the three analogues have different atoms present where the main analysis factor was the MO analysis carried out.

= Conclusion=

It has been justified and rationalised why the nitrogen analogue is more proton labile than the other two. Also, the influence of electronegativity into an aromatic ring has been analysed.

[[User:Pl1208|Fisty]] 13:13, 13 March 2011 (UTC)

=References=
<references/>

Talk:00561753 01

2011-04-08T16:09:20Z

Pl1208: Talk:00561753 01 moved to Talk:Sunkiss pl1208 01

#REDIRECT [[Talk:Sunkiss pl1208 01]]

Talk:Sunkiss pl1208 01

2011-04-08T16:09:20Z

Pl1208: Talk:00561753 01 moved to Talk:Sunkiss pl1208 01

1.2.1 Cp dimers: That’s quite a lengthy answer, it’s a good one though.

1.2.2 NAD: Good to see you presenting your minimisation graphically. Excellent analysis.

1.2.3 Taxol: A very good analysis. However, your structure for A is not fully optimised (the energy should be around 48 kcal mol-1).

1.3.1 Carbene: Good to see you’ve picked up on the bug in PM6 and that you are contrasting and comparing the results of more than one method. Another lengthy but thorough analysis.

Mini-project: Good introduction and exploration of the mechanism. A good presentation of your results as well. Some fantastically accurate calculations here. Some well presented conclusions too. An excellent project.

Overall: : We’re not looking for essays, your answers really were FAR too long and I found them rather difficult to wade through after a while; I’m sure you could have summarised the salient points into something less half the length. Although it’s a matter of personal taste, I find rotating Jmols incredibly irritating. Despite the over-enthusiastic length, a very good piece of work.

00561753 01

2011-04-08T16:09:19Z

Pl1208: 00561753 01 moved to Sunkiss pl1208 01

#REDIRECT [[Sunkiss pl1208 01]]

Sunkiss pl1208 01

2011-04-08T16:09:19Z

Pl1208: 00561753 01 moved to Sunkiss pl1208 01

='''Aim of Experimentation'''=

Specific computational methods (Molecular modelling and semi-empirical DFT) will be implemented to describe and justify the outcome of organic chemical reactions. These modelling experiments will be carried out via the use of different programmes such as MM2 force field parameters (classical method), PM6 parameters (semi-quantum method) and DFT modelling method (Quantum Mechanics) on simple organic molecules. The advantages and limitations of each method is determined via detailed analysis of the results attained. The list defines the different modelling experiments to be carried out

* Hydrogenation of cyclopentadienes to form the dimer

* Nucleophilic addition to a pyridinium ring (NAD<sup>+</sup> Ananlogue)

* Reactivity of Taxol intermediates

* Regioselective addition of Dichloro-carbene.

* To conclude the usage of the computational software, a mini-research exercise is carried out

='''Introduction''' =

The importance of computational chemistry is largely seen in modern times where it's importance is not only seen in theoretical advancements but also in allowing highly accurate predictions to be made for chemical reactions to be carried out in laboratories.<ref>

Astrophysics and Space Science Proceedings, 2010, 21-30, DOI: 10.1007/978-3-642-10322-3_3

</ref> <ref>Hudson, Matthew R., "Investigation of hydrogen bonded molecular solids by diffraction, spectroscopy, and computational chemistry" (2010). Chemistry - Dissertations and Theses. Paper 168.http://surface.syr.edu/che_etd/168</ref> Computational methods implement specific softwares such as ChemBio3D which allows the total energy of chemical reactions to be calculated.This allows the stability of the products formed to be quantified; allows accurate prediction of individual bond energies. Computational chemistry serves as the 'backbone' for chemical reactions carried out in the laboratories. Furthermore, it also helps towards predictions made for the ideal scale, reaction conditions and expected yield to be attained from the wet reaction to be carried out.

The first modelling technique considered is molecular modelling(MM).(MM) based calculations via the use of MM2 force field parameters treat the molecules classically where newtonian mechanics is implemented to model the molecular system.<ref>N.Allinger, J. Am. Chem. Soc., 1977, 99(25), pp 8127–8134:{{DOI|10.1021/ja00467a001}}</ref> The method draws parallel to 'viewing' the atoms and bonds as spheres held together by springs. The calculations carried out allow the molecule to be 'modelled' to it's lowest energy conformation. This allows the calculated results to be compared with the experimentally attained results. The overall minimum energy of the molecule is attained via a summation of the individual bond energies (''Table 1'').

The PM6 parameters is defined as the semi-classical method of modelling.<ref>doi:10.1007/s00894-008-0299-7</ref> Being a semi-classical method it still functions on the underlying principal of 'viewing' the atoms and bonds as spheres held together by springs. However, there is a specific degree of quantum influence in determining the overall energy where electronic distribution within the molecule is taken into account when calculating the specific bond energies. This enables deviations due to the influence of an electron donating or withdrawing substituent to be observed and have the calculated value to be much more similar to the experimentally attained value as compared to calculations made via the MM2 model.

The last modelling method considered is the Density Functional Theory (DFT), which is a quantum mechanics-based calculation.<ref>Density Functional Theory – an introduction, Rev. Mod. Phys. 78, 865–951 (2006)</ref> The method implements the functionals of electron density to accurately describe the excited states allowing the electron's spatial influence to be accounted. This makes the modelling method to give the most accurate optimisation of the structure relative to the other two modelling methods considered.<ref>J. Chem. Phys., 2005, 124:{{DOI|10.1063/1.2148954}}</ref>

However, the (MM) method is the most favoured as it is the least expensive method and is well adapted for the provision of excellent structural parameters such as bond distances and bond angles. Also, in the presence of a good force field, the calculated structural results resemble the experimentally determined structures more closely as compared to other computational techniques making this method of analysis accurate to a reliable extent. However, the available parameters depicted only satisfies about 20% of known molecules which limits its applicability. Furthermore, as the electrons and specific orbitals are not implemented in this method, it will not be able to evaluate chemical reactions nor predict the reactivity of specific molecules. Thus, a deviation from actual experimental results is expected.

{|align="center"| class="wikitable" border="1"
|+ <u>'''Table 1: Summarising the Individual bond energies calculated via molecular modelling'''</u>
! align="center"|No. !!align="center"| Type of bond stretch!!align="center"|Method of calculation!!align="center"|Bond Representation!!align="center"|Equation for analysis
|-
| align="center"|1 ||align="center"| Diatomic Bond stretches || align="center"|Harmonic Oscillator Approximation ||align="center"|[[Image:Stretch.bmp|centre||]] ||align="center"|[[Image:Stretch eqn.bmp|centre||]]
|-
| align="center"|2 ||align="center"| Triatomic Bond Angles (Bond Angle bending) || align="center"|Harmonic Oscillator Approximation||align="center"|[[Image:Angle bending.bmp|centre||]]||align="center"|[[Image:dihedral eqn.bmp|centre||]]
|-
| align="center"|3 ||align="center"| Tetra-Atomic bond torsions (Dihedral angle rotation) ||align="center"| cell||align="center"|[[Image:Dihedral angle.bmp|centre||]]||align="center"|[[Image:torsion eqn.bmp|centre||]]
|-
| align="center"|4 ||align="center"| Van Der Waals intections ||align="center"| cell||align="center"|[[Image:vdw.bmp|centre|300 px|]]||align="center"|[[Image:Vdw eqn.bmp|centre|300 px|]]
|-
| align="center"|5 ||align="center"| Electrostatic Interactions ||align="center"| cell||align="center"|[[Image:electrostatic.bmp|centre||]]||align="center"|[[Image:electro eqn.bmp|centre||]]
|-
| align="center"|6 ||align="center"| Hydrogen bonding ||align="center"| cell||align="center"|[[Image:H bonding.bmp|centre|width50|]]||align="center"|[[Image:h bonding eqn.bmp|centre||]]
|}

The individual atoms are treated as spheres but the overall mass is dependent on each atom's elemental property. The chemical bonds within the molecule are treated as springs which allows the implementation of Hookes Law. The bond strength ''(stiffness of the spring)'' is dependant on the type of atoms bonded and type of bond formed (single, double or triple). The individual energies when summated give the total potential energy of the molecule(''Table 1''). However as seen from table 1, modelling equations are based on bonds found in simple diatomics. This limits the modelling method to simple molecules (usually hydrocarbons) where 'complex' interactions such as Metal-halide interactions are not accounted.

There are different force fields that can be considered for the (MM) calculations such as MM2, MM3 and OPLS-AA.<ref>Toward a Better Understanding of Covalent Bonds: The Molecular Mechanics Calculation of C-H Bond Lengths and Stretching Frequencies{{DOI|10.1021/ja00092a045}}</ref> MM2 and MM3 force fields are designed for the use with organic type molecules. The MM3 force field method was generated to deal with problems surfaced in the MM2 method. Both parameter sets were developed by the Allinger group at the university of Georgia for the modelling of small organic molecules. <ref>http://europa.chem.uga.edu/allinger/mm2mm3.html</ref>

= The Formation and Hydrogenation of the Cyclopentadiene Dimer =

Cyclopentadiene has two unsaturated C=C bonds allowing it to readily undergo a [4+2] diels-alder reaction to form the cyclopentadiene dimer. From literature experimental results, it was determined that only the endo-dimer is formed exclusivly.<ref>Anil Kumar, Sanjay S. Pawar, AlCl3-catalysed dimerization of 1,3-cyclopentadiene in the chloroaluminate room temperature ionic liquid, Journal of Molecular Catalysis A: Chemical, Volume 208, Issues 1-2, 2 February 2004, Pages 33-37, ISSN 1381-1169, DOI: 10.1016/S1381-1169(03)00510-7. (http://www.sciencedirect.com/science/article/B649H1N79-3/2/b44502d1ac39a604eb092cb734e37bd1) </ref>

However, in cycloaddition reactions carried out there is the possibility for both the endo and exo products to be formed. Thus, the reasons for the formation of only the endo product will be analysed and rationalised. Secondly, the hydrogenation of the dimer formed will also be analysed since the dimer has two different alkene bonds present. Thus, there is a possibility for regioselectivity of one C=C bond over the other will be rationalised.

== Dimerisation of Cyclopentadiene via [4+2] Diels-Alder cycloadditions ==

Cyclopentadiene readily undergoes [4+2] cyclo-addition (diels-alder) reaction to form two stereoisomers (exo dimer or the endo dimer).''Diagram 1'' shows the mechanism of the expected [4+2] cycloaddition and the two dimers that could result.

[[Image:diagram 1.bmp|650px|thumb|Dig. 1 Reaction pathway for [4+2] Dimerisation of Cyclopentadiene|centre]]

{| border="1" align="center"
|+ <u>'''Table 2: Comparing Endo and Exo Cyclopentadiene Dimers'''</u>
!Endo-Dimer !! Exo-Dimer
|-
| [[Image:Endo product2 pl1208.jpg]] || [[Image:Exo product2 pl1208.jpg]]
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_product_pl1208.mol</uploadedFileContents>
<text>Load Endo-Dimer Jmol </text>
<size>600</size>
<script>spin on</script>
</jmolAppletButton>
</jmol> || <jmol>
<jmolAppletButton>
<uploadedFileContents>Exo_product_pl1208.mol</uploadedFileContents>
<text>Load Exo-Dimer Jmol </text>
<size>600</size>
<script>spin on</script>
</jmolAppletButton>
</jmol>
|}

Using Cambridgesoft software 'ChemBio3D', the two possible dimers were optimised using the MM2 force field parameters. This geometrical optimisation (lowest energy configuration) resulted in the exo-dimer to have a lower energy of 31.8777 kcal/mol as compared to the endo-dimer which has an energy of 34.0020 kcal/mol. The product with the lower energy is defined as the thermodynamic product(exo-dimer). Thus, when the dimerisation is left standing under equilibrium conditions which has thermodynamic control, it ideally should have the reaction pathway ending at the more stable product (exo-product) instead of the endo which is energetically less favoured.

However, according to experimental literature, the endo product was formed exclusively instead of the energetically lowere exo-dimer. Although cycloaddition reactions are expected to be reversible, the favourable formation of the dimer makes the backwards reaction almost negligble. This makes the reaction seem irreversible. Being an irreversible reaction, the cycloaddition takes place under kinetic control instead of thermodynamic control. Under kinetic control, the reaction pathway that is most readily attained is favoured making the relative stability of the final products irrelevant. Instead, it is the stability of the transition state that functions as the 'pre-determining factor'. The energetically more stabilised transition state gives a smaller activation energy barrier which is more readily overcomed. The faster the attainment of the transition state the more readily the product is achieved.

This explains why the higher energied endo product is formed instead of the exo product. The cyclo-addition occurring under kinetic control yields the endo-product since it has a transition state of lower energy.<ref>people.bu.edu/joneslab/ch203/endo_rule.pdf</ref>. This is represented in the energy diagram below (endo=blue exo=black).

[[Image:Reaction pathway pl1208.bmp|thumb|500px|Dig. 3 Reaction pathway for Endo Vs. Exo |centre]]

{| border="1" align="center"
|+ <u>'''Table 3: Comparing the specific bond energies of both the Endo and Exo Dimers'''</u>
|-
!Type of Dimer !! Exo !! Endo
|-
| Images || [[Image:Exo product pl1208.bmp]] || [[Image:Endo product pl1208.bmp]]
|-
| Bend (Kcal/mol) || 20.5681 || 20.8399
|-
| Angle(Kcal/mol) || '''-178.6''' || '''-45.2304'''
|-
| Torsion(Kcal/mol) || 7.6559 || 9.5143
|}

All values provided in table 3 are in Kcal/mol; the degree of accuracy provided is equal to that from the MM2 output. Although the accuracy of the MM2 calculations is expected to deviate from experimental values, the inaccuracies have been accounted for in the total energy of each dimer by keeping the value to a single decimal place. The dihedral angle across the new C-C bond formed via cycloaddition was analysed and the results for the various energies are represented in the table 3. The components representing the strain show that the endo-dimer has a smaller dihedral angle of -45.23 Kcal/mol as compared to the exo-dimer which has a value four times larger (-178.6 Kcal/mol). This is indicative that the double bonds are much closer in space for the endo dimer as compared to the exo-dimer. This results in the pi-electron systems to be spatially closer to each other resulting in more significant repulsions. The endo-dimer having greater repulsions should be destabilised to a greater extent and be less favourably formed as compared to the exo-dimer. This is indicative that the MM2 calculations are made for the expected product under thermodynamic control and not kinetic control. Thus, it can be concluded that the MM2 force field method predicts the major product if it was under thermodynamic control instead of the actual kinetic control reaction pathway observed for the cyclopentadiene dimerisation.

The endo-dimer is the kinetic product under kinetic control as it has a more stable, lower energy transition state. Thus, stereo-electronically, the endo-dimer transition state has a greater amount of HOMO-LUMO interactions as compared to the exo-dimer. This enables a greater extent of stabilisation for the endo forming transition state. The selectivity for the higher energied-endo dimer can also be explained by the Woodward-Hoffman rules <ref>www-theor.ch.cam.ac.uk/people/nch/lectures/.../node18.html</ref> <ref>Stereochemistry of Electrocyclic Reactions:{{DOI|10.1021/ja01080a054</ref> or the frontier orbital theory (diagram below).

[[Image:Orbital representation pl1208 2.bmp|thumb|500px| Dig. 4 Frontier Orbitals for Exo and Endo Dimers|centre]]

In the endo-dimer, there is a greater extent of orbital overlap between the diene and dienophile (Pi-->Pi* electron donation). This brings about a greater extent of stabilisation since the diene approaches the dienophile from the 'top' allowing greater in-phase orbital interaction. The black dotted lines represent the two C-C sigma bonds formed due to cycloaddition while the blue dotted lines represent the secondary stabilising orbital overlaps that further stabilise the transition state. However, for exo-dimer formation, the diene approaches the dienophile from a 'side along side' basis. This results in the overlapping of only the extreme end C=C orbitals. Thus, it results in the formation of the 2 C-C sigma bond without the presence of any additional orbital interactions.

The endo dimer having greater HOMO-LUMO interactions, makes it a more stable transition state and thus the kinetically driven product of the cycloaddition. The importance of the secondary orbital interactions is seen when cyclopentadiene is replaced with cyclopentene which only has 1 C=C bond. This means there will not be any pi bonds for secondary orbital interactions. This resulted in a reaction to give only 29% of the endo product even with longer reaction time <ref>M Fox, R Dardona, N Kiweit J. Org Chem 1987, 1469</ref>. This is because there are no additional interactions to make the endo-transition state more stable than the exo transition state. In conclusion, because of the secondary stabilising effect, the endo dimer has a lower energy transition state making it more readily achieved and thus the exclusive product of the cycloaddition reaction under kinetic control.

The limitation of the MM2 method is seen since the relative energies of the specific transition state's are not considered where only the final energies are considered in determining which isomer dominates. This makes the (MM) method of computational analysis more accurate only under thermodynamic conditions and not kinetic conditions. Thus limiting the types of reactions that can be considered for this ,mode of calculation.

== Hydrogenation of the Endo-Dimer ==

From Section 3.2 it has been conclusively rationalised that the dimerisation of cyclopentadiene occurs under kinetic control to give the endo-product exclusively. Thus, hydrogenation with one equivalence of H<sub>2</sub> gas will be considered only for the endo product. As the endo product has two unsaturated C=C bonds, hydrogenation is initially expected to be able to occur at either bonds as represented in the reaction scheme (Diagram 5). Based on experimental evidence, hydrogenation is regioselective and dependant if the reaction is thermodynamically or kinetically driven.

[[Image:hydrogenation pl1208.bmp|thumb|500px| Dig. 5 Reaction scheme for both possible hydrogenation|centre]]

The two hydrogenated products molecular 1 (left) and molecule 2 (right) are represented below. The formation of molecule 1 involves the hydrogenation of the cyclohexene double bond whereas the formation of molecule 2 involves the hydrogenation of cyclopentene double bond. Analysis of the molecular structures show that the mechanism and orbitals involved in the hydrogenation at both regions are very similar (Table 4). They are in very similar environments making the preference for regioselectivity to not be steric-dependent but instead be energetically dependent.

{| border="1" align="center"
|+ <u>'''Table 4: Comparing the molecular structures of both Hydrogenated Products'''</u>
! Molecule 1 !! Molecule 2
|-
| [[Image:hydrogenation 1 pl1208.bmp]] || [[Image:Hydrogenation 2 pl1208.bmp]]
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Hydorgenation_1_pl1208.mol</uploadedFileContents>
<text>Load Molecule 1 Jmol </text>
<size>600</size>
<script>spin on</script>
</jmolAppletButton>
</jmol> || <jmol>
<jmolAppletButton>
<uploadedFileContents>Hydorgenation_2_pl1208.mol</uploadedFileContents>
<text>Load Molecule 2 Jmol </text>
<size>600</size>
<script>spin on</script>
</jmolAppletButton>
</jmol>
|}

To determine the lowest energy, the hydrogenated products were subjected to (MM) via application of the MM2 forcefield parameters. This gave rise to the minimised total energy as a summation of the individual bond energies. Table 5 summarises the results attained for both molecules.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 5: Comparing the Specific Energies for both Hydrogenated Products'''</u>
! Properties !! Hydrogenated product 1 !! Hydrogenated product 2
|-
| Diagram || [[Image:hydrogenation 1 pl1208.bmp]] || [[Image:Hydrogenation 2 pl1208.bmp]]
|-
| Stretch (Kcal/mol) || align="center"| 1.1006 || align="center"| 1.2659
|-
| Bend (Kcal/mol) || align="center"| '''14.5433''' || align="center"| '''19.8063'''
|-
| Torsion (Kcal/mol) || align="center"| 12.5088 || align="center"| 10.8698
|-
| Non-1,4 Van Der Waal(Kcal/mol) || align="center"| -1.0765 || align="center"| -1.2207
|-
| Van Der Waals (VDW) (Kcal/mol) || align="center"| '''4.4978''' || align="center"| '''5.6394'''
|-
| Dipole-Dipole (Kcal/mol) || align="center"| 0.1408 || align="center"| 0.1621
|-
| Angle (C=C) (deg) || align="center"| '''116.4''' || align="center"| '''107.2'''
|-
| Total Energy (kcal/mol) || align="center"| 31.1701 || align="center"| 35.6953
|}

Comparing the overall energy of both molecules, molecule 1 is lower in energy (31.17 Kcal/mol) than molecule 2(35.70 Kcal/mol) by 4.53 Kcal/mol. Thus, the hydrogenation of the cyclohexene alkene bond gives rise to the more stable product as compared to hydrogenation of the cyclopentene alkene bond. This makes molecule 1 to be more thermodynamically favoured over molecule 2. The total energy of the endo-dimer was 34.002 Kcal/mol but the hydrogenated product is of a lower energy. When this is compared with the energy levels of both hydrogenated products it is noticed that the formation of molecule 1 is an exothermic reaction. This results in the lower energied, thermodynamically more favoured product to be formed. The formation of molecule 2 is an endothermic reaction where hydrogenation results in an increase of the total energy making it thermodynamically less favoured.

Analysis of the individual contributing bond energies in table 5 shows that the overall difference in energy is mainly due to the difference in the angle (bending) energy and Van der Waals forces of interaction. Comparing the angles across the C=C double bonds shows that molecule 1 has an angle of 116 deg while molecule 2 has a bond angle of 107 deg. The ideal angle for the SP<sup>2</sup> carbon bonds is 120 deg. The greater the deviation from the ideal angle the greater the angle strain. This explains why molecule 2 having a more strained bond angle has a higher bending energy of 19.8063 Kcal/mol. This strain can also be justified theoretically by considering the nature of the double bond which induces restricted rotation due to the pi bond. This makes the ring overall more rigid, the presence of the C=C bond in different regions influences the overall molecule's rigidity to different extents. For the endo-dimer, the 6-membered ring and the bridge in that ring influence the overall energy to a greater extent than the 5 membered ring. The presence of the C=C bond in the 6 membered ring induces greater rigidity as compared to the C=C bond being present in the 5 membered ring. This explains why molecule 2 having the C=C bond in the 6-membered ring has a higher bending energy than molecule 1 making it less stable.

In conclusion, due to the greater strain imposed and being of a higher energy than the starting material, molecule 2 is not the thermodynamically favoured product. Molecule 1 on the other hand is the product formed under thermodynamic control. As insufficient data is attained about the nature of the transition state for the reaction, the relative energies of the transition state cant be determined. Thus, it can't be concluded which would be the kinetically driven product via hydrogenation of the endo-dimer. When compared with the experimentally carried out hydrogenation, it is usually carried out with a metal catalyst (Ni or Pd complexes) that facilitates the release of ring strain. Thus, the computer modelling exercise shows that the bond strain is the key determinant for determining the thermodynamic product. This also explains why a metal catalyst is used in wet experiments.

== Futher Considerations ==

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 6: Tetrohydrogenated Product'''</u>
! [[Image:Tetrahydro pdt pl1208.bmp|centre|150px|]] !! [[Image:Tetrahydro pl1208.jpg|centre|150px|]]
|-
| |<jmol>
<jmolAppletButton>
<uploadedFileContents>Tetrahydro_pl1208.mol</uploadedFileContents>
<text>Load Tetrahydro Jmol </text>
<size>600</size>
<script>spin on</script>
</jmolAppletButton>
</jmol> ||
|}

It has been found in literature that the doubly reduced dialkenes are not hydrogenated as readily but it would happen over a longer reaction duration.<ref>SELECTIVE HYDROGENATION OF CYCLOPENTADIENE
IN THE LIQUID PHASE ON PALLADIUM CATALYSTS,React. Kinet. Catal. Lett., Vol. 19, No. 1-2, 223-226 (1982)</ref>. The overall energy of the tetrahydrogenated molecule is 32.3702 Kcal/mol. This was about 1.2 Kcal/mol higher than the starting material. Upon the first hydrogenation of molecule 1. The torsion energy of this molecule is 15.8120Kcal/mol which is significantly higher as compared to the endo-dimer's torsion. This explaining why the reaction took a much longer time to form the tetrahydrogenated product. The second hydrogenation was endothermically driven to form the higher torsion product which does not readily occur as the tetrahydrogenate product is less stable.

=Stereochemistry of Nucelophilic addition to pyridinium ring (NAD<sup>+</sup> analogue)=

This section investigates the face from which a nucleophile approaches and attacks the electron deficient centre in the pyridinium ring. Two examples will be considered where each will be using a different pyridinium compound. Conclusions about the direction from which a nucleophile attacks are drawn via the analysis and rationalisation of the two examples.

== Example 1 : Investigating the optically active derivative prolinol ('''5''') via a nucleophilic attack from a Grignard reagent==

[[Image:Nad_Eg1_pl1208_3.bmp|thumb|500px|Dig. 7 Reaction of N-Methyl Pyridoxazepinone with Methyl Magnesium Iodide|centre]]

The optically active prolinol derivative ('''5''') reacts with the grignard reagent methyl magnesium iodide to alkylate the pyridine ring at the para position. According to literature, this reaction results in the absolute stereochemistry('''6'''). The regioselectivitiy and stereoselectivity of the nucleophilic attack is rationalised to be due to chelation. The incoming grignard reagent has the electropositive Magnesium mental centre readily coordinating to the electronegative oxygen of the carbonyl.<ref>A. G. Shultz, L. Flood and J. P. Springer, J. Org. Chemistry, 1986, 51, 838.{{DOI|10.1021/jo00356a016}}</ref> It is the chelative effect that influences the face via which the methyl group attacks the pyridine ring. This addition of the methyl group occurs via a nucleophilic addition reaction (Diagram 7). Thus, the relative stereochemistry of the carbonyl group has to be taken into account towards determining the direction from which the nucleophile attacks.

[[Image:Nad_pl1208.bmp|thumb|500px|Dig. 8 Both possible nucleophilic attacks brought about by the chelation|centre]]

This brings about two possible stereoisomers prior to the nucleophilic attack. (i) The carbonyl group is above the 7-membered ring or (ii) The carbonyl group is below the 7-membered ring (Diagram 8). The two stereoisomers are of the same molecule but of different bond orientations, this makes them conformers of each other. Both conformers were analysed, where their minimum energy was computed via the use of the MM2 force field parameters. However, the MeMgI Grignard reagent was not added into the energy minimisation since ChemBio3D was unable to recognise the Mg as an metal atom. This resulted in an software error when the Grignard reagent was included for the calculations. This is due to the limitations of the MM2 model since it is based on bonds found in simple diatomic molecules.Thus the MM2 calculations is not able to model the metal-type bonding interactions. Thus, as mentioned in the introduction, the MM2 model is limited to simple molecules and would break down if it is attempted with co-ordination type or non-classical interactions.

Thus, energy of the starting material ('''5''') was minimised to determine both possible conformers. The MM2 method of minimising energy is based on the logic that it could get 'stuck' in different potential energy wells if the starting points are different. This results in giving the pseudo-minimisation which is not the lowest energy level of the reactant making it inaccurate. Thus, a different number of starting points were investigated by varying the dihedral angle between the carbonyl group and the aromatic ring (Graph 1). From the graph, the total minimum energy of the conformer is attained when the carbonyl group is 12° above the plane of the aromatic ring.

[[Image:Graph_1_pl1208.JPG|thumb|1000px|Graph 1 : Varying the dihedral angle against total energy of the molecule|centre ]]

{| border="1" align="center"
|+ <u>Table 6 : Comparing the specific energies for the 3 distinct conformers</u>
| || <CENTER>'''Conformer 1''' </CENTER>|| <CENTER>'''Conformer 2'''</CENTER>|| <CENTER>'''Conformer 3'''</CENTER>
|-
| <CENTER>Carbonyl relative position</CENTER> ||<CENTER> Above plane</CENTER> || <CENTER>Above Plane</CENTER>|| <CENTER>Below Plane</CENTER>
|-
| <CENTER>Image</CENTER> || [[Image: Conformer1_Lowest_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Conformer1_Lowest_pl1208.mol</uploadedFileContents><script>spin on</script><text>Conformer 1</text></jmolAppletButton></jmol> || [[Image: Conformer2_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Conformer2_pl1208.mol</uploadedFileContents><script>spin on</script><text>Conformer 2</text></jmolAppletButton></jmol>|| [[Image: Conformer3_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Conformer3_pl1208.mol</uploadedFileContents><script>spin on</script><text>Conformer 3</text></jmolAppletButton></jmol>
|-
| <CENTER>Stretch(Kcal/mol)</CENTER> || <CENTER>2.0276</CENTER> || <CENTER>17.1799 </CENTER> || <CENTER>8.4974 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER>14.1479</CENTER> || <CENTER> 178.1686</CENTER> || <CENTER>83.8502 </CENTER>
|-
| <CENTER>Stretch-Bend (Kcal/mol)</CENTER>|| <CENTER>0.1333</CENTER> || <CENTER>-4.6402</CENTER> || <CENTER>-2.6517 </CENTER>
|-
| <CENTER>Torsion (Kcal/mol)</CENTER>|| <CENTER>5.2352</CENTER> || <CENTER>62.5344</CENTER> || <CENTER>88.9772 </CENTER>
|-
| <CENTER>Non-1,4 VDW (Kcal/mol)</CENTER>||<CENTER>-0.6069</CENTER> || <CENTER>2.0556</CENTER> || <CENTER>7.4923 </CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER>16.5396</CENTER> || <CENTER>41.6637</CENTER> || <CENTER>35.0133 </CENTER>
|-
| <CENTER>Charge/Dipole(Kcal/mol) </CENTER>|| <CENTER>9.6711</CENTER> || <CENTER>9.2275</CENTER> || <CENTER>-15.2838 </CENTER>
|-
| <CENTER>Dipole/Dipole (Kcal/mol)</CENTER>|| <CENTER>-3.9810</CENTER> || <CENTER>-4.1944</CENTER> || <CENTER>-3.0647 </CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>'''43.1641''' kcal/mol</CENTER> ||<CENTER> '''301.9952''' kcal/mol</CENTER> || <CENTER> '''202.8303''' kcal/mol </CENTER>
|-
| <CENTER>Dihedral angle </CENTER> ||<CENTER> '''12 deg'''</CENTER> || <CENTER>'''125 deg'''</CENTER> || <CENTER>'''-105 deg''' </CENTER>
|}

From table 6, the dihedral angle of a positive value is indicative of the carbonyl bond being above the plane relative to the aromatic ring. In contrast, when the polarity of the dihedral angle is negative it is indicative of carbonyl bond being below the plane relative to the aromatic ring. From the graph 1 and table 7, it is concluded that there are no low energy conformations when the carbonyl bond is below the plane. Thus, making the product to be of an absolute stereochemistry where the methyl group via the grignard reagent always coordinates to the carbonyl group orientated above the plane of the ring. This brings about the chelating effect and has the nucleophilic addition to give product ('''6''').

{| border="1" align="center"
|+ <u>Table 7 : Comparing the specific energy for the 2 possible products</u>
| || <CENTER>'''Conformer 1''' </CENTER>|| <CENTER>'''Conformer 2'''</CENTER>||
|-
| <CENTER>Methyl relative position</CENTER> ||<CENTER> Above plane</CENTER> || <CENTER>Below Plane</CENTER>||
|-
| <CENTER>Image</CENTER> || [[Image: Product_conformer_above_ring_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Product_conformer_above_ring_pl1208.mol</uploadedFileContents><script>spin on</script><text>Above Plane</text></jmolAppletButton></jmol> || [[Image:Product_conformer_below_ring_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Product_conformer_below_ring_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton><script>spin on</script></jmol>
|-
| <CENTER>Stretch(Kcal/mol)</CENTER> || <CENTER>2.2149</CENTER> || <CENTER>2.2524 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER> 16.2579</CENTER> || <CENTER> 16.3708</CENTER>
|-
| <CENTER>Stretch-Bend(Kcal/mol) </CENTER>|| <CENTER>0.3273</CENTER> || <CENTER>0.3265</CENTER>
|-
| <CENTER>Torsion(Kcal/mol) </CENTER>|| <CENTER>5.8234</CENTER> || <CENTER>5.5423</CENTER>
|-
| <CENTER>Non-1,4 VDW(Kcal/mol) </CENTER>||<CENTER> -0.2602</CENTER> || <CENTER> -0.3180</CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER>17.5108</CENTER> || <CENTER>17.5625</CENTER>
|-
| <CENTER>Dipole/Dipole (Kcal/mol)</CENTER>|| <CENTER>-4.3629</CENTER> || <CENTER>-4.3336</CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>37.5112 kcal/mol</CENTER> ||<CENTER> 37.4029 kcal/mol</CENTER>
|-
| <CENTER>Dihedral angle</CENTER> ||<CENTER> -74 deg</CENTER> || <CENTER>70 deg</CENTER>
|}

Table 7 depicts the two possible conformers, although only conformer 1 is formed via the nucleophilic attack. The purpose of calculating the specific energy of the two products is to rule out thermodynamic control as a reason for absolute stereoselectivity. This is because both the possible products above or below the ring when minimised give rise to products of very similar total energies with a difference of only 0.1 Kcal/mol making it an insignificant difference. Thus, the energy difference is too small to reason the absolute stereoselectivity due to thermodynamic control. Instead it narrows the reason to be due to the chelate effect that results in the the methyl group to be 'depositied' on the same face as that of the carbonyl bond relative to the ring.

== Example 2 : Investigating the reaction of N-methyl Quinolium Salt with Aniline==

This example is built on the previous example where example 2 involves aniline acting as the nucleophile instead of a Grignard to react with the pyridinum ring in the N-methyl Quinolinum Salt. The reaction involves the insertion of the -NHPh group onto the para position of the ring via nucleophilic addition (Diagram 8). According to literature, the incoming nucleophile attacks from the 'face' of the plane opposite to that of the C=O carbonyl group.<ref>Leleu, Stephane; Papamicael, Cyril; Marsais, Francis; Dupas, Georges; Levacher, Vincent. Tetrahedron: Asymmetry, 2004, 15, 3919-3928.{{DOI|10.1016/j.tetasy.2004.11.004}}</ref> Another literature source defines the reaction with aniline as the nucleophile to be antropenantioselective; resulting in the complexes formed to be termed as chiral amide transfer agents.<ref>Ichinose, K.; Kodera, M.; Leeper, J.; Battersby, R. J.Chem. Soc., Perkin Trans. 1 1999, 879{DOI|10.1039/a809858a}</ref> This seteroselectivity is expected to be due to a steric control and will be investigated and rationalised via the application of molecular modelling methods.

[[Image:Reaction_pathway_nad_eg_2_pl1208.bmp|thumb|500px|Dig.8 Reaction pathway for Nucleophilic attack by Aniline|centre ]]

{| border="1" align="center"
|+ <u>Table 8 : Comparing the specific energies of both conformations of the reactant</u>
| || <CENTER>'''Conformation 1''' </CENTER>|| <CENTER>'''conformation 2'''</CENTER>||
|-
| <CENTER>Carbonyl relative position</CENTER> ||<CENTER> Above plane</CENTER> || <CENTER>Below Plane</CENTER>||
|-
| <CENTER>Image</CENTER> || [[Image: Reactant_molecule_above_plane_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Reactant_molecule_above_plane_pl1208.mol</uploadedFileContents><script>spin on</script><text>Above Plane</text></jmolAppletButton></jmol> || [[Image:Reactant_molecule_below_plane_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Reactant_molecule_below_plane_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton></jmol>
|-
| <CENTER>Stretch (Kcal/mol)</CENTER> || <CENTER>3.8973</CENTER> || <CENTER>3.9411 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER> 11.5815</CENTER> || <CENTER> 11.5979</CENTER>
|-
| <CENTER>Stretch-Bend(Kcal/mol) </CENTER>|| <CENTER>0.4032</CENTER> || <CENTER>0.4007</CENTER>
|-
| <CENTER>Torsion (Kcal/mol)</CENTER>|| <CENTER>9.9928</CENTER> || <CENTER>9.6530</CENTER>
|-
| <CENTER>Non-1,4 VDW(Kcal/mol) </CENTER>||<CENTER>4.0695</CENTER> || <CENTER> 3.5460</CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER>29.3333</CENTER> || <CENTER>29.4347</CENTER>
|-
| <CENTER>Charge/Dipole(Kcal/mol) </CENTER>|| <CENTER>9.0388</CENTER> || <CENTER>9.0174</CENTER>
|-
| <CENTER>Dipole/Dipole (Kcal/mol)</CENTER>|| <CENTER>-4.8827</CENTER> || <CENTER>-4.8857</CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>63.4337 kcal/mol</CENTER> ||<CENTER>62.7051 kcal/mol</CENTER>
|-
| <CENTER>Dihedral angle</CENTER> ||<CENTER> 21 deg</CENTER> || <CENTER>-20 deg</CENTER>
|}

Table 8 shows the total energy minimisations of both possible conformations. The difference in both formations is that (i) The carbonyl bond above the plane relative to the aromatic ring (ii) The carbonyl bond below the plane relative to the aromatic ring. The total energy of both forms varied only by 0.7 Kcal/mol making them very close in energy. Unlike example 1 where the two forms were conformers of each other, here there is negligible difference in energy between the two forms. This was indicative that in example two, both conformations are equally probable. Thus, both have to be accounted for in the reaction with the approaching nucleophile. Also, as the energies of both conformations are very similar there is no thermodynamic influence for one conformation over the other. This also suggests that the dominance for one conformer over the other lies in the energy of the products formed instead of the reactants.

{| border="1" align="center"
|+ <u>Table 9 : Comparing the specific energy of the 2 conformations of the product</u>
| || <CENTER>'''Conformation 1''' </CENTER>|| <CENTER>'''conformation 2'''</CENTER>||
|-
| <CENTER>Nucleophile attacking position</CENTER> ||<CENTER> Above plane</CENTER> || <CENTER>Below Plane</CENTER>||
|-
| <CENTER>Image</CENTER> || [[Image: Product_molecule_above_plane_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Product_molecule_above_plane_pl1208.mol</uploadedFileContents><script>spin on</script><text>Above Plane</text></jmolAppletButton></jmol> || [[Image:Product_molecule_below_plane_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Product_molecule_below_plane_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton></jmol>
|-
| <CENTER>Stretch(Kcal/mol)</CENTER> || <CENTER>4.2200</CENTER> || <CENTER>6.4949 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER> 18.1947</CENTER> || <CENTER> 25.4098</CENTER>
|-
| <CENTER>Stretch-Bend (Kcal/mol)</CENTER>|| <CENTER>0.8199</CENTER> || <CENTER>1.0082</CENTER>
|-
| <CENTER>Torsion(Kcal/mol) </CENTER>|| <CENTER>4.3644</CENTER> || <CENTER>11.5600</CENTER>
|-
| <CENTER>Non-1,4 VDW(Kcal/mol) </CENTER>||<CENTER>2.0397</CENTER> || <CENTER> 5.8101</CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER>34.1950</CENTER> || <CENTER>35.6711</CENTER>
|-
| <CENTER>Dipole/Dipole(Kcal/mol) </CENTER>|| <CENTER>-6.5875</CENTER> || <CENTER>-6.0762</CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>57.2462 kcal/mol</CENTER> ||<CENTER> 79.8779 kcal/mol</CENTER>
|-
| <CENTER>Dihedral angle</CENTER> ||<CENTER> 66 deg</CENTER> || <CENTER>-72 deg</CENTER>
|}

The nucleophile having the bulky phenyl group adjecent to the nitrogen heteroatom makes it highly 'steric-consious'. This will result in the bulky nucleophile to approach the pyridinum ring via the least hindered pathway. Based on the diagrams in table 9, it is seen that the nucleophile completely avoids the carbonyl group. It makes it favourable for the nucleophile to attack from opposite projection of the carbonyl bond. This is mainly due to the electron-electron repulsion expected from the lone pairs on both heteroatoms nitrogen and oxygen in the nucleophile and carbonyl respectively. When compared with the total energy of both product forms (table 9), it is noticed that the total energy via the nucleophile attacking from the top face is 13 Kcal/mole lesser than conformer via an attack from the bottom face. This makes the conformation via the top face to be energetically more stable and favoured. This rationalises the literature intitially discussed where the formation of 1 stereoisomer dominated over the other. Being the lower energy product it would be thermodynamically more favoured as well.

The positive dihedral angle is indicative that the nucleophile is added from the top face with respect to the aromatic ring and this in turn resulted in the dihedral angle of the carbonyl relative to the plane of the ring to be negative indicating that it is in the opposite plane relative to the nucleophile.

Comparing the magnitude of the total energy with respect to the reactant (62.7 Kcal/mol) showed that the product had a significantly lower energy level. This can be rationalised as being due to the ability of the nitrogen group to bring about further stabilisations via dipole dipole interactions with the electrophilic pyridinum ring. Also. the product formation was thermodynamically more favoured as it also allowed the (+) on the electronegative nitrogen to be removed.

It is concluded that the attack of the nucleophile is under steric control and readily occurs under thermodynamic control. It is not determined if it is the same coformer formed under the kinetic control. Thus, additional analysis and experiments should be carried out on the transition state to determine its energy level. It is key in determining if the product formed is also kinetically inclined.

Lastly, as the experiment is to be carried out in wet conditions, the influence of solvent should also be taken into consideration for a more accurate understanding on which product is more stable. This is important for this compound since the presence of electronegative groups would allow interactions with protic solvents (H-bonding) that could greatly stabilise one product isomer over the other. Thus, the influence of solvents should also be looked into for future considerations.

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

The total synthesis of Taxol (an important drug in the treatment of ovarian cancers) involves the formation of a key intermediate which is readily isolated due to its overall ring stability. According to literature, the intermediate exhibits atropisomerism which results in formation of 2 different stereoisomers with respect to the orientation of the carbonyl bond (Diagram 9).<ref>S. W. Elmore and L. Paquette, Tetrahedron Letters, 1991, 319; DOI:10.1016/S0040-4039(00)92617-0 10.1016/S0040-4039(00)92617-0 10.1016/S0040-4039(00)92617-0</ref> Being atropisomers, they are not readily interconverted in situ; the correct experimental conditions have to be imposed to ensure the correct intermediate is formed so as to allow the successive stereochemistry-orientated reactions to occur and produce the target molecule of the correct stereochemistry. Thus, computational modelling efforts will be adopted to determine the specific conditions that allows one isomer to be the predominant product. This will allow the ideal wet conditions to be established towards effective target synthesis of the taxol molecule. Also, the effect known as 'Hyperstable Alkenes' will be investigated with reference made to the intermediate of Taxol.

==Key Intermediates towards the Synthesis of Taxol==

[[Image:Intermediates_pl1208.bmp|thumb|500px|Dig.9 Atropisomers of the Key intermediate towards Taxol synthesis|centre ]]

According to literature,(A) and (B) isomers of the key intermediate would have isomer(A) interconverting to (B) when left standing under equilibrating conditions.<ref>

K. C. Nicolaou, J.-J. Liu, Z. Yang, H. Ueno, E. J. Sorensen, C. F. Claiborne, R. K. Guy, C.-K. Hwang, M. Nakada, P. G. Nantermet

J. Am. Chem. Soc., 1995, 117 (2), pp 634–644

</ref> This implies that (B) is thermodynamically more stable than (A) making it the favoured conformation under thermodynamic control. This is verified by means of molecular modelling where the MM2 forcefield parameters are implemented to determine the minimum energy of the 2 isomers (Table 10). Similar to earlier sections the minimisations were carried out from different 'starting' points to avoid the trapping in a potential energy wells.

{| border="1" align="center"
|+ <u>Table 10 : Comparing the specific energies of the 2 atropisomers of the Intermediate</u>
| || <CENTER>'''Isomer A''' </CENTER>|| <CENTER>'''Isomer B-1'''</CENTER>|| <CENTER>'''Isomer B-2'''</CENTER>
|-
| <CENTER>Carbonyl relative position to adject H atom</CENTER> ||<CENTER> Same Side</CENTER> || <CENTER>opposite side</CENTER>|| <CENTER>opposite side</CENTER>
|-
| <CENTER>Image</CENTER> || [[Image: MoleculeA_Aboveplane_pl1208_2.jpg]]<jmol><jmolAppletButton><uploadedFileContents>MoleculeA_Aboveplane_pl1208_2.mol</uploadedFileContents><script>spin on</script><text>Above Plane</text></jmolAppletButton></jmol> || [[Image:MoleculeB_BELOWplane_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>MoleculeB_BELOWplane_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton></jmol>|| [[Image:MoleculeB_BELOWplane2_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>MoleculeB_BELOWplane2_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton></jmol>
|-
| <CENTER>Stretch(Kcal/mol)</CENTER> || <CENTER> 3.1417</CENTER> || <CENTER>2.5752 </CENTER> || <CENTER>2.4811 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER> 20.4401</CENTER> || <CENTER> 10.7213</CENTER> || <CENTER>10.8893 </CENTER>
|-
| <CENTER>Stretch-Bend(Kcal/mol) </CENTER>|| <CENTER>0.5049</CENTER> || <CENTER>0.3195</CENTER> || <CENTER>0.2942 </CENTER>
|-
| <CENTER>Torsion(Kcal/mol) </CENTER>|| <CENTER>21.9620</CENTER> || <CENTER>19.6097</CENTER> || <CENTER>17.3102 </CENTER>
|-
| <CENTER>Non-1,4 VDW (Kcal/mol)</CENTER>||<CENTER> -0.6339</CENTER> || <CENTER> -1.31460</CENTER>|| <CENTER>-1.7058 </CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER> 14.3452</CENTER> || <CENTER>12.5536</CENTER>|| <CENTER>12.4299 </CENTER>
|-
| <CENTER>Dipole/Dipole (Kcal/mol)</CENTER>|| <CENTER>0.0091</CENTER> || <CENTER> -0.1838</CENTER> || <CENTER>0.1427 </CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>59.7690 kcal/mol</CENTER> ||<CENTER>44.2809 kcal/mol</CENTER>|| <CENTER>41.8416 kcal/mol </CENTER>
|}

Table 10 shows the lowest energy configuration for both isomers. To facilitate better understanding, the 6 membered ring is coloured black and the hydrogen adjecent to the carbonyl is in yellow. Comparing the total energy of (A) against (B) shows that (B) which has the carbonyl in the opposite plane to that of the hydrogen (yellow) is significantly more stable than (A) quantitatively by 18 Kcal/mol. Thus, (B) has an carbonyl orientation that is thermodynamically more stable as compared to that of (A) making (B) the dominating isomer under thermodynamic control. This is in agreement with earlier mentioned literature which states that (A) isomerises to (B) when left standing.

Analysis of Isomer B showed that there were two possible chair conformations adopted by the cyclohexane ring. Thus, the chair conformations were swapped to give both forms (Table 10) which showed that one conformatino more was slightly more stable (3 Kcal/mol) than the other. This could be attributed to the sterics involved where one form of the chair orientation resulted in a lesser steric influence as compared to the other.

(B) was overall a much lower energy than (A) which was rationalised to be due to the steric influence imposed. The carbonyl group having an orientation to be on the same plane as that of the adjecent hydrogen (yellow) will result in a greater degree of steric clash with the hydrogen atoms on the same side of the plane and the bridge head methyl group. Thus, this steric clash is avoided when the carbonyl group is in the opposite plane of the bridge head such as that in (B). Drawing parallels with the bend energy, (A) has a significantly much higher bend of 20 Kcal/mol as compared to (B) of 10 Kcal/mol. This could be due to the relative position of the carbonyl bond. In (A), the bond is above the plane and experiences the strain of the cyclic system and the steric clashes earlier discussed. This results in the sp<sup>2</sup> angle to be 109.4 deg but in (B) it would have the cyclic strain but not the steric clashes which results in an angle of 118.2 deg. Thus, (B) has a angle closer to the ideal angle of 120 deg. This results in lesser strain which is depicted by a smaller bend energy giving an overall lower total energy.

The difference in the total energy was 17.9274 Kcal/mol when carried out under the MM2 force field parameters and when it was carried out under the MMFF94 parameters the individual values were significantly different but the difference in total energy was 18.9458 Kcal/mol which is very close to the MM2 calculated difference. Thus both showed (B) to be the thermodynamically more stable isomer.

In conclusion, (B) is the thermodynamically more favourable product and when the reaction is done under reversible equilibrating conditions, the formation of (B) over (A) as the key intermediate can be expected. However, insufficient analysis is done on the transition state energy levels to determine which predominates under kinetic control as well.

==Extraordinary Reactivity of Alkenes==

The C=C alkene bond having a bond order of two is expected to be an electron dense reaction site. Being electron dense, it is nucleophilic making it ideally susceptible to an electrophilic attack or an addition reaction across the double bond as the pi-bonds are more reactive than the sigma bonds due to weaker orbital overlaps. However, in the environment such as that of the Taxol intermediate, C=C bond is experimentally observed to have slower reaction rates.<ref>Concerning attempts to synthesize out-bicyclo[4.4.4]tetradec-1-ene derivatives

David P. G. Hamon, Guy Y. Krippner

J. Org. Chem., 1992, 57 (26), pp 7109–711

DOI: 10.1021/jo00052a024

</ref> Reacting slower would mean the C=C is considerably less reactive which makes it an 'Hyperstable Alkene'. A hyperstable alkene is one that experiences a lesser strain as compared to the parent hydrocarbon due to the adjecent bridgehead. <ref>W.F. Maier, P.v.R. Schleyer, ''J. Am. Chem. Soc.'', '''1981''', ''103'', pp 1891-1900</ref> Qualitatively, this is observed by calculating the Olefin Strain Energy (OSE). The OSE is calculated via comparing the torsion energy of the alkene with the torsion energy of the hydrogenated equivalent (Parent). When the torsion is lesser than that of the parent, it is hyperstable (Table 11).

{| border="1" align="center"
|+ <u>Table 11 : Comparing the torsion energy of the hyperstable alkene against the parent hydrocarbon</u>
| || <CENTER>'''Intermediate (Alkene)''' </CENTER>|| <CENTER>'''Parent Hydrocarbon'''</CENTER>||
|-
| <CENTER>Image</CENTER> || [[Image: MoleculeB_BELOWplane2_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>MoleculeB_BELOWplane2_pl1208.mol</uploadedFileContents><script>spin on</script><text>Above Plane</text></jmolAppletButton></jmol> || [[Image:Parent_molecule_BELOWplane2_pl1208.jpg]]<jmol><jmolAppletButton><uploadedFileContents>Parent_molecule_BELOWplane2_pl1208.mol</uploadedFileContents><script>spin on</script><text>Below Plane</text></jmolAppletButton></jmol>
|-
| <CENTER>Stretch (Kcal/mol)</CENTER> || <CENTER>2.4700</CENTER> || <CENTER>4.9197 </CENTER>
|-
| <CENTER>Bend(Kcal/mol)</CENTER> || <CENTER> 10.8592</CENTER> || <CENTER> 23.9431</CENTER>
|-
| <CENTER>Stretch-Bend(Kcal/mol) </CENTER>|| <CENTER>0.2925</CENTER> || <CENTER>1.2070</CENTER>
|-
| <CENTER>Torsion (Kcal/mol)</CENTER>|| <CENTER>'''17.3565'''</CENTER> || <CENTER>'''23.4667'''</CENTER>
|-
| <CENTER>Non-1,4 VDW(Kcal/mol) </CENTER>||<CENTER>-1.6846</CENTER> || <CENTER> 2.1497</CENTER>
|-
| <CENTER>1,4 VDW(Kcal/mol)</CENTER> || <CENTER>12.4177</CENTER> || <CENTER>18.9016</CENTER>
|-
| <CENTER>Dipole/Dipole (Kcal/mol)</CENTER>|| <CENTER> 0.1418</CENTER> || <CENTER>0.0000</CENTER>
|-
| <CENTER>Total Energy</CENTER> || <CENTER>41.8532 kcal/mol</CENTER> ||<CENTER>74.5877 kcal/mol</CENTER>
|}

The hyperstable alkene present in the intermediate has a lower torsion energy of 17.4 Kcal/mol than the parent hydrogenated molecule of 23.5 Kcal/mol. The olefin strain energy (OSE) is calculated as (17.3565-23.4667)= -6.1102 Kcal/mol which is lesser than zero. The intermediate is indeed of a lesser strain than the parent hydrocarbon showing that the C=C bond is less reactive due to the adjecent bridgehead in the intermediate.

= Modelling Using Semi-empirical Molecular Orbital Theory =

== Introduction to Semi-empirical methods with comparisons draw against Molecular modelling ==

In the previous sections, calculations were carried out via molecular modelling which was purely based on classical mechanics. This classical method of modelling was observed to significantly deviate from experimental result as it did not take into considerations the electron interactions. Thus resulting in inaccurate results when electron density is able to play a major role in determining the progress and outcome of the chemical reaction. This failure from the molecular modelling was clearly seen in '''<u>section 3 : Dimerisation of cyclopentadiene</u>''' where the MM2 calculations of total minimised energy predicted the exo-product to be more stable; product of the reaction. But upon consideration of electron density and the frontier orbitals, it was rationalised that due to secondary orbital interactions the less stable endo product that was instead attained exclusively.

This brings about the importance of Semi-Empirical methods. Being semi-classical, it is able to take into account electronic behaviour by adding the valence shells s, p<sub>x</sub> and p<sub>y</sub> orbitals to the already present p<sub>z</sub> orbitals from Huckel theory. The semi-empirical method that will be implemented for this section is the Hartree Fock (HF) approximation. The approximation carried out will implement a single basis function for each orbital present. This is done via the minimal basis sets known as Slater Type Orbitals(STO-3G).<ref>Computational Chemistry, David Young, Wiley-Interscience, 2001. pg 86.</ref> They are defined as the basis set for the 1s, 2s and 2p orbitals via the linear combination of 3 primitive gaussian functions.<ref> Chemical Modeling From Atoms to Liquids, Alan Hinchliffe, John Wiley & Sons, Ltd., 1999. pg 294</ref> However, the minimal basis set (STO) gives rather rough results that are not accurate enough for research-quality publications; more advanced Pople-basis sets will be considered instead. Implementation of the 6-31G basis set will be considered to provide more accurate results.<ref>Leach, Andrew R. (1996). Molecular Modelling: Principles and Applications. Singapore: Longman. pp. 68–77. ISBN 0-582-23933-8.</ref>

The main limitations of the HF approximations is that the energy of the system calculated will always be higher than the actual energy of the real system due to the implementation of the variational theory and the independent electron approximation. Thus, the HF limit will be the smallest limiting value of the system which is almost always slightly higher than that of the actual molecule's lowest energy.

There are four main approximations made via this method of calculation that accounts form its deviation from the expected results. <ref>Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems; David C. Young; Copyright ( 2001 John Wiley & Sons, Inc. ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic))</ref>

* The Born-Oppenheimer Approximation

* Central Field Approximation

* Omitting Core electrons resulting in the incomplete basis set

* Absence of relativistic effects

However, despite the assumptions made which results in some deviation from the actual results, the deviations are minimal as compared to the inaccuracy attained in results via molecular modelling calculations.

== Regioselective Addition of Dichlorobenzene ==

[[Image:Reaction scheme dicl pl1208.bmp|thumb|500px|Dig.10 Reaction scheme for Dichlorocarbene with napthalene|centre]]

9-Chloro-1,4,5,8-tetrahydro-4a,8a-methanonaphthalene undergoes a regioselective addition with dichlorocarbene (diagram 10). The starting reagent is an asymmetric molecule with the presence of two alkene bonds. This induces the potential for regioselectivity for one C=C bond over the other. The calculations carried out facilitate the rationalisations for why one C=C bond is electrophilically more favoured over the other. Based on the mechanism, electrophilic addition could either be added endo or exo to the plane. However, experimentally only the endo isomer was isolated bringing about absolute regioselectivity to the reaction.<ref>B. Halton, R. Boese and H. S. Rzepa., J. Chem. Soc., Perkin Trans 2, 1992, 447.{{DOI|10.1039/P29920000447}</ref>

=== Classical Calculation : MM2 Optimisation ===

{| border="1" align="center"
<u>'''|+ Table 12: MM2 Optimised energies'''</u>

| || ''' (9-Chloro-1,4,5,8-tetrahydro-4a,8a-methanonaphthalene)'''
|-
| Image || [[Image:MM2 calculation pl1208.jpg|350px|centre]]
<jmol>
<jmolAppletButton>
<uploadedFileContents>MM2_calculation_pl1208.mol</uploadedFileContents>
<text> MM2 optimised Methanonaphthalene </text>
<size>500</size>
<script>spin on</script>
</jmolAppletButton>
</jmol>
|-
| Stretch kcal/mol || 0.6218
|-
| Bend kcal/mol || 4.8116
|-
| Stretch-Bend kcal/mol || 0.0410
|-
| Torsion kcal/mol || 7.6246
|-
| Non-1,4 VDW kcal/mol || -1.0910
|-
| 1,4 VDW kcal/mol || 5.7852
|-
| Dipole/Dipole kcal/mol || 0.1116
|-
| Total Energy || 17.9048 kcal/mol
|}

Analysis of the JMOL (Table 12) shows that the exo-double bond has a shorter distance of 2.541 amstrongs towards the bridging carbon while the Endo-double bond distance is 2.564 amstrongs. This shows that an incoming electrophile should favourably attack the Endo-double bond instead of the Exo-double bond. Being further from the bridging carbon group, it imposes a lesser amount of sterics. This is the shortfall of the MM2 model where the interacting orbitals and electron density cannot be accounted for where only the relative sterics can be considered.

=== Semi-Classical Calculation : MOPAC/RM1 & mMOPAC/PM6 Semi-Empirical Optimisation ===

When RM1 and PM6 energy optimisations were carried out, it resulted in the total energy to be 1945.65404 eV (44867.836601 Kcal/mol). Being a different modelling method, it does not facilitate any total energy-based comparisons. Instead, comparisons of the bond distances between the Endo C=C bond (same side as the Cl bond) and Exo C=C bond is carried out.

{| class="wikitable"{| border="1" align="center"
|+ Table 13 : Comparing the C=C bond distances
! Type of Bond !! MM2 model!! MOPAC/RM1 model!! MOPAC/PM6 model
|-
| Endo C=C bond (Angstrom) || 2.564 || 2.504 || 2.543
|-
| Exo C=C bond (Angstrom) || 2.541 || 2.492 || 2.539
|}

There is a slight change in distance where the Exo bond results in a slight reduction. This could be attributed to an interaction between the C-Cl antibonding orbital and the C=C pi orbital. But this is a weak comparison as the two methods are totally different and only serves as a preliminary assumption.

==== Analysis of Molecular Orbitals ====

{| align=center
|+ '''<u>Dig 11 - Molecular Orbitals of 9-Chloro-1,4,5,8-tetrahydro-4a,8a-methanonaphthalene via the MOPAC/PM6</u>'''
| [[Image:HOMO-1 PL1208 2.JPG|thumb|250px|HOMO-1]]
| [[Image:HOMO PL1208 2.JPG|thumb|250px|HOMO]]
| [[Image:LUMO PL1208 2.JPG|thumb|250px|LUMO]]
| [[Image:LUMO+1 PL1208 2.JPG|thumb|250px|LUMO+1]]
| [[Image:LUMO+2 PL1208 2.JPG|thumb|250px|LUMO+2]]
|}

Based on the PM6 calculating model and reference to the HOMO molecular orbital, it shows that the endo-alkene has a higher electron density as compared to the exo-alkene. The exo C=C bond is defined as the alkene bond on the opposite side of the C-Cl bond (yellow). The smaller lobe size is indicative of a smaller electron density. Thus, the HOMO has the endo alkene to be more electron rich than the exo alkene. The endo alkene is more nucleophilic than the exo alkene making it more readily able to interact with the incoming electrophile as compared to the exo C=C bond. The larger lobes are indicative of sterically more accessible pi-orbitals for an electrophilic attack as compared to the exo-alkene which is sterically less accessible.

[[Image:HOMO-1 PL1208 2 edited.bmp|thumb|350px|Dig. 12 Annotated electron density in the HOMO|centre]]

Analysis of the HOMO-1 shows that the exo orbitals are more readily stabilised as compared to the endo orbitals. This is attributed to the Cl-C sigma* orbital having an antiperiplanar interaction with the exo pi-orbital. This explains the larger lobe of the Endo alkene; indicative of higher electron density. The Cl-C sigma * orbitals are readily seen as the LUMO +1 orbitals (Diagram 13).

{| class="wikitable" border="1" align=center
|+ Diagram 13 Annotated diagrams for HOMO-1 & LUMO+1
|-
| [[Image:LUMO+1 PL1208 2 edited.bmp|thumb|350px|]] || [[Image:HOMO-1 PL1208 2 edited .bmp|thumb|350px|]]
|}

* The above rationalisations show that endo C=C is more electrophilic and accessible to an approaching electrophile. The endo product is more readily formed as compared to the exo product. However, when this is compared with the literature molecular orbitals, deviations are seen from the above orbitals generated.

* The literature molecular orbital was ran via the MOPAC/PM3 parameters. The conformation of the molecule was bent but in diagram 12, the molecule displays plane symmetry and is flat.

* The HOMO orbitals are defined as filled Pi-orbitals while the LUMO are defined as pi* anti-bonding orbitals. But from diagram 11, it is seen that there is discrepancy where the LUMO+1 is infact sigma anti-bonding instead. This is rationalised to be due to possible orbital mixings as the C-Cl sigma* is expected to be strongly anti-bonding and a much higher energy level than LUMO+1. But mixing with a higher sigma* results in the C-Cl sigma* to be stabilised and lowered in energy to be the LUMO+1. This is highly probably due to the high degree of symmetry in the molecule since the bond lengths on both sides from the 'central carbon' is relatively equivalent (Table 13).

In efforts to minimise the discrepancy, a different calculating method was adopted to compare and contrast the molecular orbitals that resulted. This also helps to determine how detrimental the presence of the bug is in using the PM6 calculations.

{| align=center
|+ '''<u>Dig 14 - Molecular Orbitals of 9-Chloro-1,4,5,8-tetrahydro-4a,8a-methanonaphthalene via the MOPAC/RM1</u>'''
| [[Image:HOMO-1 PL1208.JPG|thumb|350px|HOMO-1]]
| [[Image:HOMO pl1208.JPG|thumb|350px|HOMO]]
| [[Image:LUMO PL1208.JPG|thumb|350px|LUMO]]
| [[Image:LUMO+1 PL1208.JPG|thumb|350px|LUMO+1]]
| [[Image:LUMO+2 PL1208.JPG|thumb|350px|LUMO+2]]
|}

Comparing and contrasting the molecular orbitals in diagram 14 against the corresponding ones in diagram 12, the following observations are made

* The HOMO orbital in RM1 is similar to the HOMO orbital via the PM6. Both show that the electron density is more concentrated about the endo-alkene as compared to the exo-alkene. Thus, endo alkenes are more nucleophilic as compared to the exo alkenes. This explains why the endo product was formed under experimental conditions.

* The endo alkene region is further made nucleophilic due to the stabilisation of the exo pi-orbitals via interactions with the Cl-C sigma* antibonding oribtals. This interaction was strongly stabilising since it was of an antiperiplanar orientation; allowing maximum overlap between the orbitals. Thus, bringing about a huge extent of stabilisation for the C=C exo alkenes( Pi-->sigma*).

In conclusion, with reference to the HOMO and HOMO-1 both have considerable electron density about both the C=C bonds of the molecule. However, the endo C=C bond is of greater electron density making it more readily attacked by the electrophile. However, this does not explain the exclusive nature of the reaction as the lack of steric bulk from the top face could allow both the exo and endo C=C bonds to be attacked by the incoming electrophile. Thus, the MO diagrams are not sufficient in rationalising the regioselectivity of the molecule.

=== Vibrational Frequencies of the Molecules ===

[[Image:Both molecules pl1208.bmp|thumb|250px|Dig. 15 Molecules compared for the vibrational frequencies|centre]]

This sections aims to vibrationally rationalise the influence the C-Cl bond has on key bond vibrations present in the molecule. The main bond vibrations to be analysed in the dialkene (A) and the monoalkene (B) are the ''(i) C-Cl bond stretches'' ''(ii)Endo C=C bond stretches'' ''(iii) Exo C=C bond stretches''. The smaller the vibrational wavenumber the weaker bond.

The calculations for the bond energies were carried out in the following specific steps to result in the respective sections below.

1. The molecular structures were optimised by the use of the MM2 force field parameters (Classical)

2. The MOPAC/PM6 energy minimisations were carried out (Semi-classical)

3. The structures were then exported to Gaussview where they were subjected to B3LYP / 6-31 G(d,p) geometric optimisation and successive vibrational frequency calculation

==== Dialkene vibrational frequencies analysis ====

The approach taken towards this analysis was to analyse the IR spectra by focusing on the key stretches initially highlighted. Then the stretches, magnitude and its intensity were tabulated in Table 14.

[[Image:Ir dialkene pl1208.jpg|thumb|800px|Dig. 16 Infrared Spectrum of the Dialkene molecule with specific bond stretches annotated |centre]]

{| class="wikitable" border="1" align="center"
|+ Table 14 : The different Bond stretching vibrations and their vibrational wave numbers
! Type of Vibration !! Image with arrows !! Vibrational wavenumber (cm<sup>-1</sup>) !! Intensity !! Energy (kJmol<sup>-1</sup>)
|-
| width=175px | <CENTER> C-Cl stretch </CENTER> || [[image:C-cl stretch pl1208.jpg|325px|centre]] || <CENTER> 770.921 </CENTER> || <CENTER> 25.115 </CENTER> || <CENTER> 9.22 </CENTER>
|-
| <CENTER> exo C=C stretch </CENTER> || [[image:Exo C=C stretch pl1208.jpg|325px|centre]] || <CENTER> 1737.14 </CENTER> || <CENTER> 4.2078 </CENTER> || <CENTER> 20.79 </CENTER>
|-
| <CENTER> endo C=C stretch </CENTER> || [[image:Endo C=C bond pl1208.jpg|325px|centre]] || <CENTER> 1757.37 </CENTER> || <CENTER> 3.9338 </CENTER> || <CENTER> 21.02 </CENTER>
|-
|}

Analysis of Table 14 and the corresponding Ir spectra in diagram 16 shows the 3 key peaks indicative of the dialkene. The C=C bond stretches are seen as two seperate peaks as they are in chemically inequivalent environments so the bond stretch of the endo C=C bond is not equivalent to that of the exo C=C bond. There is significant difference in the electron density of the two C=C bonds (This was established in the earlier MO section). The literature Stretching frequency for C-Cl stretches is 780 cm<sup>-1</sup> which is very close to the experimentally attained value of 770.921 cm<sup>-1</sup>. Thus, the C-Cl bond is proven to be present via IR methods.

Similarly, C=C bond stretches has a literature value of 1670-1870 cm<sup>-1</sup>. Both values in table 14 are within the range. However, the Exo has a smaller stretching frequency magnitude as compared to the endo C=C bond. This can be re-considered with the conclusion made in the earlier MO section. The C-Cl bond is anti-periplaner to the exo C=C pi orbitals allowing an effective overlap of the C-Cl sigma* antibonding orbitals with the C=C exo pi bonding orbitals. Thus, there will be a significant amount of electron density from the C=C bond directed towards the sigma* bond. But C=C endo is not antiperiplanar to the C-Cl orbital and will not have this secondary orbital interactions. Thus, the electron density of the C=C endo is larger than the C=C exo allowing the endo to have a greater vibrational frequency and thus a slightly higher magnitude by +20 cm<sup>-1</sup>.

==== Monoalkene vibrational frequencies analysis ====

A similar approach taken towards this analysis by focusing on the key stretches initially highlighted. The stretches, magnitude and its intensity were tabulated in Table 15. This will allow easier comparisons to be made between this section and the previous one; facilitating conclusions to be drawn.

[[Image:Monoalkene Ir pl1208.jpg|thumb|800px|Dig. 17 Infrared Spectrum of the Dialkene molecule with specific bond stretches annotated |centre]]

{| class="wikitable" border="1" align="center"
|+ Table 14 : The different Bond stretching vibrations and their vibrational wave numbers
! Type of Vibration !! Image with arrows !! Vibrational wavenumber (cm<sup>-1</sup>) !! Intensity !! Energy (kJmol<sup>-1</sup>)
|-
| width=175px | <CENTER> C-Cl stretch </CENTER> || [[image:Ccl STRETCH PL1208.jpg|325px|centre]] || <CENTER> 779.931 </CENTER> || <CENTER> 21.3911 </CENTER> || <CENTER> 9.35 </CENTER>
|-
| <CENTER> endo C=C stretch </CENTER> || [[image:EndoC=Cstretch pl1208.jpg|325px|centre]] || <CENTER> 1753.76 </CENTER> || <CENTER> 5.0423 </CENTER> || <CENTER> 21.11 </CENTER>
|-
|}

=== Conclusions Drawn from vibrational Analysis ===

The C-Cl vibrational stretch according to literature is defined as a strong sharp peak at 780 cm<sup>-1</sup>.<ref>G. Socrates, Infrared and Raman Characteristic Group Frequencies, Third Edition, 2001, Pg. 65</ref> When this value is compared with those of (A) and (B), it is concluded that they are in good agreement with literature.

Molecule (A) which is the dialkene, has two asymmetric alkenes which have slightly different C=C bond stretches. The Endo C=C bond has a vibrational frequency of '''1757 cm<sup>-1</sup>''' while the Exo C=C bond have a vibrational stretch of 1737 cm<sup>-1</sup>. But molecule (B) having only 1 C=C bond (Endo) would have only 1 value expected and indeed only 1 peak was seen in the IR '''1753.76 cm<sup>-1</sup>'''. Both the Endo stretches are of a similar value (in bold) indicative that they are in rather similar environments and have minimal influence to the C=C electron density due to the hydrogenation of the exo C=C bond. However, in general when compared with literature (1670-1870 cm<sup>-1</sup>)<ref>G. Socrates, Infrared and Raman Characteristic Group Frequencies, Third Edition, 2001, Pg. 65</ref> it is noticed that both molecules have the C=C bonds to be of a slightly greater energy level. This is indicative of strong bonds which could be due to possible orbital mixings that further stabilises the pi bonds and hence account for the slightly higher C=C bond magnitude.

Another reason could be attributed to the earlier conclusions drawn in the Molecular orbital analysis where it was concluded that the C-Cl sigma* has an anti periplanar overlap with the Pi orbitals of the exo C=C bond. This will allow the overlap to be strong and so greater stabilisation of the bonding orbital (Pi orbital of the Exo C=C bond). The lowering of the orbital makes it a stronger more stable bond which is indicated by a higher magnitude vibrational frequency. (Diagram 18 ) This stabilisation will only be seen in (A) where there is the availability of the C=C exo. But in (B), the absence of the C=C exo will not allow this secondary stabilisation to be present.

[[Image:MO diagram pl1208.bmp|thumb|600px|Dig.18 MO diagram showing the Stabilisation of the Exo C=C pi bond |centre]]

== Additional Exercise ==

This exercise investigates the influence a substituent on the Exo-alkene has towards the bond strengths of the three key bonds considered in the earlier sections (i) C-Cl bond stretch (ii) Endo C=C bond stretch and (iii) Exo-bond stretch. This was done by varying between 4 different substituents that were made up of two distinct nature. The first type were electropositive substituents namely SiH<sub>3</sub> and BH<sub>2</sub>. The second group of substituents were electronegative substituents namely the OH and CN. Thus, the electron donating and electron withdrawing influences brought about by the respective substituents were investigated by measuring the respective stretching frequencies as mentioned to allow a compare and contrast to be carried out. Diagram 19 shows the four different substituents considered and the sub-groups they fall under.

[[Image:Additional reactants pl1208.bmp|thumb|800px|Dig. 19 Summary of the Different substituents considered |centre]]

{| class="wikitable" border="1"
|+ '''Table 15: Summarising the different molecules and the influence upon the C=C and C-Cl bonds'''
! R Substituent !! '''Nature of substituent''' !! <CENTER>'''Image''' !!'''Jmol''' !! '''MM2 Optimisation''' (Kcal/mol) </CENTER> !!'''Endo C=C stretch''' cm<sup>-1</sup>!!'''Exo C=C stretch''' cm<sup>-1</sup>!!'''C-Cl stretch''' cm<sup>-1</sup>
|-
| <CENTER> SiH<sub>3</sub> </CENTER> || <CENTER> Electropositive</CENTER> || [[Image:Sih3 img pl1208.jpg|thumb|330px|]] ||
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|| <CENTER> 17.6081 </CENTER>|| <CENTER>1756.22 </CENTER>|| <CENTER>1689.57 </CENTER>|| <CENTER>761.73 </CENTER>
|-
| <CENTER> BH<sub>2</sub></CENTER> || <CENTER>Electropositive </CENTER> || [[Image:Bih3 img pl1208.jpg|thumb|330px|]] ||
<CENTER>
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| | <CENTER> 15.8271 </CENTER> || <CENTER>1756.56 </CENTER> || <CENTER>1657.2 </CENTER> || <CENTER>759.042 </CENTER>
|-
| <CENTER> OH </CENTER> || <CENTER>Electronegative </CENTER> || <CENTER> [[Image:OH img pl1208.jpg|thumb|330px|]] ||
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|| <CENTER> 17.9484 </CENTER>|| <CENTER>1756.27 </CENTER>|| <CENTER>1776.6 </CENTER>|| <CENTER>766.786 </CENTER>
|-
| <CENTER>CN </CENTER> || <CENTER>Electronegative </CENTER> || <CENTER>[[Image:CN img pl1208.jpg|thumb|330px|]] ||
<jmol>
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| | <CENTER> 16.7286 </CENTER> || <CENTER>1756.54 </CENTER> || <CENTER>1706.31 </CENTER> || <CENTER>765.79 </CENTER>
|}

When an electropositive R group is introduced as the exo-substituent such as BH<sub>2</sub> substituent or the SiH<sub>3</sub> they are able to influence the different bonds in the molecule electronically. Being electropositive groups they were electron-donating groups by nature and resulting in readily donating electron density towards the C=C pi orbitals of the ring. This allowed the antiperiplanar orbital overlap with the C-Cl sigma* orbitals to become more significant. As the sigma* oribtal became more stronger it leads to the C-Cl bond to become weaker as the antibonding orbital is more readily occupied. This is seen in the vibrational frequency of the C-Cl bond which by literature should boe 780 cm<sup>-1</sup>. But due to the increased Pi-->Sigma* interaction the stretching when R=SiH<sub>3</sub> is reduced to 761.73 cm<sup>-1</sup> and in BH<sub>2</sub> 759.042 cm<sup>-1</sup>. Similarly as the C=C exo has the electron density directed towards the interaction with the C-Cl sigma* orbital, the C=C bond also becomes less electron dense and is also expected to have a lower than literature stretching frequency. This was also observed as predicted where the C=C exo bond frequency i SiH<sub>3</sub> is reduced to 1689.57 cm<sup>-1</sup> and in BH<sub>2</sub> 1657.2 cm<sup>-1</sup>. This was significantly lower than the endo C=C bonds which was part of the same molecule but did not have the antiperiplanar interactions with the C-Cl antibonding oribtals. This allowed the endo C=C bonds to be of greater electron density and so a higher magnitude vibrational frequency.

In contrast when an electronegative R group is introduced as the exo-substituent such as OH or CN, they are able to influence the different bonds within the molecule electronically. The electronegative groups being electron-withdrawing would make the C=C exo bond less electron dense as compared to the endo C=C bond. The reduction in the pi orbitals electron density would mean the pi orbital are not readily able to have secondary orbital overlaps with the C-Cl antibonding orbital as readily. As the antibonding interaction becomes weaker, the C-Cl sigma bond is stronger and would have a higher magnitude for the vibrational frequency which is determined to be 766.786 cm<sup>-1</sup> when R=OH and 765.79 cm<sup>-1</sup> when R=CN which is much closer to the literature value of 780 cm<sup>-1</sup> as compared to the other two substituents. When the C=C bond vibrational frequency is analysed, it is seen that down the table the magnitude of the exo C=C bond becomes more closer to that of the endo since it is less readily available to functinon as a electron donating orbital and instead functions as double bond with no secondary orbital interactions as it does not have sufficient electron density for such interactions. However due to the inductive effect it is still slightly weaker in bond strength as compared to the endo C=C bonds. Thus, the endo C=C when R=OH or CN is of a slightly higher magnitude than the exo C=C bond.

= Structure based Mini Project Using DFT-based Molecular Orbital methods =

The purpose of the mini project was to implement the different modelling techniques adopted in the previous sections towards organic reactions that give rise to a mixture of different isomers. The different modelling techniques would ideally, be able to rationalise the reaction carried out where it should be able to provide sufficient conclusions to justify the formation of the isomers and possibly lead to determining the ideal conditions where one isomer could dominate and be isolated for further reactions. This plays a key role in stereospecific/regiospecific reactions where the wrong isomer formed could result in the expected chemical reaction to not occur. This could be translated to a biological effect if the wrong isomer of a drug is administered resulting in adverse reactions. Thus, computational chemistry serves as a tool in accurately identifying the specific isomers in a mixture.

==Introduction to the Mini-Project: Cu(I)-Catalyzed Highly Exo-selective and Enantioselective [3 + 2] Cycloaddition of Azomethine Ylides with Acrylates==

This reaction has been chosen as it involves the formation of highly substituted 5-membered heterocycles via a very common 1,3-Dipolar cycloaddition method. The 5 membered rings being highly substituted, are very useful towards the synthesis of biologically active molecules and functioning as organic catalysts.<ref>Harwood: L. M.; Vickers, R. J. In The Chemistry of Heterocyclic Compounds: Synthetic Applications of 1,3-Dipolar Cycloaddition Chemistry
Toward Heterocycles and Natural Products; Padwa, A., Pearson, W. H., Eds.; Wiley and Sons: New York, 2002.</ref> <ref>Kunz, R. K.; MacMillan, D. W. C. J. Am. Chem. Soc. 2005, 127,3240</ref> Thus, it is of paramount importance that the correct isomer is isolated for its further use in the synthesis of catalysts and biological molecules. In most cycloaddition reactions that involves the combination of N-methalated azomethine yildes and electron deficient alkenes, the endo adduct was either the major product or it was the absolute product formed. In this paper however, the Cu-(I)/P,N-ligand catalyzed [3 + 2] cycloaddition of N-metalated
azomethine ylides with acrylates was able to yield the exo adduct as the major product with high enantioselectivity. Furthermore, the studies carried out were the first to have use the Cu(I) salt as a catalyst towards 1,3-Dipolar cycloaddition of azomethine ylides.<ref>Cu(I)-Catalyzed Highly Exo-selective and Enantioselective [3 + 2] Cycloaddition of Azomethine Ylides with Acrylates Org. Lett., 2005, 7 (19), pp 4241–4244 {{DOI|10.1021/ol0516925}}</ref>

[[Image:Reaction_scheme_2_PL1208.bmp|thumb|1000x400px|Diagram A- Reaction scheme for Cu(I)-Catalyzed 1,3-Dipolar Cycloaddition |centre ]]

The reaction is a novel Cu(I)/P,N-ligand catalyzed asymmetric 1,3-dipolar cycloaddition of azomethine ylides with acrylates, providing exo products of polysubstituted proline derivatives in up to 98% enantiomeric excess.

=== Purpose of Computational Efforts ===

The 1,3 Dipolar cycloadditions cycloaddition reaction having said to be a novel reaction drew my interest. Thus, computational techniques are employed to validate the authenticity of the results published and inevitably determine the reliability of the results and the findings made towards 1,3 Dipolar cycloadditions. This will be quantitatively carried out by comparing the calculated <sup>13</sup>C NMR against the literature. Similarly the same method will be carried out with the less accurate <sup>1</sup>H NMR spectra as well.

The typical cycloaddition usually yields the endo product as the major or only product in previous reactions and could be justified due to favourable secondary stabilisations present. But in this reaction the exo was the significant majority of 96%. Thus, the individual energies of the endo and exo isomers are to be determined to rule out thermodynamic control. Also, the possible influence of any secondary stabilisations is to be looked into; in-efforts to rationalise why the exo is favoured over the endo product.

=== Underlying Mechanism<ref>Cu(I)-Catalyzed Highly Exo-selective and Enantioselective [3 + 2] Cycloaddition of Azomethine Ylides with Acrylates Org. Lett., 2005, 7 (19), pp 4241–4244 {{DOI|10.1021/ol0516925}}</ref>===

[[Image:MECHANISM2.bmp|thumb|800x500px|Diagram (B)- Proposed reaction mechanism towards the absolute synthesis of the Exo-product (Adapted from literature source)|centre ]]

With reference to Dig. B, the first step involves the formation of complex '''A''' via the complexation of the Imine ('''1''') to the copper(I) catalyst. The Copper(I) is able to have interactions with both the N and O hetero atoms to form the complex.

The addition of the base triethylamine facilitates deprotonation of the alpha proton which is highly acidic due to being in-between two electron withdrawing groups. This allows the formation of the highly reactive azomethine ylide-Copper(I) complex '''B'''.

The reactive species formed is able to react with different dipolarphiles to facilitate the formation of intermediate '''C'''. Upon interaction with the ammonium salt formed upon proton abstraction, the copper catalyst and base are regenerated. The exo is formed as the major product while the endo is formed as the minor in a ratio of 96:4.

====Mechanistic Rationalisation====

Based on the mechanism, a preliminary rationale for the selectivity of the exo over the endo can be made. The intermediate B (Diagram B), has the transition metal centre forming the isolable complex. The copper(I) metal from group 11 has a +1 oxidative charge making have a d<sup>10</sup>orbital configuration. Thus, it can have a tetrahedral coordination with four bulky L-type ligands to achieve the stable 18 electron complex. The interaction with the carbonyl (O) and the Imine (N) allows this stable electron configuration to be achieved.

Phosphino-Oxazoline ligands were used for coordination to the Cu(I) centred complex, these in combination with Cu(OAc) were readily able to catalyse the [3+2] cycloaddition. They were also determined to bring about strong exo-selectivity over the endo product. This was because the large bulky ligands added on to the steric bulk around the metal complex (Diagram C). This made the steric repulsion between the ligands and the dipolarophiles to become more significant and thus the endo approach was highly not favoured. The exo approach minimised the steric repulsion making it much more favourable.

[[Image:Steric_clash_pl1208.bmp|thumb|800x400px|Diagram C- Significant steric clash present in the formation of the endo as compared to exo|centre ]]

== Rationalising the issue of Selectivity via modelling programmes ==

The following section will look to prove the conclusions drawn from the mechanisatic analysis carried out. The minimial energy of the 2 possible products will be attained and compared to determine if there is any thermodynamic control for the dominance of the exo over the endo isomer. Following that the respective IR and NMR spectra will be analysed and compared with the literature values to determine the accuracy of the literature.

=== Conformational Analysis of Considered Isomers ===

{| class="wikitable" border="1" align="center"
<u>''' Table A: MM2 Optimised and PM6 Optimised energies'''</u>

| || ''' Exo Product '''|| ''' Endo Product'''
|-
| Image upon MM2 optimisation || [[Image:Exo_Product_pl1208.jpg|350x200px|centre]]
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|| [[Image:Endo_Mm2.jpg|350x200px|centre]]
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|-
| Stretch kcal/mol || 1.5834 ||1.6641
|-
| Bend kcal/mol || '''7.8428'''|| '''8.2561'''
|-
| Stretch-Bend kcal/mol || 0.4767|| 0.5462
|-
| Torsion kcal/mol || -3.9131|| -1.6198
|-
| Non-1,4 VDW kcal/mol || -7.1173||-7.6094
|-
| 1,4 VDW kcal/mol || 17.6254 ||18.0415
|-
| Dipole/Dipole kcal/mol || 4.2308|| 4.2522
|-
| Selected Dihedral Angle via MM2 (deg) || -85|| 34
|-
| Total Energy (MM2) || '''20.8285''' kcal/mol|| '''23.5309 ''' kcal/mol
|-
| Selected Dihedral Angle via PM6 (deg) || -101|| 6
|-
| Image Upon Optimisation (PM6) || [[Image:Exo_pm6_Product.jpg|350x200px|centre]]
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|| [[Image:Endo_pm6.jpg|350x200px|centre]]
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|-
| Heat of Formation (PM6) || -187.60199 kcal/mol|| -182.01345 Kcal/Mol
|-
| Heat of Formation (DFT-631(d,p)) ||-1476.71913280 a.u. || -1476.74522110 a.u.
|}

Both the endo and exo products were optimised under the MM2 parameters (classical), The PM6 semi-classical parameters and finally the DFT (quantum) parameters. All three modes of calculations could not be inter-related as they were of different modes. But the difference in energy between the endo and the exo across all three systems brought about the same conclusion. Both products had realtively similar energy levels such as in the DFT where they only differed by a 0.03 a.u. difference from each other. The DFT optimisation could be carried out as it only too 45 mins for the geometry optimisation which was relatively short.

This allows the conclusion to be made that the reason for the exo to be preferred over the endo is not due to thermodynamic control as they are both of relatively similar energy if formed. Thus it should be a non-thermodynamic reason that reasons in the exclusive formation of the exo instead of the endo. Comparing the Jmol for the PM6 against the MM2 would show that they were of relatively similar orientations w.r.t dihedral angles thus should that the optimum starting point was chosen for the mm2 modelling.

=== NMR comparisons ===

==== <sup>1</sup>H NMR {{DOI|10.1021/ol0516925}} ====

<u>'''Literature <sup>1</sup>H NMR for Exo product'''</u>

From the Literature paper under study, the <sup>1</sup>H NMR for the Exo product is given as 1H NMR (360MHz, CDCl3) δ 7.39 (d, J = 8.5 Hz, 2 H), 7.28 (d, J = 8.5 Hz, 2 H), 4.31 (d, J = 8.6 Hz, 1H), 4.00 (dd, J = 8.8 and 5.3 Hz, 1 H), 3.76 (s, 3 H), 2.74 (q, J = 8.8 Hz, 1 H), 2.45 (dt, J= 13.0 and 8.8 Hz, 1 H), 2.35-2.25 (m, 2 H), 1.35 (s, 9 H).

<u>'''Calculated <sup>1</sup>H NMR for the Exo product'''</u> {{DOI|10042/to-7346}}

[[Image:H_NMR_PL1208.JPG|thumb|1000X800px|Diagram D- representing calculated H NMR peaks for Exo product|centre]]

<u>'''Comparisons made for the Calculated Against the Literature value'''</u>

[[Image:Annotated_Exo_diagram_H_nmr.bmp|thumb|250px|Diagram E- for Numbered Hydrogens|centre ]]

[[Image:Table exo pl1208.JPG|thumb|800x600px|Table B- representing calculated against Literature values|centre ]]

[[Image:Graph_comparing_values.JPG|thumb|800x600px|Graph A- for Calculated Vs Literature Chemical Shifts|centre ]]

Although, it was stated that the <sup>1</sup>H NMR were less reliable, the <sup>3</sup>J<sub>H-H</sub> coupling constants were compared with the literature attained coupling constants. This was done by the utility of the Jancchio software to provide the computationally calculated coupling constants. The results were summarised in table B along with the calculated chemical shifts 'tagged' onto the proton expected of that chemical shift.

It was noticed that literature did not 'label' the protons corresponding to the different chemical shifts so this was done to allow better rationalisation of the spectra. For eg. the identification of H-27 being more deshielded than H-28 could have be mistaken as the opposite since H-28 is bonded to the electron-withdrawing N heteroatom.

When a comparison of the literature chemical shifts are done against the calculated chemical shifts (Graph A),there are slight deviations in the overall alighnment of the plots. This slight deviations is indicative of significantly small differences between the calculated and literature values. From the table, it is noticed that computationally, a chemical shift is assigned for every proton even if they are in the same chemical environment unlike in literature which has a lesser. This is because, in the computational 'environment' the molecule is frozen with the protons in their specific positions. Thus the protons of a methyl functional group will be seen as being in different but similar chemical environments. But in the actual NMR carried out, the molecules are free to rotate about the sigma bonds and thus there will be chemically equivalent environments present such as the methyl or the t-butyl which has all 9 protons in the same chemical environment and presented by a single peak of an integration of 9. As both the molecules are sterochemically different from each other, they are expected to have relatively similar chemical shifts by the different functional groups.

Thus, the calculated spectra is in good correlation with the computationally calculated values for the Exo H nmr.

<u>'''Literature <sup>1</sup>H NMR for Endo product'''</u>

From the Literature paper under study, the <sup>1</sup>H NMR for the Endo product is given as 1H NMR (360 MHz, CDCl3) δ 7.35-7.25 (m, 4 H), 4.44 (d, J = 7.8 Hz, 1 H), 3.94 (t,J = 8.2 Hz, 1 H), 3.80 (s, 3 H), 3.24 (q, J = 7.8 Hz, 1 H), 2.55 (br, 1 H), 2.43 (dt, J = 13.3
4 and 8.2 Hz, 1 H), 2.35-2.25 (m, 1 H), 1.06 (s, 9 H)

<u>'''Calculated <sup>1</sup>H NMR for the Endo product'''</u> {{DOI|10042/to-7343}}

[[Image:H_nmr_endo_pl1208.JPG|thumb|1000X800px|Diagram F representing calculated <sup>1</sup>H NMR peaks for Endo product|centre]]

<u>'''Comparisons made for the Calculated Against the Literature value'''</u>

[[Image:Annotated_Endo_diagram_H_nmr.bmp|thumb|250px|Diagram G for Numbered Hydrogens|centre ]]

[[Image:Table_1_endo_pl1208.JPG|thumb|800x600px|Table C representing calculated against Literature values|centre ]]

[[Image:Graph_1_endo.JPG|thumb|800x600px|Graph B for Calculated Vs Literature Chemical Shifts|centre ]]

Similar to the exo case the less number of peaks seen in the literature values as compared to the computational would be due to the computing method looking at the molecule in a 'frozen' state where rotations about the c-c bonds are not considered. However, for the endo H nmr comparisons there is a less 'fit' of the 2 plots indicative of a more significant deviation from the literature values. This could be seen at H-42 which computationally is a -0.23ppm chemical shift while literature is of 1.06 ppm. This would be the draw back as by theory, the last 9 protons in the table should be chemically equivalent and have the same chemical shift.

But overall the graphs do have a significant number of plots to correspond to each other well with minimal deviations making it relatively accurate.

==== <sup>13</sup>C NMR ====

<u>'''Literature <sup>13</sup>C NMR for Exo product'''</u>

13C NMR (90 MHz, CDCl3) δ 175.0, 172.6, 140.7, 133.7, 129.0, 128.8, 81.5, 66.4, 59.5, 52.8, 52.6, 34.5, 28.4;

<u>'''Calculated <sup>13</sup>C NMR for the Exo product'''</u> {{DOI|10042/to-7346}}

[[Image:C_NMR.JPG|thumb|1000X800px|Diagram H representing calculated 13C NMR peaks for Exo product|centre]]

<u>'''Comparisons made for the Calculated Against the Literature value'''</u>

[[Image:Annotated_Exoo_diagram_C_nmr.bmp|thumb|400px|Diagram I for Numbered Carbons|centre ]]

[[Image:Table_exo_13C_pl1208.JPG|thumb|800x600px|Table D representing calculated against Literature values|centre ]]

[[Image:Graph_comparing_values_c13.JPG|thumb|800x600px|Graph C for Calculated Vs Literature Chemical Shifts|centre ]]

As the carbons are all accounted for individually via both the literature and computationally, unlike in the H NMR analysis, here the total number of data points correspond to each other. However, as the position of the different carbons influences the extent of deshielding and this is to be taken into account a number of error corrections are carried out. These error corrections as labelled in the table above show a much greater similarity to the literature values which makes the 13C NMR much more accurate as compared to the H nmr. This is well observed by the plot of the graph where there is a large degree of overlap between the two plots. This is indicative of very minimal deviations for the calculated from the literature chemical shifts. This is quantitatively seen where all the differences are less than 3 ppm.

Thus the correlation for the exo product is good and can be concluded that the exo-product has indeed been attained as stated in literature.

<u>'''Literature <sup>13</sup>C NMR for Endo product'''</u>

13C NMR (90 MHz, CDCl3) δ 174.0, 171.9, 138.5, 133.5, 129.1, 128.6, 81.2, 65.2, 60.2, 52.6, 50.5, 34.2, 27.9

<u>'''Calculated <sup>13</sup>C NMR for the Endo product'''</u> {{DOI|10042/to-7343}}

[[Image:Endo_c13_spectra.JPG|thumb|1000X800px|Diagram J representing calculated 13C NMR peaks for Endo product|centre]]

<u>'''Comparisons made for the Calculated Against the Literature value'''</u>

[[Image:Annotated_Endo_diagram_C_nmr.bmp|thumb|250px|Diagram K for Numbered Carbond|centre ]]

[[Image:Endo_c13_table_pl1208.JPG|thumb|800x600px|Table E representing calculated against Literature values|centre ]]

[[Image:Endo_c13_graph_pl1208.JPG|thumb|800x600px|Graph L for Calculated Vs Literature Chemical Shifts|centre ]]

On the exact points as for the exo 13C NMR, the correct product has been formed due to the almost perfect overlap of the two plots with only 1 data point to be over the 3 ppm difference. Thus it can be concluded that the correct endo product has been formed in literature as well.

==== Conclusions Drawn ====

The 13C NMR is indeed more accurate in comparing against the literature chemical shifts as seen by the degree of overlap between the plots in the respective graphs. From both the 13C plots that compared with the literature paper, it is conclusively said that the correct products have been isolated and formed as reported. However, there was some deviation in the calculated values from that of the literature. The reasons have been summarised as below

The computational method observes the molecule from a 'frozen' view; the conformation the molecule is in is the conformation in which it is analysed. This could differ from reality where in reality where not bound by potential energy wells, it would indeed adopt the lowest energy conformation whereas in the computer form it could have adopted a pseudo-lowest energy due to being trapped by an energy well as in the case of the MM2 optimisation.

In the computational model the influence the solvent (CDCl3) is not taken into account if it has an influence in the conformation. Thus it can be expected that the real conformation will deviate slightly from that modelled in the computer. Thus the error in determining the lowest energy conformation despite using the DFT method could have led to the slight deviations seen in the tables above.

A significant amount of difficultly was faced in assigning the chemical shifts to the contributing protons against literature chemical shifts. This is expected to be due to the resoultion power of the NMR machine. If it is at low resolution, the expected peaks of slightly different chemical shifts will be seen as a single overlapping peak or as a multiplet that will be hard to distinguish. Thus explain why the expected quartet via computational modelling is only seen as a double or a triplet as the very small coupling constants result in the peaks to be seen as one.

=== IR Analysis ===

Both the literature IR values were taken from the exact literature source as for the NMR section. For the computationally calculated IR values, they were attained by first carrying out both the MM2 and PM6 energy minimisations. Following that, the DFT gaussian calculations were carried out to optimise the geometry for more accurate vibrational frequencies to be acquired. The B3LYP model with the 6-31G(d,p) basis set was used both the exo and endo ir calculations.

==== IR comparison with Literature for the Exo Product ====

[[Image:Annotated_Ir.jpg|thumb|1000x600px| Diagram M Annotated IR spectra for the Exo Product|centre ]]

[[Image:Ir_table_pl1208.JPG|thumb|800x600px|Diagram Table F comparing the calculated stretches against literature|centre]] [[Image:Ir_comparing_graph_pl1208.JPG|thumb|800x600px| Graph E comparing Experimental against Calculated|centre]]

{| align=center
|+ '''<u>Dig N - Annotated diagrams of the key bond stretches</u>'''
| [[Image:C-Cl_stretch_pl1208.bmp|thumb|250X250px|C-Cl stretch]]
| [[Image:C=C_in_ring_bond_stretch_pl1208.bmp|thumb|250X250px|C=C in ring stretch]]
| [[Image:C=O_OME_stretch_pl1208.bmp|thumb|250X250px|C=O OMe stretch]]
| [[Image:C=O_OtBu_stretch.bmp|thumb|250X250px|C=O Ot-Bu stretch]]
| [[Image:C-H_aLKANE_stretch.bmp|thumb|250X250px|C-H Alkane stretch]]
| [[Image:C-O_ester_stretch..bmp|thumb|250X250px|C-O Ester stretch]]
|}

==== IR comparison with Literature for the Endo Product ====

[[Image:ENDO_iIR.jpg|thumb|1000x600px| Diagram O Annotated IR spectra for the Endo Product|centre ]]

[[Image:Ir_TABLEENDO.JPG|thumb|800x600px| Table F comparing the calculated stretches against literature|centre]] [[Image:Comparing_graph_pl1208.JPG|thumb|800x600px|Graph comparing the calculated results against the experimental|centre ]]

{| align=center
|+ '''<u>Dig P - Annotated diagrams of the key bond stretches</u>'''
| [[Image:C-cL_pl1208ENDO.bmp|thumb|250X250px|C-Cl stretch]]
| [[Image:C=C_stretchingENDO.bmp|thumb|250X250px|C=C in ring stretch]]
| [[Image:C=O_ome_StretchENDO.bmp|thumb|250X250px|C=O OMe stretch]]
| [[Image:C=O_OTBU_StretchENDO.bmp|thumb|250X250px|C=O Ot-Bu stretch]]
| [[Image:C-O_stretch_Otbuendo.bmp|thumb|250X250px|C-O Ester (Ot-Bu stretch]]
| [[Image:C-O_stretch_OmeENDO.bmp|thumb|250X250px|C-O Ester (OMe) stretch]]
| [[Image:CN_stretchENDO.bmp|thumb|250X250px|CN stretch]]
|}

==== Conclusions Drawn ====

When both the Ir spectras, corresponding table and graphs were compared, it is seen that the IR stretching frequencies for both the molecules are almost identical. Both the calculated IR spectra encompassed the key vibrations that were expected from the molecule such as the N-H, and the 2 distinct carbonyl stretches. The stretching frequency of the methoxy carbonyl C=O stretch is slightly higher than the stretching frequency of the O-tBu C=O stretching in both the compounds which is indicative that there is a slightly stronger C=O bond in the methoxy carbonyl as compared to the O-tBu. But as both are very similar with minimal deviations from the literature range, the computational geometry of the two products can be said to be rather accurate to that of the molecule in reality.

===Overall Conclusion Drawn===

In conclusion, the NMR assignment attained computationally correlate well to the literature <sup>1</sup>H and <sup>13</sup>C NMR spectras. Furthermore, comparisons made via the IR spectras show equally good correspondence with literature IR values. However as rationalised earlier, there is large similarity is both the exo and endo <sup>13</sup>C chemical shifts. This would suggest that another method is required besides NMR and IR to differentiate the two compounds from each other. The separation of the enatiomers could readily take place via the use of CaSH which gave a literature yield of in ratio of 1.5:1 for exo:endo products. Also physical methods such as chiral chromatography could be considered for the complete isolation of the 2 compounds in the 96:4 ratio of the exo:endo with the exo in 97% ee excess.

[[User:Pl1208|IceBurn]] 16:13, 1 March 2011 (UTC)

=References=
<references/>

Talk:Excalibur

2011-04-08T16:04:44Z

Pl1208: Talk:Excalibur moved to Talk:Sunkiss pl1208 3

#REDIRECT [[Talk:Sunkiss pl1208 3]]

Talk:Sunkiss pl1208 3

2011-04-08T16:04:44Z

Pl1208: Talk:Excalibur moved to Talk:Sunkiss pl1208 3

Grade A:

Cope Rearrangement:
Presentation:
Extremely easy to read. All data was tabulated. Good use of Jmol, animations. All figures and tables given clear legends. Good use of file sharing. Overall this report was an excellent read.
Results:
Generally v.good. Some results for energies given to 3 or 4 d.p. (be careful as 0.0001 Hartree is equivalent to ~0.063kcal/mol!) Optimized all gauche and anti isomers and correctly noted gauche 3 lowest in energy. V.good analysis of why this could be, considering MOs etc. For graph 1 perhaps this would have a greater meaning if it showed the dihedral scan (i.e. the energy of 1,5-hexadiene as you rotate the dihedral angle).
Good individual thought given to the lowest energy isomers and the reaction path. Stated ‘Gauche-3 conformer would be the most abundant in the reaction mixture’ – good, although should note its abundance is dependant on temperature. Also ‘the Cope rearrangement is expected to occur via the Anti-conformer 2 due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species’, however via this thought process wouldn’t it go through the anti-conformer 3 which is highest in energy? Good thought process to try and rationalize why we are focussing on anti-2 in our further calculations – however is in fact due to when optimized with a higher level of theory, anti 2 is actually slightly lower in energy than the gauche 3 isomer. By definition, activation energy is calculated from the lowest energy/most stable reactant to the transition state.
Good understanding you can’t really compare energies between different methods (HF and DFT), they’re calculated on different energy scales. But analysis of the difference in the geometries is v. thorough.
Vibrational analysis good – Output from frequency analysis of HF shows only a freq calculation, not freq+opt. What was the input structure for here? Should have been the optimized anti-2 conformer as shown in the section above, which when you look in the output has optimised to a local minimum. V.good investigation of ZPE/thermal contributions at 0K and 298K and understanding of this matter.
Boat and chair TS: optimisations explained well and imaginary frequencies given and animated. IRC carried out and many methods considered and attempted – shown a minimum reached. Activation energies calculated for 0K and 298K and compared to literature - v.good. No need for graph (2) as doesn’t really show additional information – only 2 points on line.
Understanding:
Outstanding introduction, and v. good understanding of the steps taken. E.g. A good understanding of the benefits of carrying out a rough optimisation before carrying out higher-level optimisation. Good understanding to why such methods were applied, the pros and cons, and the use of keywords in the input. Gauche/anti conformers and their energies analysed thoroughly. A great deal of individual thought given!
Diels alder:
Presentation:
Excellent, as above. Excellent use of animations and figures to display results.
Results:
Generally very good. MOs for reactants and TS correct, and a very good understanding of their interaction. Very thorough investigation into TS, investigating a number of methods and even considering IRC analysis.
Exo/Endo: correct alignments and MOs for these. Structural analysis given in great detail. Endo correctly noted as lowest in energy and good explanation given. A better angle of the MO figures may make the secondary orbital overlap apparent, as is hard to see.
Understanding:
Very good. A thorough explanation to the steps given throughout. Good understanding of HOMO/LUMO interactions and a very nice explanation of allowed and forbidden reactions. All results explained scientifically, in very good detail.
Well done! You have an outstanding understanding of the topic and your work was a pleasure to read. Do consider a computational research project in the future.

Rep:Sunkiss pl1208 3

2011-04-08T16:04:44Z

Pl1208: Excalibur moved to Sunkiss pl1208 3

<u>'''MODULE 3 COMPUTATIONAL CHEMISTRY REPORT BY PRATHAP LATCHMANAN 00561753 (Hope you enjoy reading it as much as i did doing it :) )'''</u>

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product''' !! '''Endo-Exo (Energy Difference)/ Polarity'''
|-
| Diagram of Products || [[image: Exo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42191.mol</uploadedFileContents><text>Exo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || [[image: Endo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42190.mol</uploadedFileContents><text>Endo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || -
|-
| Electronic Energy || -612.75578460|| -612.75829022 || -0.0025056 (Negative)
|-
| Sum of electronic and zero-point Energies at 0 K || -612.569372 || -612.572070 || -0.002698 (Negative)
|-
| Sum of electronic and thermal energies 298.15 K || -612.559972 || -612.562604 || -0.002632 (Negative)
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8049 |D-Space Exo]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8048 |D-Space Endo]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed. But all the polarity of the differences in the specific energy being negative is indicative that the endo product is the more stable isomer of the two. As the energy difference is small there is a mixture, however being the more stable product it would be the major product of the mixture. This agrees with literature so the computationally attained results are accurate to that of literature and the 'Endo-Rule'.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

All the calculations in the diels alder reactions agree well with that expected of literauture. The secondary orbital interactions were proven to be existant for the Endo product via MO analysis. Thus this module was a highly exciting one while explain my possible over-enthusiasm to provide as much detail as possible. Advanced apologies for that.

[[User:Pl1208|Fisty]] 23:12, 22 March 2011 (UTC)

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T23:14:09Z

Pl1208:

Rep:Sunkiss pl1208 3

2011-03-22T23:13:41Z

Pl1208:

= MODULE 3 COMPUTATIONAL CHEMISTRY REPORT BY PRATHAP LATCHMANAN 00561753 (Hope you enjoy reading it as much as i did doing it :) )=
= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product''' !! '''Endo-Exo (Energy Difference)/ Polarity'''
|-
| Diagram of Products || [[image: Exo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42191.mol</uploadedFileContents><text>Exo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || [[image: Endo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42190.mol</uploadedFileContents><text>Endo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || -
|-
| Electronic Energy || -612.75578460|| -612.75829022 || -0.0025056 (Negative)
|-
| Sum of electronic and zero-point Energies at 0 K || -612.569372 || -612.572070 || -0.002698 (Negative)
|-
| Sum of electronic and thermal energies 298.15 K || -612.559972 || -612.562604 || -0.002632 (Negative)
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8049 |D-Space Exo]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8048 |D-Space Endo]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed. But all the polarity of the differences in the specific energy being negative is indicative that the endo product is the more stable isomer of the two. As the energy difference is small there is a mixture, however being the more stable product it would be the major product of the mixture. This agrees with literature so the computationally attained results are accurate to that of literature and the 'Endo-Rule'.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

All the calculations in the diels alder reactions agree well with that expected of literauture. The secondary orbital interactions were proven to be existant for the Endo product via MO analysis. Thus this module was a highly exciting one while explain my possible over-enthusiasm to provide as much detail as possible. Advanced apologies for that.

[[User:Pl1208|Fisty]] 23:12, 22 March 2011 (UTC)

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T23:12:16Z

Pl1208: /* Conclusion */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product''' !! '''Endo-Exo (Energy Difference)/ Polarity'''
|-
| Diagram of Products || [[image: Exo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42191.mol</uploadedFileContents><text>Exo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || [[image: Endo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42190.mol</uploadedFileContents><text>Endo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || -
|-
| Electronic Energy || -612.75578460|| -612.75829022 || -0.0025056 (Negative)
|-
| Sum of electronic and zero-point Energies at 0 K || -612.569372 || -612.572070 || -0.002698 (Negative)
|-
| Sum of electronic and thermal energies 298.15 K || -612.559972 || -612.562604 || -0.002632 (Negative)
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8049 |D-Space Exo]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8048 |D-Space Endo]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed. But all the polarity of the differences in the specific energy being negative is indicative that the endo product is the more stable isomer of the two. As the energy difference is small there is a mixture, however being the more stable product it would be the major product of the mixture. This agrees with literature so the computationally attained results are accurate to that of literature and the 'Endo-Rule'.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

All the calculations in the diels alder reactions agree well with that expected of literauture. The secondary orbital interactions were proven to be existant for the Endo product via MO analysis. Thus this module was a highly exciting one while explain my possible over-enthusiasm to provide as much detail as possible. Advanced apologies for that.

[[User:Pl1208|Fisty]] 23:12, 22 March 2011 (UTC)

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T23:10:05Z

Pl1208: /* Energetic Analysis of Transition States */

Rep:Sunkiss pl1208 3

2011-03-22T23:07:26Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product''' !! '''Endo-Exo (Energy Difference)/ Polarity'''
|-
| Diagram of Products || [[image: Exo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42191.mol</uploadedFileContents><text>Exo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || [[image: Endo_product_diagram_pl1208.JPG|centre|200px]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42190.mol</uploadedFileContents><text>Endo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || -
|-
| Electronic Energy || -612.75578460|| -612.75829022 || -0.0025056 (Negative)
|-
| Sum of electronic and zero-point Energies at 0 K || -612.569372 || -612.572070 || -0.002698 (Negative)
|-
| Sum of electronic and thermal energies 298.15 K || -612.559972 || -612.562604 || -0.002632 (Negative)
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8049 |D-Space Exo]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8048 |D-Space Endo]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T23:01:20Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product''' !! '''Endo-Exo (Energy Difference)/ Polarity'''
|-
| Diagram of Products || [[image: Exo_product_diagram_pl1208.JPG|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42191.mol</uploadedFileContents><text>Exo Product</text><color>black</color></jmolAppletButton></jmol></CENTER> || [[image: Endo_product_diagram_pl1208.JPG|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_42190.mol</uploadedFileContents><text>Endo Product</text><color>black</color></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.75578460|| -612.75829022 ||
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

= References =
<references/>

File:Checkpoint 42190.mol

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Pl1208:

File:Checkpoint 42191.mol

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Pl1208:

File:Endo product diagram pl1208.JPG

2011-03-22T22:59:39Z

Pl1208:

File:Exo product diagram pl1208.JPG

2011-03-22T22:59:16Z

Pl1208:

Rep:Sunkiss pl1208 3

2011-03-22T22:47:28Z

Pl1208: /* Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

m== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that do not form primary bonds but are able to bring additional stabilisation to the molecule. But for this type of stabilisation to occur, the orbitals have to be within a certain distance of each other and are to have the correct phase symmetry to allow the effective orbital overlap. From the Diagram above it can be seen that the endo product has the fragments arranged to be much more sterically unfavourable. However, it allows the other set of p orbitals to be close enough to have the secondary orbital interactions that has been seen present in the above analysis.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

From the above molecular orbitals which has the isolobal value decreased to 0.004 to allow the secondary orbital interactions (if any) to be observed. Comparing the HOMO of the two transition states, it is seen that the Endo transition state is of a lower energy level (more negative) than the HOMO of the exo. This can be rationalised by looking at the cyclohexadiene fragment which shows the presence of the secondary orbital overlapping being present but absent in the exo transition state. Thus the secondary orbital interactions being present has been proven via MO analysis and explains why the endo although more sterically hindered it is more stable

== Conclusion ==

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T22:16:27Z

Pl1208: /* Conclusion */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==

= References =
<references/>

Rep:Sunkiss pl1208 3

2011-03-22T21:58:11Z

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_2.JPG|200px]]||[[Image:Lumo_exo_pl1208_2.JPG|200px]]||[[Image:Homo_pl1208_endo_secondary.bmp|200px]]||[[Image:Lumo_endo_pl1208_2.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==

File:Homo pl1208 endo secondary.bmp

2011-03-22T21:58:07Z

Pl1208:

File:Lumo exo pl1208 2.JPG

2011-03-22T21:54:25Z

Pl1208:

File:Homo exo pl1208 2.JPG

2011-03-22T21:54:04Z

Pl1208:

File:Lumo endo pl1208 2.JPG

2011-03-22T21:53:22Z

Pl1208:

File:Homo endo pl1208 ts 2.JPG

2011-03-22T21:52:44Z

Pl1208:

Rep:Sunkiss pl1208 3

2011-03-22T21:32:57Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals. The difference in the energy between the two transition states is significant (electronic = 2.3 Kcal/mol). This difference would explain why under kinetic control there is the possibility of attaining only the endo isomer.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

Analysis of the energies of the two types of products show that the two products have a small difference in their total electronic energy. Although the endo product is of a smaller magnitude, it is not that big a difference to rule it as the thermodynamic product. This would explain why in the experimental reactions carried out under thermodynamic control, there is a mixture of the products being formed.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_ts.JPG|200px]]||[[Image:lumo_exo_pl1208_ts.JPG|200px]]||[[Image:Homo_endo_pl1208_ts.JPG|200px]]||[[Image:lumo_endo_pl1208_ts.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==

Rep:Sunkiss pl1208 3

2011-03-22T20:59:42Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 27 - Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 28 - Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 29 - Activation energies for the Exo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 30 -Activation energies for the Endo Transition State'''
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table 31 - Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_ts.JPG|200px]]||[[Image:lumo_exo_pl1208_ts.JPG|200px]]||[[Image:Homo_endo_pl1208_ts.JPG|200px]]||[[Image:lumo_endo_pl1208_ts.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==

Rep:Sunkiss pl1208 3

2011-03-22T20:55:30Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ Activation energies for the Exo Transition State
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ Activation energies for the Endo Transition State
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals.

To conclude the energy analysis, the energies of the two different products will also be analysed to determine if the endo is indeed the more thermodynamically favoured product.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table Comparing the Specific Energies for the Exo and Endo Products'''
! '''Specific Energy (a.u.) !! Exo Product !! Endo Product'''
|-
| Diagram of Products || [[image: exo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Exo_product_for_jmol.mol</uploadedFileContents><text>Exo- Product</text></jmolAppletButton></jmol></CENTER> || [[image: endo product of diels alder.jpeg|centre|200px]] |<CENTER><jmol><jmolAppletButton><uploadedFileContents>Endo_product_for_jmol.mol</uploadedFileContents><text>Endo- Product</text></jmolAppletButton></jmol></CENTER>
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_ts.JPG|200px]]||[[Image:lumo_exo_pl1208_ts.JPG|200px]]||[[Image:Homo_endo_pl1208_ts.JPG|200px]]||[[Image:lumo_endo_pl1208_ts.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==

Rep:Sunkiss pl1208 3

2011-03-22T20:48:57Z

Pl1208: /* Energetic Analysis of Transition States */

= Introduction to Module 3 =

The previous modules (both 1 and 2) often involved the energy minimisation of isomeric products. The minimisation was carried out for the reactants as well as products to determine which would be the more stable product in equilibrium; determining the thermodynamic product of the reaction. However, there was a limitation to the analysis (diels-alder reaction via module 1) (Molybdenum complex via module 2) whereby the predominance of an isomer under kinetic control could not be determined. This was because the transition state energies were not determined and could not be used to determine which isomer had a smaller activation energy barrier towards product formation under kinetic control. Module 3 is expected to give insight into transition state energy determination relative to that of the reactants to determine the reaction profile under kinetic control.

[[Image:Introduction_pl1208_phy.bmp|thumb|500x500px|'''Diagram 1- Reaction profile comparing different activation energy pathways''' |centre]]

The energy of the transition state can determine the specific kinetics of the reaction profile. Route B imposes a lower barrier and under kinetic conditions yields the more favoured transition state(Diagram 1). Computational efforts are expected to allow an accurate prediction of the actual geometry and energy of the transition state. This allows predictions to be made about the specific product and which would be formed under kinetic control.

= Aims of Module =

* The Cope rearrangement reaction will be analysed to determine the preferred reaction mechanism by which the [3,3] sigmatropic shift rearrangement occurs. The reaction mechanism will be drawing attention to the specific type of transition state that will be reaction to allow the rearrangement reaction to occur (chair transition state Vs. boat transition state)

* By modelling the 1,5 hexadiene molecule, shortfalls of molecular mechanics and the harmonic oscillator are expected to be highlighted.

The second part of the module looks into the reaction mechanism of two different diels-alder reactions. The first reaction is one that is in-between the dienophile cis-Butadiene and the diene ethylene.

* The computational modelling exercise is expected to shed some light into how the concerted cycloaddition reaction occurs and to determine if it corresponds well to the predicted orbital theory.

* The second diels alder reaction looks into the reaction of maleic anhydride and 1,3-Cyclohexadiene where there is the possibility of both the endo and the exo transition states. This would allow the comparisons to be made against experimentally attained results and draw conclusions.

= Cope Rearrangement with specific reference to the 1,5-Hexadiene =

[[Image:Cope_reaction_pl1208.bmp|thumb|300x300px|'''Diagram 2 - Cope Rearrangement Mechanism''' |centre]]

The Cope Rearrangement is defined as a [3,3]-Sigmatropic rearrangement if the 1,5-dienes.<ref>Williams, R. V., Chem. Rev. 2001, 101 (5), 1185-1204.</ref> The Cope rearrangement reactions was developed by Arthur Cope in the 1930-40s where it has been shown to be a purely intramolecular process for the hexadiene. <ref>Arthur C. Cope; et al.; J. Am. Chem. Soc. 1940, 62, 441.</ref> However, there has been much controversy about the specific mechanism via which the rearrangement occurs where a sigma bond is broken at the [1,1] position and reformed at the [3,3] position. The reaction has been proposed to be a concerted reaction. The computationally modelled system will be used to prove and rationalise it being a concerted reaction.

== Introduction ==

The breaking and reforming of the σ bond is proposed to be possible due to the conjugated π system. Thus, the reaction mechanism (Diagram 2) can be further elaborated as seen in diagram 3 which shows that there is the possibility for two possible transition states. There are two energetically inequivalent conformations for the transition state. This is because due to the conjugated system, there is no intact C=C bonds in the transition state; making it non-rigid to allow both the chair and boat conformations.

[[Image:Cope_reaction_2_pl1208.bmp|thumb|500x500px|'''Diagram 3 - Detailed Cope Rearrangement Mechanism'''|centre ]]

For the calculation of the individual reactants and products, the Molecular Mechanics(MM2 or MM3) as learnt from module 1 will not be used as it is an empirical method that is specific towards determining the total energy of the molecule. In determining the total energy, it would average the electron density of the molecule instead of calculating the specific electron density about each atom within the molecule. Thus although it would give a realistic final energy value, it would not explicitly point out the bond breaking or bond making which we are mechanistically interested in. The schrodiner equation should be solved to allow the specific changes in electron density to be observed. This would help in explaining the orbitals at the transition state and the product. Thus the quantum methods of calculation will be considered for the following optimisations such as Density Functional Theories (DFT).

== Optimisation of the Reactants and Products ==

[[Image:Conformers_pl1208)frst.bmp|thumb|500x500px|'''Diagram 4 - Different conformers of the Reactant/Product'''|centre ]]

The diagram above shows the Newman projections for the various conformers of 1,5-hexadiene. <ref>Hehre, W. J.; Radom, L.; Schleyer, P. v. P.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley: New York, 1986</ref> The molecule is expected to be only slightly rigid as it has three free rotating C-C bonds between the two extreme C=C bonds. Based on literature, there are 27 conformations for the reactant but the relatively high symmetry, of the molecule would rule out 17 of the conformers to give only 10 energetically distinct conformers. Rotation about the C-C central bond would show that conformers 4 and 6 make up an enantiomeric pair. The first 3 conformers are rotamers of each other about the C-C central bond with the vinyl groups fixed to be antiparallel to each other. The next three (4,5 and 6) are similar rotamers of each other but instead have the vinyl groups parallel to each other.

The next 3 conformations, (7 to 9) have one of their C<sub>sp2</sub>-C<sub>sp3</sub> bonds to be in S-cis conformation which results in the C-C bond to eclipses the C=C bond. For the last three conformers both the C<sub>sp2</sub>-C<sub>sp3</sub> bonds eclipse the two C=C bonds. However, from literature it is known that the last 6 conformers are of a very high energy level and would have a low population of states during the reaction as the population of states is expected to comply to a boltzmann distribution. Only the first 6 conformations will be considered as they are the more energetically populated isomers of the reactant.

===Process towards Optimisation===
<br />
1. The 1,5 hexadiene was first drawn in gaussview as a linear chain possessing the anti linkage about the centre 4 carbon atoms. The structure was then cleaned to prepare the molecule for optimisation calculations. For the first set of optimisations, the antiperiplanar conformations were analysed so the dihedral angle between the central 4 atoms was fixed at 180.0 degrees.

[[Image:Process_1_pl1208.JPG|thumb|300x300px|'''Diagram 5- Process 1 for fixing the Anti orientation'''|centre ]]

2. As per instructions, the optimisation was carried out via the Hartree Fock(HF) method under the 3-21G basis set. However, the initial settings under the 'Link 0' tab of %mem=250MW resulted in the last link of the calculation via Gaussian 3 to be aborted. This was rectified and changed to %mem=250MB which fixed the error. This could be due to the lack of memory or the wrong units needed for the run.

3. This optimisation was carried out for all 6 conformations and the respective summary of the optimisations were taken note. This was done by opening the .fchk files from each of the optimisations.

4. For some of the summary files, there was no point group dictated. Thus, the edit tab was opened and 'Symmetrise' was selected to determine the point group of the conformer. The results were tabulated in table 1 below.

This was the standard procedure adopted in optimisation efforts for all 6 conformers. For the antiperiplanar conformers, the dihedral angle was initially set to 180 deg. The terminal vinyl groups were rotated in different orientations while keeping the central 4 carbons in the same 180 degree dihedral angle. The results were recorded as per '''table 1'''.

A similar set of procedures were adopted for the Gauche conformers where the Gauche linkage between the central 4 carbon atoms was established by setting the dihedral angle as 60 deg. The results were recorded in '''table 2'''

[[Image:Process_2_pl1208.JPG|thumb|300x300px|'''Diagram 6- Process 2'''|centre ]]

The different variations were made, optimised and compared against the structures in [[Mod:phys3#Appendix 1|Appendix 1]]. To arrange the atoms to the different conformations, the bonds were first removed to allow the different spatial arrangements of the atoms. Once the varying orientations were made, the bonds were re-formed and then optimised via Gaussian 3.

=== Results and Analysis ===

[[Image:Difference_in_anti_and_gauche_pl1208.bmp|thumb|500x500px|'''Diagram 7 - Initial rationalisation for APP to be more stable than Gauche'''|centre ]]

The large steric clash expected in the gauche conformer is expected to result in the most stable conformation to be one of the antiperiplanar conformers (Diagram 7). This is because the largest groups being the vinyl groups relative to the hydrogens are furthest apart in the anti conformation. This makes them sterically in a more favourable orientation as compared to the Gauche conformers which have a more positive energy than the most stable antiperiplanar conformer due to greater destabilisation effects brought about by increased steric repulsion and greater torsional strain due to the vinyl groups being much closer to each other.

<br />
{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 1 - Optimised conformations of the antiperiplanar Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
<br />
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar 1'''''
|
[[Image:Anti_1_diagram_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_1_c2_symmetry_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| C<sub>2</sub>
| -231.69260
| -231.69260
|
[[Image:Anti_1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/f/f1/ANTI_1pl1208.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 2'''''
|
[[Image:Anti_image_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_2_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>i</sub>
| C<sub>i</sub>
| -231.69254
| -231.69254
|
[[Image:Anti_2_pl1208.JPG|150px]]| [[https://wiki.ch.ic.ac.uk/wiki/images/3/33/ANTI_2_pl12082.LOG |Anti 1 Log File]]
|- align="center"
| '''''Antiperiplanar 3'''''
|
[[Image:Anti_3_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Anti_3_pl1208.mol</uploadedFileContents> <text>Anti-Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2h</sub>
| C<sub>2h</sub>
| -231.68907
| -231.68907
|
[[Image:Anti_3_summary_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/c/c3/ANTI_3_pl1208_2.LOG |Anti 3 Log File]]
|}

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 2- Optimised conformations of the Gauche Conformers compared against literature'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="100" | '''Literature Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Literature Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Gauche 1'''''
|
[[Image:Gauche_1_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_1_pl1208.mol</uploadedFileContents> <text>Gauche-Conformer 1</text> <color>Black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69167
| -231.69167
|
[[Image:Summary_gauche1_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/8/81/GAUCHE_1_pl1208.LOG |Gauche 1 Log File]]
|- align="center"
| '''''Gauche 2'''''
|
[[Image:Molecule_gauche2_pl1208.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_2_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 2</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>/C<sub>1</sub>
| C<sub>2</sub>
| -231.69153
| -231.69153
|
[[Image:Summary_gauche4_pl1208.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/e/e0/GAUCHE_2_pl1208.LOG |Gauche 2 Log File]]
|- align="center"
| '''''Gauche 3'''''
|
[[Image:Molecule_pl1208_3.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>Gauche_3_pl1208.mol</uploadedFileContents> <text>Gauche Conformer 3</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>1</sub>
| C<sub>1</sub>
| -231.69266
| -231.69266
|
[[Image:Summary_pl1208gacuhe3.JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/images/6/65/GAUCHE_4_pl1208.LOG |Gauche 3 Log File]]
|}

Comparisons made from both tables (1 and 2) show that the Gauche-3 conformer is the most stable conformer. This contradicts with the initially rationalised steric argument. Thus, it can be concluded that steric interactions is not the only determining factor on which is the most stable conformation. Graph 1 shows that Gauche 3 (plot 6) is the lowest and is followed by the C2 anti conformer (plot 1).

''(Error in graph-legend labelling : the second lowest is Anti-conformer 1 not anti-conformer 2)''

[[Image:Graph_comparison_pl1208_1.JPG|thumb|500x500px|'''Graph 1 - showing the different conformations'''|centre ]]

Analysis of the HOMO and LUMO of the two conformers indicates that the Gauche 3 conformer has additional stabilisation from the effective overlap of orbitals as they are in the correct orientation, correct phase and most importantly close proximity.<ref>B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446 </ref>However, there is a significant amount of steric repulsion as initially anticipated but the stabilising effects of the orbital overlap between the C=C π orbital and the C-H σ* orbital is much more stabilising than the destabilising effects of the steric repulsions in the Gauche 3. Thus, the gausche conformation being in the ideal orientation and proximity is more stable than the antiperiplanar conformation which does not have the availability for such an orbital overlap due to greater proximity between the C=C and C-H orbitals. This explains the deviation from the initially expected trend specifically for the Gauche 3 conformer.

From the table, it is noted that '''it is only Gauche 3''' that is of a more stable energy level than the antiperiplanar conformations due to the favourable orbital interactions. The others Gauche conformations do not have this stabilising interaction and the initial steric rationale shows that usually the anti conformers are more stable conformer (usually).

The most stable conformer has the largest activation energy barrier to 'reach' the transition state (Diagram 1). Thus, it can be expected that the Gauche-3 conformer would be the most abundant in the reaction mixture as it would have the highest activation energy barrier and least readily 'activated'. Thus, the Cope rearrangement is expected tp occur via the '''Anti-conformer 2''' due to the correct orbital orientation of the vinyl groups and as it is of a slightly higher energy level (lower energy barrier) making it the most reactive species relative to Gauche 3.

This makes perfect sense from a boltzmann distribution argument where by the Cope rearrangement is in equilibrium and the anti-2 conformer not being the most stable is not the most populated. But being the most reactive, it would be readily driven towards product formation and would have the implication of an equilibrium shift to occur to maintain the equilibrium population of states. This would drive the reaction foreward towards product formation making anti-2 the ideal conformer as the reacting species.

=== Comparison of HF optimisation against DFT optimisation ===

It has been determined that the anti-2 conformer is the reactant towards product formation. However, the HF method of optimisation is expected to only result in a approximation of the optimised energy level. To gain further accuracy, the anti-2(C<sub>i</sub>) is further optimised with Density Functional Theories (DFT) via the B3LYP/6-31G(d) basis set. This is expected to provided a better representation of the optimised geometry of the conformer as it is via a more accurate basis set and method.

{| border="1" cellpadding="5" align="center"
|+ <u>'''Table 3- Optimised conformations of the anti conformer via the DFT B3LYP/6-31G (d)'''</u> <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1</ref>
|- align="center"
| width="150" | '''Conformer'''
| width="150" | '''Structure'''
| width="100" | '''Point Group'''
| width="200" | '''Energy/Hartrees <br />HF/3-21G'''
| width="200" | '''Energy/DFT <br />6-31G (d)'''
| width="200" | '''Summary of Optimisation'''
|- align="center"
| '''''Antiperiplanar Conformation C<sub>i</sub>'''''
|
[[Image:Molecule_pl1208_dft.JPG|150px]]<jmol> <jmolAppletButton> <uploadedFileContents>6-31gd_ci_pl1208.mol</uploadedFileContents> <text>DFT optimised Anti Ci conformation</text> <color>black</color></jmolAppletButton> </jmol>
| C<sub>2</sub>
| -231.693
| -234.612
|
[[Image:Summary_6-31G(d).JPG|150px]] [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41531_pl1208.out |Anti-Ci DFT Log File]]
|}

The energy of the conformer attained via the DFT optimisation method is significantly lower than the HF/6-31G method of optimisation. However the use of different methods of calculation and different basis sets makes the energies not a reliable parameter for comparison. However, as it is a different method, the energies of the molecule are distinctly different as anticipated. Other properties of the molecules will be looked into to compare and rationalise the differences.

{| class="wikitable" border="1" align="center"
|+ <u>'''Table 4- Comparing the bond lengths and bond angles via the DFT B3LYP/6-31G (d) against the HF/3-21G'''</u>
! Parameter of Comparison !!Method of Optimisation : HF/3-21G !! Method of Optimisation : DFT B3LYP/6-31G (d)
|-
|width=150px|<CENTER> '''C-C Bond lengths (Å)''' </CENTER>|| [[image:Anti_211_PL1208.bmp|centre|250px]] || [[image:Anti_211232_PL1208.bmp |centre|250px]]
|-
|<CENTER> '''C-H Bond lengths (Å)''' </CENTER>|| [[image:C-h_bond_pl1208.bmp |centre|250px]] || [[image:Anti_2-_C-H_BOND_PL1208.bmp|centre|250px]]
|-
|<CENTER> '''C-C-C Bond angles (<sup>o</sup>)''' </CENTER>|| [[image:C-c_bond_pl1208.bmp |centre|250px]] || [[image:C-c_bond_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''H-C-H Bond angles(<sup>o</sup>)''' </CENTER>|| [[image:C-h_angles_pl1208_1.bmp |centre|250px]] || [[image:C-h_angles_pl1208_2.bmp |centre|250px]]
|-
|<CENTER> '''C-C-C-C Dihedral Angle(<sup>o</sup>)''' </CENTER>||<CENTER> '''179.99 deg''' </CENTER> || <CENTER> '''180 deg''' </CENTER>
|-
|<CENTER> '''Job Run time''' </CENTER>|| <CENTER> '''56.0 seconds''' </CENTER> || <CENTER> '''7 mins 39.4 seconds''' </CENTER>
|}

There is no change in the point group symmetry via both methods of optimisations. However, the corresponding bond lengths
show slight changes upon the DFT optimisation. This shows that although the HF method of optimisation is not the most accurate representation some credit can be given to that method of optimisation as there is minimal differences between the bond lengths as seen in table 4.

Comparisons of the C=C bond lengths; they are very similar to their counterparts. But on average, the C=C bond lengths in the DFT optimised structure is larger than the HF optimised structure. As for the C-H bond lengths, a similar trend is seen where by the DFT optimised structure has a larger bond length as compared to the corresponding bond lengths in the HF optimised conformer. Analysis of the bond angles and the dihedral angles would show that they remain relatively constant across both optimisations which further rationalises the reason for both conformations to have the same point group symmetry.

It would also show the importance of carrying out a rough optimisation before carrying out the more accurate optimisation since the rough optimisation brings the molecule to the lowest energy conformation. This allows the accurate optimisation to only further improve on it; by which reducing the computational time and resources.

=== Vibrational Frequency Analysis of the Reactants and Products ===

The vibrational frequency analysis carried out is useful as it determines if the optimisation has been successful in attaining the lowest energy configuration of the conformer. This can be determined via two ways that have been annotated as follows.

* The largest low frequency should always be almost 1 order of magnitude larger than the lowest normal vibration given. This can be attained from the .log file; thus indicating that the ground state has been attained instead of the transition state.

* The other way is to check for a negative frequencies in the output where by if there is one negative frequency it is indicative that there is a maximum turning point instead of a minimum and is that of the transition state instead of the group state. It there is more than one negative value then the optimisation has failed to reach completion. If all are positive then the minimum point has been achieved :)

Using the two methods as a basis of comparison, both methods of optimisation (HF and DFT) will be compared to determine if the optimisation had reached completion. This will help determine which one will be more reliable for further vibrational analysis.

{|align="center"
| [[Image:lowfreq_failed_pl1208.bmp|thumb|500px|'''Diagram 7 - Annotated Diagram indicative of incomplete Optimisation under HF/3-21G'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41536_pl1208.out |Failed Freq Analysis HF/3-21G]]
|}

Diagram 7 shows the extract from the .log file of anti-2 conformer via the HF/3-21G optimisation + Freq method. The presence of the negative frequencies (highlighted) is indicative that the minimum point was not reached. Optimisation did not reach a completion under the HF method.

{|align="center"
| [[Image:Summary_adPL1208.JPG|thumb| 520px |'''Diagram 8 - Annotated Diagram indicative of zero negative frequencies under DFT/6-31G
(d)'''|centre]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41590.out |Freq Analysis DFT/6-31G(d)] ]
|}

From the .log file attained via the DFT optimsation, it is observed that there are no negative frequencies. This is indicative that the optimisation had resulted in the ground state configuration to be achieved for the anti-2 conformation (C<sub>i</sub>). Thus, the optimisation under the DFT method has been successful and further analysis can be extracted from that frequency file. Since the HF method led to an incomplete optimisation the analysis of that log file will be left out.

==== Thermochemistry Analysis via DFT/6-31G (d) ====

Analysis of the Log file under the DFT method was carried out where by under the <u>vibrational temperatures</u> section there was a list of energies. There were 4 distinct rows which are denoted as follows with their significance.

* the '''sum of electronic and zero-point energies''' - the potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)

* the '''sum of electronic and thermal energies''' - the energy at 298.15 K and 1 atm of pressure which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>)

* the '''sum of electronic and thermal enthalpies''' - an additional correction for RT (H = E + RT) which is particularly important when looking at dissociation reactions

* the '''sum of electronic and thermal free energies''' - the entropic contribution to the free energy (G = H - TS)

Analysis of the .log file revealed that the initial calculations were carried out at 298.15 K (room temp). These calculations were taken into account and a further calculation was done by re-running the data at an temperature of 0 K. This was done by re-running the frequency optimisation with the same basis set and method as the previous run with the additional keywords ‘Freq=ReadIsotopes’. At the end of the input file, a manual addition of <u>(blank line) 0.0 1.0</u> was added to indicate the 0 K and 1 atm pressure. The energies were tabulated as found below in table 5.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5- Showing the different sum of energies '''
| Energies (a.u.) || '''298.15 K/a.u.''' || '''0 K/a.u.'''
|-
| '''Sum of Electronic and Zero-point Energies''' || -234.469154 (-234.4692) || -234.4499
|-
| '''Sum of Electronic and Thermal Energies''' || -234.461788 (-234.4618) || -234.4499
|-
| '''Sum of Electronic and Thermal Enthalpies''' || -234.460886 (-234.4609) || -234.4499
|-
| '''Sum of Electronic and Thermal Free Energies''' || -234.500704 (-234.5007) || -234.4499
|-
| Brackets refer to the rounding off to 4 decimal places
|}

Switching from the 298K to 0K operating temperature, allowed the determination of the zero point energy (purely electronic energy). Although this is less useful since all experiments are done at room temperature, it would allow the following observation to be made via comparisons. Also it would allow the zero point energy to be determined and the entropic contribution to be determined at above 0K temperatures.

The results show that as the temperature decreases, there is a decrease in the magnitude for all 4 specific energies. This signifies that the stability of the conformer also decreases as the temperature goes down. This can be rationalised as the reduction in the entropic contribution to the overall Gibbs free energy which results in less stable energetics. All the values at 0K are the same as they are indicative of the zero thermal contribution. Thus, the value at 0K is representative solely of the electronic and zero point energies of the system. This could rationalise why the energy level at 0K is higher (less stable) since more energy will be required to overcome the higher vibrational,rotational and translation energies due to the absence of entropic contributions. Also, it can be noted that the entropic term is expected to be temperature dependent where by with a higher temperature a higher entropic contribution is expected.

== Optimizing the "Chair" Transition Structures ==

From the earlier sections, the anti-Conformer with the C<sub>i</sub> symmetry was determined to be the reactive species and was then optimised and analysed. This section aims to determine the transition state for the earlier discussed cope rearrangement reaction for the 1,5 hexadiene molecule. In the introduction it was discussed that there are two possible transition states that could arise (chair and boat). Two different techniques will be applied towards determining the transition state and a cross analysis of the two will be carried out.

=== Method 1 : Attaining the Chair Transition state via the Force Constant Matrix (Hessian) ===

The force constant matrix works on the principal that if the starting geometry of the molecule fragment is close enough to that expected of the transition state, then the Hessian method of computation can be used as the first step. The matrix if successful will determine the direction of a negative curvation in the potential energy surface (PES). This would be the expected transition state that is to be determined.

==== Process Under taken towards Hessian Implementation ====

* The Allyl Fragments of CH<sub>2</sub>CHCH<sub>2</sub> was drawn in Gaussview 5 and was optimised using the HF/3-21G method and basis set to give the lowest energy conformation. [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7988 |initial optimisation]]

* The optimised structure was then copied twice into a new mol group where the two fragments were re-orientated such that they eclipsed each other and had an inter fragment distance (terminal-terminal) of 2.2 Å. This was shown in the below table.
{| class="wikitable" border="1" align="center"
|+ ''' Table 6 -Schematic and Arranged allyl Fragments'''
! <CENTER>Schematic of Transition State</CENTER> !! <CENTER>Calculated Transition State</CENTER>
|-
| [[image: Schematic_pl1208.bmp|250px|centre]] || [[image: calculated chair transition state.jpeg|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>Arranged fragments into chair Ts State</text><color>black</color></jmolAppletButton></jmol></CENTER>
|}

* The next step was to click the '''Gaussian''' menu, under '''Calculate''' clicked the '''Job Type''' tab. The '''Opt+Freq'''was selected and then change to '''Optimization'''''' to a ''''''Minimum to Optimization to a TS (Berny)''''''. Subsequently, in the Additional keyword box at the bottom, '''Opt=NoEigen''' was typed . The keywords are implaced to stop the calculation from crashing if more than one imaginary frequency is detected during the optimization. This can often happen if the guess transition structure is not good enough. The Hessian implementation is successful only if the geometry of the 'guessed' state is close to the actual transition state.

====Analysis of Results Via the Hessian Implementation ====

The successful optimisation to the transition state, was vibrationally confirmed via the presence of a negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7 - Summarising the total energy and vibrational frequency for the Force Matrix method'''
| '''Parameter''' || '''Optimised Through the Hessian Optimisation'''
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_berney.JPG |300px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -817.95 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7989 |Hessian Optimisation Log file]]
|}
From table 7, it is evident that the transition state has been reached as the log file shows a single negative frequency. As we have initially suggested the [3,3] mechanism of the cope rearrangement (Diagram 2), the animation of the -818 cm<sup>-1</sup> can be put into context.

The animation shows that one end of the terminal carbon bonds move closer to each other at the expense of the other-end of carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter shows a representation of bond breaking which are anticipated in the transition state for the hexadiene.

=== Method 2 : Attaining the Chair Transition state via the Frozen coordinate Method ===

This method of finding the transition state via frozen coordinates is the other method previously discussed. The main difference between the two methods is that in the frozen coordinate method of bond making and bond breaking distances have been set to a constant of 2.2 Å. The implementation of the '''Redundant coordinate editor''' should result in the fully optimised transition state with respect to differences in the bond lengths. This method is ideal in the event the 'guessed' transition state geometry is very different from the actual transition state geometry which then results in the Hessian method to fail. This is because, the curvature of the surface is significantly different at points far removed from the transition structure.

==== Process Under taken towards Frozen coordinate Implementation ====

The distances between the two fragments were fixed to 2.2 Å via the Gaussview 5.0 Redundant Co-Ordinate Editor. Once this was set, the optimisation was again carried out via the HF/3-21G method and basis set.

After the initial optimisation, the terminal carbon fragment's distance were measured to ensure the distance was still 2.2 Å (it was !). This was indicative that the bonds were still frozen.

Following this the lengths between the two ally fragments of the transition state were 'unfrozen' and optimised via the Editor again. This time the '''Optimisation + Frequency optimisation''' was carried out. A change in the previously frozen C-C distance is expected to change.

====Analysis of Results Via the frozen coordinate Implementation ====

Successful optimisation to the transition state was then carried out and the frequency analysis was carried out. The maximum turning point is vibrationally confirmed via the presence of a single negative vibration (-818 cm<sup>-1</sup>). This is indicative of the maximum turning point on a reaction profile which fits that of a transition state.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8 - Summarising the total energy and vibrational frequency for the Frozen Method'''
| '''Parameter''' || '''Optimised Through the Redundant editor Optimisation'''
|-
| '''Pre Opt+Freq Inter-frag distance''' || [[Image:Pre_distance_pl1208_2.JPG |200px|centre]]
|-
| '''Post Opt+Freq Inter-frag distance''' || [[Image:Post_distance_pl1208.JPG |200px|centre]]
|-
| '''Total Energy Summary''' || [[Image:Summary_pl1208_frozen.JPG |200px|centre]]
|-
| '''Total Energy (a.u.) ''' ||-231.6193
|-
| '''Imaginary Frequency (cm<sup>-1</sup>)''' || -818.043 (-818)
|-
| '''Animation of Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|-
| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Frozen Method Optimisation Log file]]
|}

It is evident that the transition state has been reached as the log file shows a single negative frequency (Table 8). This is indicative of the successful optimisation towards the transition state. As we know the [3,3] mechanism of the cope rearrangement and drew the schematic earlier of that transition state, the animation of the -818 cm<sup>-1</sup> can be put into context similar to how it was done via method 1.

The animation shows that one end of the terminal carbon bonds are moving closer to each other at the expense of the other end carbon atoms to be moving further away from each other. This complements the conclusion that the transition state has been achieved as the former shows the bond formation and the latter is a representation of bond breaking which are anticipated in the transition state for the hexadiene.

Comparing the initially set distances between the terminal allyl fragments would show that the fragments were brought closer together to each other from the initial distance of 2.20 Å to a much closer distance to 2.02 Å. Analysis of the gradient, shows a value of 0.00005 which is much smaller than 0.001. This was also indicative that the optimisation had reached a completion.

==== Comparison of Structures attained via Both Methods ====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9 - Comparing Specific Parameters via the Transition state by both methods '''
| Parameter || '''Optimised Through a TS (Berny)''' || '''Optimised Through a Frozen Coordinate Method'''
|-
| '''Terminal C-C bond length (Å)''' || 2.02 ||2.02
|-
|'''Energy/a.u.''' || -231.6193 || -231.6193
|-
| '''Imaginary Frequency/cm<sup>-1</sup>''' || -817.95 (-818) ||-818.043 (-818)
|-
| '''Animation''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.35;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41598_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol>
|}

The detailed comparison shows that both the methods result in the same chair transition to be formed with the same inter-fragment bond distance upon the optimisation. As the difference in the total energy is more than 4 decimal differences, it has been assume to be of a negligible difference. As both the vibrational frequency result in a single negative wavenumber that are of highly similar magnitudes they are shown via the animal to also give the same type of vibrational stretch to be observed (Table 9).

Thus, it can be concluded that the 'guessed' conformation was significantly close to the geometry of the actual chair transition state which allowed the Force matrix method of optimisation to be successful as it has been compared against the 'Frozen' method of optimisation which is not as 'geometry-sensitive' as the force matrix method and both result in the same values.

=== Intrinsic Reaction Coordinate (IRC) Method For the Chair Transition State ===

As it is almost impossible to predict which conformer the reaction pathways from the transition structures will result in, the IRC method serves as a medium to follow the reaction mechanism. This is defined as the monitoring of the minimum energy path taken from the transition state to the local minimum on the potential energy surface (towards product). This monitoring is done by generating a series of points that represent the small geometry changes in the direction where the gradient on the potential energy surface is the largest (steepest). The greater the number of points the more accurately small geometric changes will be accounted for.

When the calculation does not reach the minimum geometry (anticipated), three different routes towards geometry optimisation will be considered and subsequently compared with each other to compare and contrast the respective methods.

* The first method upon incomplete minimisation would result in taking the last point on the IRC and run a normal minimization via the HF/3-21G method and basis set to achieve the minimisation. This approach is advantageous since it is the fastest, but if you are not close enough to a local minimum, you may end up in the wrong minimum

* The second method would be to restart the IRC and specify a larger number of points until it reaches a minimum point geometry is achieved.is more reliable but if too many points are needed, could also veer off in the wrong direction after a while and end up at the wrong structure.

* The final method would be to redo the IRC specifying that you want to compute the force constants at every step. This is the most reliable but also the most expensive and is not always feasible for large systems

==== Process for IRC calculations ====

To allow a sound basis of comparison, all the optimisation efforts will be carried out by the use of the HF/3-21G method and basis set. As the transition is expected to subtend between the reactant and products, the movement from transition state to products is same as the movement from transition state to reactants. Thus, a symmetric reaction profile only needs the calculations to be done for the foreward reaction since the foreward calculations of geometries and energies will be equal to the backwards reaction in a symmetric reaction profile.

The .ckh file for the optimized transition chair structure (TS Berny) under the redundant coordinate method was opened via Gaussview 5. For the first step, the '''IRC job type''' was selected and the number of points along the IRC was set to 25. The job were then submitted to scan and analysed in the table below.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10 - Summarising IRC carried out at n=25 points'''
! Diagram for 1st point !! Diagram for last point!! Summary!! IRC Pathway!! IRC Gradient
|-
| [[Image:First_n=25.JPG|thumb|200px]]|| [[Image:FinalN=25.JPG|thumb|200px]]|| [[Image:SummaryN=25.JPG|thumb|200px]]|| [[Image:Total_energyN=25.JPG|thumb|200x300px]]||[[Image:Rms_gradientN=25.JPG|thumb|200x300px]]
|-
| colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41651.out |IRC n=25 log File]] || colspan="2"| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41651.fchk |IRC n=25 fchk File]]
|}

The RMS gradient is 0.00124 which was well above the 0.001 (table 10). This was the first indication that the calculation had not reached a completion yet. Thus, the above energy was not a representation of the lowest energy of the transition state. Furthermore, when this was compared with the conformers in appendix 1, it is evident that the lowest energy conformation was not reached since the energy of the optimisation was -231.6860 a.u. but the conformers in the appendix have more negative energies (more stable). This is indicative of a even lower energy conformation being possible.

A similar IRC calculation was then carried out for same transition state but instead with n=50 points. This gave a total energy of -231.6880 and an gradient of 0.00192. This was very similar to the above table and was indicative that the minimisation was not complete even with twice the number of points. Thus the three different methods were then considered.

===== <u>Method 1</u> =====

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11 - Summarising the Last point Optimisation Via HF/3-21G'''
! Summary !! Diagram of Optimised Transition state !! RMS Gradient and total energy
|-
| [[Image:SummaryLASTOPT_PL1208.JPG|thumb|300px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.27</script> <uploadedFileContents>Log_41600_pl1208.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Total_energy_pl1208_edfw.JPG|thumb|400px ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41600.out |HF/3-21G Optimisation log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41600.fchk |HF/3-21G Optimisation fchk File]]
|}

[[Image:Reference_APPENDIX2.JPG|thumb|700px|'''Diagram 9 - Gauche 2 conformer extracted from appendix 2 for comparison'''|centre ]]

From table 10, it is observed that the RMS gradient has been reduced significantly to a value of 0.00000219 which is well below the threshold of 0.001 for the gradient. Thus, it can be seen from the gradient diagram that there is a horizontal line close to zero obtained. This is indicative that the optimisation to the lowest geometry for the transition state has been achieved. When the conformer was symmetrised under the Gaussview option, it resulted in a C2 symmetry point group. The minimised energy was given as -231.691667 a.u. and had an dihedral angel of -64.2 for the centre 4 Carbons. This suggested a gauche type conformation as seen earlier and was further confirmed by comparing with the Gauche 2 values (Diagram 9). This shows that the optimisation was successful in attaining the Gauche 2 conformation for the transition state.

===== Method 3 =====

This being the most reliable method and having a small transition state system allowed it to be highly feasible to be carried out. This is because having every force constant calculated for a large molecule would take up a large amount of computational resource and time. But here the molecule being rather small would allow it to be readily carried out.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12 - Summarising the Constant force constant (Method 3)'''
! Summary !! Diagram of Optimised Transition state !!First point !! last point !!
|-
| [[Image:Summary_pl1208_mthd3.JPG| 200px ]] || <jmol><jmolApplet><color>BLACK</color> <size>200</size> <script>zoom 100;frame 1.48</script> <uploadedFileContents>Log_41710_2.txt</uploadedFileContents> </jmolApplet>
</jmol>||[[Image:Firstpl1208sd.JPG| 200px ]] ||[[Image:Finalpl1208sdf.JPG | 200px ]]
|}

{| class="wikitable" border="1" align=center style="text-align:center"
! RMS Gradient !! Total energy
|-
| [[Image:Graident.JPG|thumb|450px|centre ]] ||[[Image:Total_energyzcsc.JPG|thumb|450px|centre ]]
|-
| [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Log_41710.out |Constant Force IRC .log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Checkpoint_41710.fchk |Constant Force IRC .fchk File]]
|}

From table 10, the total energy of the conformation is given as -231.691664 a.u. The gradient of the minimisation gave a value of 0.00001 which is smaller than 0.001 and is indicative of completion of the minimisation. The dihedral angle calculated across the 4 central carbons is given as -64.6 deg. When this total energy is compared against the appendix A, the value is very close to the energy of the Gauche conformer 2. Also, when this is compared with the single force constant calculation, the gradient yielded a better result and is more accurate to the minimal value of the gauche 2 conformer.

==== Comparing Method 3 (Constant Force Calculation against Single Force Calculation ====

{| align=center style="text-align:center"
| [[Image:Total_energyzcsc.JPG|thumb|450px|'''Diagram 10 - Total energy via constant force (Method 3)'''|centre]] ||[[Image:Total_energyN=25.JPG|thumb|400px| '''Diagram 11 - Total energy via single force (Method 1)'''|centre]]
|-
|[[Image:First_image_pl1208_asdas.JPG|thumb|200px|'''Diagram 12 - Final point via Method 1'''|centre]] ||[[Image:Final_image_pl1208_sdfse.JPG|thumb|200px| '''Diagram 13 - Final point via Method 3'''|centre]]
|-
| [[Image:Graident_PL1208_)WER.JPG|thumb|550px|'''Diagram 14 - RMS Gradient via constant force calculation''' ]] || [[Image:Rms_gradient_pl1208_dsfe.JPG|thumb|550px|'''Diagram 15 - RMS Gradient via single force calculation''' ]]
|}

From the comparisons made, it is seen that the method that utilised the single force constant calculation (Method 1) only needed 24 steps to 'fully converge' before the 50th point was reached. This was because there was a loop between the 24th and 25th point that resulted in the early termination.

Comparing the total energies (Diagram 10 and 11) would show that the energy of the system decreases from the first point towards the final point (n=50). The closer the RMS gradient value is to zero would be indicative that the energy of the system gets closer to the minimum value and would start to plateau. The more it reaches a plateau, the closer the gradient value gets to zero. This is distinctly seen when method 3 is compared against method 1 where by method 3 would show the 'plateau-ing' of the total energy as compared to method 1 which has yet to plateau.

Comparing the RMS gradients, the same effect is seen where by the Method 3 (Diagram 14) brings about a much more closer to zero value as compared to the single force calculation (Diagram 15) which is still significantly above the 0.001 a.u. value.

From this it is seen that the problem of determining the conformer that leads to transition state is no more. The IRC method of calculation allows the energy of the transition state to be determined as it would be shown to be that of the Gauche 2 conformer. This has been proven in both the sections that the final energy via Method 3 and Method 1 that the final energy corresponds well to -231.6917 a.u. (Diagram 9).

However, there is a slight amount of contradiction that has to be considered. The Gauche 2 has an energy level of -231.69167 a.u. which was deemed by IRC to be the lowest minimised conformation. But analysis of the Appendix A would show that the Gauche 3 due to the orbital stabilisations has the even lower energy configuration of -231.69266 a.u. . This would show that maybe more points are needed for the IRC caluclation and possibly the calculation of the force constants should be constant as well. Thus there has to be a balance between the number of points along the IRC as well as calculation of force constants. This has to be a balance as more accurate results would mean higher computational resources being needed.

== Optimizing the "Boat" Transition Structures ==

Upon optimising the Chair like transition state and determining which conformer contributes towards the formation. The other transition will now be considered. This transition state will be utilising a different technique known as the QST2 calculation. For the optimisation of the boat transition state, both the reactant and products are defined. By explicitly defining the reactant and product structures, the Gaussian 03 is to be used to interpolate between the two states. This is utilised to determine the transition state between the two 'extremes'.

=== Process for the optimisation via QST2 ===

The first part of the reaction involved the sequential numbering of the carbon fragment. This was to ensure that the reactants will be directly mapped onto the products upon the [3,3] rearrangement. Thus the molecule in the Anti-Ci conformation from a previous .chk file was opened. Upon opening, a second window was opened and a Copy the optimized reactant molecule was added into the new window. To allow the schematic carbon labelling to correspond with the Molgroups in Gaussview, and for the reactant numbering to correspond to that of the product's, the Edit tab was opened to alter the atom labelling (Diagram x).

{| class="wikitable" border="1" align="center"
|+ '''Table - 13 Representing the Carbon labelled Reactant (1) and Product (2) for QST2 calculations'''
! Schematic representation !! Gaussview Diagram
|-
| [[Image:Schemiatic.bmp|thumb|400px|Schematic Representation]] || [[Image:Process_1_pl1208_boat_2.JPG|thumb|400px|Re labelled Reactant (1) and Product (2)]]
|}

The file will then be set up to be run a TS (QST2) file via the Optimisation + Frequency option using the HF/3-21G method and basis set. However, as anticipated in the script, the calculation had failed. It resulted in the formation of a chair-like transition state instead of the boat expected. As rationalised by the script, the calculation linearly interpolated between the two structures, translated the top allyl fragment and did not even consider the possibility of a rotation around the central bonds.

Instead the initial geometry file was opened and a further change to the angles was carried out to make it more 'like' a boat orientation. This was done by changing the dihedral of the centre 4 carbons to 0 deg and the inside C-C-C bond to 100 degrees (Diagram x)

[[Image:Process_2_pl1208asd.JPG|thumb|400px| '''Diagram 16 - showing the further modified Bond angles to be more boat like''' |centre]]

The new-geometry of the input was then resent for optimisation+Frequency via the QST2 mode and the HF/3-21G basis set.

=== Analysis of Results for Boat Optimisation via QST2 method ===

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table - 14 Summarising Post Optimisation to Boat Transition state '''
| ''' Boat Optimised Using QST2''' ||'''Terminal C-C bond length/Å'''
|-
| [[Image:Boat_pl1208sdfs.JPG|300px|centre]] <CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>Successful Boat Transition State</text></jmolAppletButton></jmol></CENTER>|| [[Image:Terminal_cc_pl1208.JPG|thumb|300px|'''Terminal C-C bond length = 2.14'''|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Energy/a.u.''' || '''RMS Gradient and Total Energy'''
|-
| [[Image:Summary_pl1208_wgf.JPG|thumb|300px|'''Summary from optimisation/ Total Energy = -231.60280'''|centre ]] || [[Image:Gradient_pl1208_rth.JPG|thumb|400px|RMS gradient|centre ]]|[[Image:Energy_pl1208sdfds.JPG|thumb|400px|Total Energy|centre ]]
|}

<br />

{| class="wikitable" border="1" align=center style="text-align:center"
| '''Imaginary Frequency/cm<sup>-1</sup>''' || '''Animation for Imaginary Frequency'''
|-
| [[Image:Vibratinoa.JPG|thumb|400px|Showing 1 single negative Frequency vibration @ -840|centre ]] || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| Source File || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |D-space Link to Optimisation]]
|}

It can be seen that the slight change in the dihedral angle and inner c-c-c bond angles allowed the gaussian software to successfully optimise the geometry towards that of the boat transition state. Analysis of the Summary table above shows that the total energy was slight less stable than that attained via the Chair transition state optimisation. This can be rationalised due to the two CH2 groups being on the same side of the plane would result in a greater extent of steric influence that brings about a slightly greater destabilisation effect relative to the chair transition state which has an optimised energy via HF/3-21G of -231.69 a.u. Thus, being more less negative would make the boat conformation (-231.60 a.u.) the less stable transition state. This would further explain why the C-C bond has a much longer bond length of 2.14Å as compared to the chair transition state which has 2.02Å. The shorter the bond length, the stronger and more stable the bond thus the more negative total energy for the chair relative to the boat.

The Optimisation efforts were deeemed to have reached completion as seen from the RMS gradient to be very much smaller than 0.001 a.u. Thus the convergence was evident. This was further reconfirmed from the Vibrational frequency log file which was indicative of a single negative frequency that was indicative of a maximum turning point. This was indicative that the optimisation have progressed successfully towards the transition state.

The imaginary frequency attained via the chair transition state optimisation was -818 cm<sup>-1</sup>. But from the table above it is seen that the imaginary frequency is of a slightly higher magnitude of -840 cm<sup>-1</sup>. The type of vibrations seen is very similar to that of the chair stretching vibrations. Here, it is showing the C-C bond formation at one end and at the other terminal end it is the C-C bond breaking. These are represented by the shortening and lengthening of the bonds respectively.

Thus it can be seen that bond the Ts(BERNY) method and the Ts(QST2) method are able to bring about relatively accurate optimisation to the transition state. This comparison can be made since both had used the same method (HF) and the same basis set (3-21G). But as seen the Ts(BERNY) method was a much less tedious method as compared to the QST2 method. This is because alot of 'pre-calculations' had to be done for the latter such as the atom re-numbering to ensure the reactants map onto the products if not the extrapolation would fail via the QST2. For this hexadiene the re-numbering was done relatively readily but if a much larger reactant was to be done the re-numbering of the atoms will prove to be a challenge which would make the Ts (berny) a much more prefered method of calculating.

== Determining the Activation Energies for the [3,3] Cope rearrangement reaction for 1,5-Hexadiene ==

From previous modules (1,2), and the initial sections it is very evident that the HF method and the 3-21G basis set are not the most accurate way of determining the energy level of a conformer. Thus to be able to accurately determine the transition stay energy level and hence the activation energy (Transition state energy - reactants energy level) would be more accurately determined. Thus a better approximation to the energy level is expected to be attained via the use of the Density Functional Theory method instead (DFT) with the Larger more accurate 6-31G(d) basis set instead. As it is a different method and basis set, significant changes in both the chair and boat transition states are anticipated.

=== DFT Optimisation for the Chair Transition State compared against the HF method ===

The table below shows the comparison between the HF method against the DFT where the latter uses the more accurate basis set for the optimisation process. Using the chair transition state initially optimised by the HF/3-21G, the DFT method was used to further optimised the chair transition state.

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>HF/3-21G </CENTER> !!<CENTER>DFT B3LYP/6-31G (d) </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Post_distance_pl1208.JPG|300px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>arranged_allyl_pl1208.mol</uploadedFileContents><text>HF optimised Chair Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagram_pl1208_activationchiar.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Log_41743.mol</uploadedFileContents><text>DFT optimised Chair Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.02 </CENTER>||<CENTER> 1.97 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 120.49<sup>o</sup> </CENTER>||<CENTER> 119.95<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> 68.8<sup>o</sup> </CENTER>||<CENTER> 68.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -818 </CENTER>||<CENTER> -566 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.25;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41599_pl1208.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41743_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_frozen.JPG |thumb|450px| '''Total Energy = -231.6193 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208_iuyt.JPG |thumb|450px| '''Total Energy = -234.5570 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |D-Space link to DFT/6-31 G(d) chair optimisation ]]
|}

As anticipated, there is a significant different in the total energy of the transition state. The more accurate basis set results in a more stable energy representation of -234.6 a.u. as compared to the HF method which gives the energy to be -231.6 a.u. Similarly with the DFT a smaller inter-allyl fragment distance is observed where there is a reduction by 0.05 Å. The shorter the distance the greater the bond strength which also usually adds up to a more stable (more negative) total energy. However, there is a slight increase in the C-C bond length within the fragements which could be due to the slightly smaller bond angle seen as well.

The most distinct difference is seen in the Imaginary vibrational wave number as seen in the table above. The larger the magnitude of the wavenumber is an indication of a higher energy. Thus the DFT method has a smaller magnitude for the wavenumber since it is the more stable optimisation (more negative). This could be rationalised based on the fragments being closer. This is because, when closer there will be more attraction which would bring better stabilisation and so the vibrations should be lesser which is indicative by a smaller magnitude of -556 relative to the -818 by the HF method.

=== DFT Optimisation for the Boat Transition State compared against the HF method ===

Again in the table below the optimisation for the boat geometry was carried out similar to that done previously

{| class="wikitable" border="1" align="center"
|- Table Comparing the Various parameters for the two methods of optimisation
! <CENTER>Parameter </CENTER> !!<CENTER>'''HF/3-21G''' </CENTER> !!<CENTER>'''DFT B3LYP/6-31G (d)''' </CENTER>
|-
|<CENTER> '''Structure''' </CENTER>|| [[image: Hk_diagram.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41734.mol</uploadedFileContents><text>HF optimised Boat Ts</text></jmolAppletButton></jmol></CENTER> || [[image: Diagrma_pl1208.JPG|250px|centre]]<CENTER><jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41744.mol</uploadedFileContents><text>DFT optimised Boat Ts</text></jmolAppletButton></jmol></CENTER>
|-
|<CENTER> '''Inter-fragment Distance (Å)''' </CENTER>||<CENTER> 2.14 </CENTER>||<CENTER> 2.21 </CENTER>
|-
|<CENTER> '''C-C bond length (Å)''' </CENTER>||<CENTER> 1.39 </CENTER>||<CENTER> 1.41 </CENTER>
|-
|<CENTER> '''C-C-C Bond Angle''' </CENTER>||<CENTER> 121.684<sup>o</sup> </CENTER>||<CENTER> 122.270<sup>o</sup> </CENTER>
|-
|<CENTER> '''C1-C2-C3-C4 Dihedral angle''' </CENTER>||<CENTER> -64.6<sup>o</sup> </CENTER>||<CENTER> -64.1<sup>o</sup> </CENTER>
|-
|<CENTER> '''Wavenumber of Imaginary Vibration (cm<sup>-1</sup>)''' </CENTER>||<CENTER> -840 </CENTER>||<CENTER> -530 </CENTER>
|-
|<CENTER> '''Animation of Imaginary Vibration ''' </CENTER>||<CENTER> <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.9;vectors 4;vectors scale 2.0;color vectors purple; vibration 3;
</script><uploadedFileContents>Log_41739_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>||<CENTER><jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>300</size>
<script>zoom 100;frame 1.13;vectors 4;vectors scale 2.0;color vectors yellow; vibration 3;
</script><uploadedFileContents>Log_41744_2.txt</uploadedFileContents></jmolApplet></jmol> </CENTER>
|-
|<CENTER> '''Energy of Chair Transition State (amu)''' </CENTER>||<CENTER>[[Image:Summary_pl1208_wgf.JPG |thumb|300px| '''Total Energy = -231.603 a.u.'''| centre]]</CENTER>||<CENTER> [[Image:Summary_pl1208webds.JPG |thumb|300px| '''Total Energy = -234.543 a.u.'''| centre]]</CENTER>
|-
|Source File|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |D-Space link to DFT/6-31 G(d) boat optimisation ]]
|}

From the table it can be seen that there is minimal changes in the geometry which is indicative that the HF optimisations were relatively accurate and the DFT was just a further accuracy enhancement. The same can be said by looking at the bond lengths which did not change by much (0.02Å). However, looking at the energies there is again a very large difference relative to each other but is mainly due to the different method being used to calcualte. Thus they are not really on the same basis to be compared. Thus the other parameters will be looked at. But it must be said that as expected the DFT did result in the total energy to be resulting in a slightly more negative one.

Similar to the chair, the biggest change is seen in the vibrational frequency of the imaginary mode. However the same rationalised used for interfragment distance against wavenumber completely fails here. Through the optimisation here the fragment distance increases by 0.07Å. Thus by the previous section's rationale with a larger distance there should be lesser interaction and the stretches would be of a much higher magnitude. But the opposite is seen where by the vibrational magnitude is similar to that seen in the chair. This concludes that the initially made rationalisation is wrong and a correct one is yet to be determined.

=== Calculating the Activation Energies ===

As the reoptimisation of the chair and boat structures using the B3LYP/6-31G (d) basis set was carried out, the frequency calculations were carried out for both the HF/3-21G and the B3LYP/6-31G(d) optimised structures.

{| border="1" cellspacing="1" cellpadding="10" align="center"
|+ '''Table 15 - summarising the respective energies for both transition states via different basis sets and temperature'''
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
| width="125" align="center" | '''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
| width="125" align="center" | '''Electronic energy (amu)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (amu)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (amu)'''
|-
| ''' '''
| ''' '''
| 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" | -231.6193
| align="center" |-231.46640
| align="center" |-231.46670
| align="center" |-234.55700
| align="center" |-234.42113
| align="center" | -234.4090
|-
| '''Boat TS'''
| align="center" |-231.60280
| align="center" |-231.45093
| align="center" |-231.44530
| align="center" |-234.54309
| align="center" |-234.40234
| align="center" |-234.39601
|-
| '''Reactant (''anti-2'')'''
| align="center" |-231.69254
| align="center" |-231.53954
| align="center" |-231.53241
| align="center" |-234.61171
| align="center" |-234.46897
| align="center" |-234.46186
|}

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7998 |Link to D-Space for HF3-21G CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7997 |Link to D-Space for HF3-21G Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7990 |Link to D-Space for HF3-21G Boat TS @ 298.15k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8001 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-7999 |Link to D-Space for B3LYP/6-31G (d) CHAIR TS @ 298.15k]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8002 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 0k ]]

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8000 |Link to D-Space for B3LYP/6-31G (d) Boat TS @ 298.15k ]]

The table above would be computing the specific energies from the two transition states with varying types of basis sets used as well as the different temperatures they are carried out at. When these values are compared with those provided it [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_2 | Appendix 2 ]]. It can be seen that they are very much alike with slight deviations from the 4th decimal place onwards. This would be deemed to be of a negligible deviation. It can be seen that using the larger basis set (6-31G(d)) would result in the energies to be significantly different. This could be rationalised to be due to the different method being used to calculated and cant be compared directly against each other as they are based on different approximations.

From the table it can be seen that as the temperature is increased the energies of the reactant and the transition states are closer. However, the transition state of the boat would still be higher than the energy of the chair transition state. This is indicative that the chair transition state being more stable would impose a smaller potential energy barrier towards product formation as compared to that of the boat. This will be more distinctly show in the table below.

{| class="wikitable" border="1" align="center"
!height=70px| Transition State !! Activation Energy at 0 K (kcal<sup>-1</sup>) [HF/3-21G] !! Activation Energy at 0 K (kcal<sup>-1</sup>) [DFT/6-31G(d)]!! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[HF/3-21G] !! Activation Energy at 298.15 K (kcal<sup>-1</sup>)[DFT/6-31G(d)]!! Experimental Activation Energy at 0 K (kcal<sup>-1</sup>)
|-
|width=175px height=70px|<CENTER> '''Chair''' </CENTER>||width=200px|<CENTER> 45.890 (46) </CENTER>||width=200px|<CENTER> 30.0 </CENTER> || width=200px|<CENTER> 41.23 (41) </CENTER>|| width=200px|<CENTER> 33.17 (33) </CENTER>|| width=200px|<CENTER> 33.5± 0.5<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|-
|height=70px|<CENTER> '''Boat''' </CENTER>||<CENTER> 55.6 (56) </CENTER>||<CENTER> 41.81 (42) </CENTER>|| width=200px|<CENTER> 54.66 (55) </CENTER>|| width=200px|<CENTER> 41.32 (41) </CENTER>|| width=200px|<CENTER> 44.7 ± 2.0<ref>M.J. Goldstein, M.S. Benzon, J. Am. Chem. Soc., 1972, 94, 7147. DOI:10.1021/ja00775a046</ref> </CENTER>
|}

The values in the table are calculated by individually determining the activation energies (E<sub>A</sub>) for both the transition states at the different temperatures. For the calculations at 0 K this was done by taking the difference between the '''(sum of the electronic and zero-point energies) of reactant anti2 and the chair/boat transition state''', multiplied by 627.509 kcal/mol. Similar for the calculations at 298.15 K was done by taking the difference between the '''(sum of the electronic and thermal energies) of reactant ''anti1'' and the chair/boat transition state''', multiplied by 627.509 kcal/mol.

[[Image:Graph2_pl1208.JPG|thumb|500px|Graph 2 - Comparing the DFT method values against literature|centre ]]

From the table and graph it can be seen that the activation energy as predicted for the Ts Boat is always higher than that for the Ts Chair. With increasing temperature, the difference still remains significantly large. This is because the chair conformation has the staggered structure which would have the minimum steric repulsion as the CH<sub>2</sub> ends are on opposite sides of the plane. But for the boat both the CH<sub>2</sub> ends are on the same side of the plane which results in increased steric repulsion due to the eclipsed conformation. This destabilises the transition state and thus would be of a much higher energy level than the Chair.

The values in the above table correspond well to the 'model' results provided in the Appendix 2. This shows that the calculations were successful. However when the results were compared with literature, it can be seen there is about a 35% deviation at 0K. This would be indicative that the computationally generated values were not optimised to most ideal transition state conformations. Thus were not of the most minimum energy level. Thus further methods and different, more accurate basis sets could be considered to get the calculated values to be as close to that of experimental. When that is achieved, it would be indicative of an even more effective optimisation. However, for that degree of accuracy alot more computer resources can be anticipated and will not be considered for now.

= Diels Alder Cycloaddition Reactions =

In the previous section, the [3,3] Cope rearrangement reaction was analysed and the activation energies was determined through analysis of the transition states. The same concepts will now be translated onto the pericyclic type reactions. These are defined as Diels-Alder reactions which involves the 4π diene system to have a concerted reaction with the 2π dienophile.

For this report, two types of cycloaddition reactions will be looked at, the first will be a Prototypical Reaction between Cis-Butene and ethylene where there are no secondary orbital effects to consider (Diagram 17). The second reaction is a more complex diels alder reaction which has the involvement of the substituents that bring about secondary orbital effects (Diagram 18).

{| align="center"
| [[Image:Prototypical_pl1208e.bmp|thumb|500px|'''Diagram - 17 Prototypical Reaction between Cis-Butene and ethylene''']] || [[Image:Substituted_complex_dsgfs.bmp|thumb|500px|'''Diagram - 18 Complex Diels Alder reaction with substituted diene and dienophile''']]
|}

Pericyclic reactions are expected to have the [4+2] type cycloaddition which is a concerted rearrangement reaction. This has the simultaneous π-bond breaking and σ-bond formation. Through this analysis, the bond breaking and bond formation is expected to be better understood via analysis of the frontier molecular orbitals.<ref>http://pubs.acs.org/doi/abs/10.1021/ja01474a023</ref>

The reactions occuring in the concerted stereospecific fashion can be an allowed or forbidden reactant; depending on the number of π electrons involved. The reaction will be allowed if the HOMO of one species can interact favourably with the LUMO on the other. <ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref> For this to occur there has to be a favourable molecular orbital overlap between the two MOs

== Cycloaddition reaction for the Prototypical Reaction (Cis-Butene and ethylene)==

The diene is a conjugated π-system consisting of 4 π-electrons while the ethene acts as the dienophile consisting of a 2 π-electron system. The reaction between 1,3-butadiene and ethene to give cyclohexene is described as a [4+2] cycloaddition reaction. This type of cycloaddition is also called a Diels-Alder reaction.

In a Diels-Alder reaction the 4 π-electron system is referred to as "the diene" and the 2 π-electron system as the "dieneophile". These terms are used in related [4+2] reaction systems even when the functional groups are not actually dienes or alkenes.<ref>http://www.meta-synthesis.com/webbook/49_pericyclic/pericyclic.html</ref>

=== HOMO-LUMO Analysis ===

The diene and the dienophile were optimised via the 'first round' of optimisation using the HF/3-21G method [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8005 |link to 3-21G optimisation for Butadiene]]. A more accurate optimisation was then carried out by the DFT/B3LYP method to ensure the minimal geometry was achieved [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8006 |link to B3LYP opt for Butadiene]] [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |link to B3LYP opt for Ethylene]]. Both the optimisations resulted in the final outcome structure to be planar and having the C<sub>2V</sub> point group symmetry. The same process of optimisation was then carried out for the ethylene as well. The table below would be summarising the total energy upon the optimisation. The RMS gradient for both reactants is well below the value of 0.001 (a.u.); indicative of the convergence to the present threshold (Table 16). Thus, it could be concluded that the optimisation was successful for both reactants.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 16 - Summarising the Optimisation for 1,3 Butadiene and Ethylene'''
! 1,3 Butadiene !! Ethylene
|-
| [[Image:SUMMARY_CORRECTION_PL1208.JPG|thumb|300px|'''Summary for Diene via B3LYP/6-31G(d)''']] || [[Image:Summary_pl1208_ethylene.JPG|thumb|300px|'''Summary for Dienophile via B3LYP/6-31G(d''')]]
|}

Using the DFT optimised file, a further molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 17 - Summarising the MO analysis for 1,3 Butadiene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''1,3-Butadiene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_-0.343.JPG|THUMB|HOMO of 1,3-Butadiene|200px]]||[[Image:Lumo_0.017_PL1`208.JPG|LUMO of Butadiene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.343||0.017
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41775.mol</uploadedFileContents><text>1,3-butadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Butadiene MO analysis]]
|}

From the table above it can be seen that the cis-Butadiene has the HOMO to be of an Anti-symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane, it has a symmetric distribution.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table - 18 Summarising the MO analysis for Ethylene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Ethylene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_ethylene.JPG|HOMO of ethylene|200px]]||[[Image:Lumo_pl1208_ethylene.JPG.JPG|LUMO of ethylene|200px]]
|-
|'''Energy of MO (a.u.)'''||-0.387||0.052
|-
|'''Symmetry'''||Symmetric||Antisymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41772.mol</uploadedFileContents><text>ethylene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8004 |D-space link to Ethylene MO analysis]]
|}

From the table above it can be seen that the ethylene would have the HOMO to be of a symmetric orbital distribution with respect to the reflection plane(σ<sub>v</sub>). For the Lumo with respect to the same plane it has a asymmetric distribution.

The HOMO of the cis-butadiene and LUMO of the Ethylene are both anti symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap here would be 0.395 (a.u.)

Similarly comparing the LUMO of the cis-butadiene and HOMO of the Ethylene are both symmetric which allows them to have a favourable orbital overlap. Thus orbital interactions between this pair can be anticipated. The HOMO-LUMO gap would be 0.404 (a.u.)

Where by the smaller HOMO-LUMO gap is expected to be more favoured as it would allow a greater splitting which brings about a greater extent of stabilisation.

==== Further Analysis and MO analysis ====

[[Image:MO_summary_pl1208_23.bmp|thumb|450px|'''Diagram 19 - HOMO-LUMO Orbital diagram with specific orbital energies'''|centre ]]

The diagram above would facilitate the rationalisation on why the reaction is allowed. The reaction happening with 2N+2 electrons where n=2 would be indicative of a thermodynamically occurring reaction. There will be a suprafacial type addition of the two fragments to form the cyclic compound. From the above diagram it is seen that the orbitals would be able to overlap as they are of the similar symmetry and the energy gap is not very big allowing effective orbital overlap to form the molecular orbitals of the cyclic ring. The HOMO-LUMO gap where butadiene is acting as the HOMO is of a smaller magnitude making that overlap more favoured as compared to the other HOMO-LUMO combination. Thus this fits with theory where the butadiene functions as the diene and the ethylene functions as the dienophile.

=== Transition state Analysis ===

To deviate from normality, my approach to the optimisation of the transition state, would be via the QST2 method which proved reliable but tedious in the earlier sections will be used and also by the Frozen Coordinate method. Similar to processes followed before, the reactants and the products were drawn as follows.

The terminal carbon carbon bond length (not connected for the reactants but connected for the products) were set at 2.10 Å. The transition state is expected to have the diene approaching from above or below the plane in an envelope type fashion. This is becuase, theoretically, it would allow a maximisation of the orbital overlap between the ethylene π orbitals (π/π*) and the π system of 1,3-butadiene. Thus the ethylene is drawn slightly above the plane of the butadiene.

[[Image:Arraangement_pl1208_qst2.JPG|thumb|500px| '''Diagram - 20 Arrangement of Reactants (1) and Products (2) with atomic re-numbering'''|centre]]

==== Method 1 : QST2 via HF/3-21G and DFT/B3LYP/6-31G(d) ====

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 19 - Comparing the QST2 via 2 different basis sets and method'''
! '''Method''' !! '''QST2 via HF/3-21G'''!! '''QST2 via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagramdsgfsw.JPG|150x150px]] || [[Image:Failed_pl1208sf.JPG|150x150px]]
|-
| '''Energy/a.u.'''|| [[Image:Failed_summary_pl1208.JPG|thumb|200px|Total energy = -231.603 a.u.]] || [[Image:FAILED_2.JPG|thumb|200 px|Total energy = -234.548 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-818 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.61;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41836_2.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41837_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8007 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8008 |DFT/6-31G(d)]]
|}

There is a single imaginary frequency which is indicative of the transition state (Table 19). This is because, being a single negative frequency it is indicative of a maximum turning point in the reaction profile and so the optimisation has been successful towards the transition state instead of the ground state. From the table, it has been observed that that inter-fragment distance is different via the two methods. This will be analysed in the section to come. But comparing the total energy of the two methods, it can be seen that the more accurate b3lyp method results in a more negative value. However, as they are different methods of calculation they cant be compared quantitatively.

Looking at the RMS gradient, both the optimisations can be concluded to have reached a convergence. This is because both have a gradient significantly lesser than 0.001 a.u. of 0.00003 a.u. and 0.00007 a.u. respectively. Thus the results would be deemed to be accurate and would be expected to have successfully optimised towards the transition.

The biggest difference in values between the two methods is seen in the imaginary wavenumber. The difference is almost twice in-magnitude but this difference could be attributed to the different type of method and basis set used. This is because analysis of the animation below would show that they are of the same type of vibrational stretches.

Anlysis of the animation shows that the ethylene is approaching the terminal end of the butadiene, and when this happens there is the back-bending of the hydrogens. This could be due to the loss of bond order (2 to 1) which results in sp2 towards sp3 carbon centre so the bond angles increase. The ethylene approaching in the synchronous fashion is indicative of the reaction being a concerted mechanism (As expected for a cycloaddition !). The elongation of the double bonds on the butadiene are indicative of a lower bond order; bond breaking occurs. Similarly the c-c (2 and 3) are seen to get shorter as the ethylene approaches which is indicative of higher bond strength and hence a higher bond order (double bond). From this it can be seen that the QST2 has successfully optimised towards the envelope-shaped transition state of the cycloaddition.

==== Method 2 : Ts (Berny) via HF/3-21G and DFT/B3LYP/6-31G(d) ====

The transition having known to be an envelop type structure, would be carried out in two steps where the first step would be the frozen coordinate type condition. This was implemented to 'lock' the distance between the terminal carbons at 2.10 Å which the result of the transition state bonds relax. The molecule was arrange to be locked at such a distance with the ethylene to be orientated above the butadiene so that it significantly mimicks the expected transition state orientation. Upon the frozen optimisation, the Ts(Berny) mode was selected to allow the reactants to get closer towards formation of the transition as rationalised in the earlier sections. Both the less accurate HF/3-21G method and the DFT/B3LYP/6-31G(d) method were implemented. The table below summarises the results and findings.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 21 - Comparing the Ts(Berny) via different methods and basis set'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_tsberny.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_asf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41834.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_tsberny_hf.JPG|thumb|200px|'''Total energy = -231.603 a.u.''']] || [[Image:Summary_pl1208)as.JPG|thumb|200 px|'''Total energy = -234.544 a.u.''']]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-820 ||-524
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.45;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41834_20.txt</uploadedFileContents></jmolApplet></jmol>
| <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.49;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41835_2.txt</uploadedFileContents></jmolApplet></jmol>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8009 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8011 |DFT/6-31G(d)]]
|}

The table above would have a very similar representation of values to that of method 1. It can be concluded that this method of the Ts (BERNY) was similarly accurate in deriving with the almost same results as those in method 1. Thus, it could be concluded that the optimisation efforts towards the transition state was successful. This was further confirmed from the animation of the imaginary vibrations which were also synchronous of the ethylene approaching the butadiene and were the same as that of method 1, indicative of the concerted cycloaddition transition state.

==== Reconfirming the attainment of the transition state ====

===== IRC Analysis =====
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8014 |IRC analysis of Transition state n=30]]

{| class="wikitable" border="1" style="text-align:center" align=center
|-
| [[Image:TOTAL_ENERGY_IRC_PL1208.JPG|thumb|500px|'''Diagram 21 - showing the total energy at each step''']] || [[Image:FIRST_irc_n=170.JPG|thumb|200px| '''Diagram 22 - Fragments orientation at first optimisation step''']]||
[[Image:FINAL_pl1208_n=170.JPG|thumb|200px| '''Diagram 23 -Fragments orientation at last optimisation step''']]
|}

The IRC optimisation was carried out to complement the earlier made conclusions that the optimisation was successful towards the transition. It is seen that the total energy plateau off to indicate the convergence. Furthermore, analysis of the first and the final steps in the optimisation would show that the final step has the conformation to be exactly that of the expected transition state. Thus it can be concluded that the optimisation was successful in both methods above as they lead to the same energy level.

===== Bond Length Analysis =====

{| class="wikitable" border="1" style="text-align:center" align=center
|'''Diagram with labelling''' ||colspan="3"|[[Image:Atomic_numvering_pl1208.JPG]]
|-
| Type of Method || B3LYP/6-31 G(d) || HF/3-12G || Literature HF/3-12G<ref>C. Spino, M. Pesant and Y. Dory, Angew. Chem. Int. Ed. Engl., 37, 3262 (1998)</ref>
|-
|Bond lengths C1-C6/ C4-C5 (Å)||2.27230 (2.27) || 2.20927 (2.21) ||2.2-2.3
|-
|Bond length C1-C2/ C3-C4 (Å)||1.38305 (1.38) || 1.37007 (1.37) || 1.37
|-
|Bond lengths C5-C6 (Å)||1.38597 (1.39) || 1.37603 (1.38) || 1.39
|-
|Bond length C2-C3 (Å)||1.40721 (1.41) || 1.39407 (1.39) || 1.40
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="3"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="3"|1.50
|}

The DFT method of optimisation results in the bond lengths to be slightly longer than those of the HF method of optimisation. But comparing the literature HF values against those from the computationally attained value, it is almost a perfect match ! This is indicative that the calculations towards optimisation especially via the HF/3-21G method were highly successful.

Comparing between the different C-C bond lengths show that the forming '''C-C sigma bonds (1-6 and 4-5)''' are of slightly greater than the typical C-C bond length of 1.54Å. This is because, the bond lengths calculated are that of the transition state where the bond has not been completely formed. It is definitely lesser than 1 which shows it is not fully formed yet. Similarly, the '''C=C bonds (1-2 and 3-4)''' are slightly longer than the typical bond length of a double bond. This lengthening is indicative of the loss in bond order; loss in bond strength. This would be explained by the cycloaddition process which breaks the pi-bonds to form the 2 new sigma bonds. In this case, the pi-bonds are partially broken and hence longer than expected. But as the C=C bond lengths are closer to the typical C=C instead of the typical C-C bond length. It would mean the transition state more closely resembles the reactants than the products. Thus it would be an early activation energy barrier expected.

==== Molecular Orbital Analysis : HOMO and LUMO of the Transition State ====

[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8018 |MO analysis via B3LYP]]

{| class="wikitable" border="1" align="center" style="text-align:center" align=center
! '''Parameter''' !! '''HOMO of Transition State''' !! '''LUMO of Transition State'''
|-
| Diagram of MOs || [[image: HOMO_TS_PL1208_B3LYP.JPG|center|300px]] || [[image: LUMO_TS_PL1208_B3LYP.JPG|center|300px]]
|-
| Energy of MO (a.u) ||-0.323 || 0.023
|-
| Symmetry || Asymmetric || Symmetric
|}

Comparing the Above molecular orbitals with the individual HOMO and LUMO of butadiene and ethylene. The specific reactant HOMO and LUMO contributions can be determined.

* For the Asymmetric HOMO of the transition state, it can be seen to be an overlap between the LUMO of Ethylene and the HOMO of the 1,3-Butadiene.

* For the Symmetric LUMO of the transition state, it is an overlap between the HOMO of ethylene and the LUMO of 1,3-Butadiene.

The above observations would allow the conclusion to be made that the two reactants do form the product via a concerted cycloaddition. The LUMO are made up of the π* orbital of C=C bonds on both alkene functionalities while the HOMO are π orbitals on both alkene functionalities as well. This justfies the expected reaction where the '''electron rich 1,3-butadiene (HOMO)''' would have the favourable π-orbital overlap with the '''empty π* orbital of ethylene (LUMO)'''. Another point that justifies this is that this combination of HOMO-LUMO has a smaller orbital energy difference and thus would allow a greater overlap and hence larger splitting which is more favourable as it brings greater MO (bonding) stabilisation.

== Cycloaddition reaction between 1,3-Cyclohexadiene and Maleic Anhydride ==

Experimentally it has been shown that there are two possible products that result from the [4+2] Diels-Alder pericyclic reaction of 1,3-cyclohexadiene (diene) with the Maleic Anhydride (dienophile). This [4+2] cycloaddition would be highly similar to the prototypical reaction carried out. However, the main differences expected to be seen are those due to the influence the substituted reactants bring about.

A similar approach would be taken to first optimise the structures of both the reactants and transition states. This would then allow the total energy of the molecules involved to be determined as well as to carry out a Molecular Orbital analysis to rationalise the HOMO-LUMO interaction. The overall aim would be to computationally generate results to justify and rationalise why there is a regioselectivity for the endo product instead of the exo product.

=== Optimisation and MO analysis of Reactants ===

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 22 - Summarising the Optimisation of both reactants'''
|-
|rowspan="2"| ||colspan="2"|'''Diene = Cyclohexa-1,3-diene'''||colspan="2"|'''Dienophile = Maleic Anhydride'''
|-
| '''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''||'''Semi-Empirical method (AM1)'''||'''DFT/B3LYP/6-31G(d)'''
|-
|'''Summary Table'''||[[Image:Summary_am1_pl1208.JPG|250px]]||[[Image:Summary_pl1208_b3lyp.JPG|250px]]||[[Image:Summary_pl1208_am1.JPG|250px]]||[[Image:Summarty_pl1208_byl3p_maleic.JPG|250px]]
|-
|'''RMS (a.u.)'''|| 0.00001||0.00000006 ||0.00002 || 0.0000007
|-
|'''Source'''|| [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8024 |D-Space]] ||[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8026 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-Space]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8027 |D-Space]]
|}

The optimisation of the two reactants were first carried out via the semi-empirical AM1 method and was subsequently further optimised by the DFT/B3LYP method that was expected to give a more accurate optimisation to the lowest energy configuration. As they are totally different methods and working on different assumptions, the total energy cant be compared against each other. However the RMS gradient can be compared where by when the value is lesser than 0.001 a.u. it is taken to have converged and would have been successfully optimised. Comparing the AM1 method against the B3LYP, both method appear to have successful optimisation as their gradients are significantly lower than the present threshold. But for the B3LYP method, the gradient is much more closer to zero which is indicative of an even 'more' horizontal plot. This could be attributed to the more accurate basis set which allowed a more accurate optimisation.

Using the DFT optimised file, a molecular orbital calculation was carried out via the Semi-Empirical method defined as AM1. The molecular orbitals were then displayed with specific attention towards the HOMO and LUMO for each reactant and their symmetry of orbital lobes. The results are summarised in the two tables below.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 23 - Summarising the MO analysis for Cyclohexa-1,3-diene'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''Cyclohexa-1,3-diene'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_cyclohexa.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.321||0.016
|-
|'''Symmetry'''||Antisymmetric||Symmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41895.mol</uploadedFileContents><text>1,3-Cyclobutadiene</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8003 |D-space link to Cyclohexa-1,3-diene MO analysis]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 24 - Summarising the MO analysis for maleic anhydride'''
|-
|rowspan="2"|'''Parameter''' ||colspan="2"|'''maleic anhydride'''
|-
| '''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_pl1208_maliec.JPG|THUMB|HOMO |200px]]|| [[Image:Lumo_pl1208_maleic.JPG|THUMB|HOMO |200px]]
|-
|'''Energy of MO (a.u.)'''||-0.441||-0.059
|-
|'''Symmetry'''||Symmetric||Asymmetric
|-
|||<jmol> <jmolAppletButton><uploadedFileContents>Checkpoint_41945_anhydride.mol</uploadedFileContents><text>maleic anhydride</text><color>black</color></jmolAppletButton></jmol> || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8025 |D-space link to maleic anhydride MO analysis]]
|}

As the HOMO of the Cyclohexa-1,3-diene and and LUMO of the maleic anhydride are both Asymmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbitals. The HOMO-LUMO energy gap in this possible overlap would be 0.262 (a.u.)

Similarly comparing the LUMO of the Cyclohexa-1,3-diene and and HOMO of the maleic anhydride are both symmetric, this would mean they would have a favourable orbital overlap and thus would be able to have the interactions between the orbital. The HOMO-LUMO energy gap in this possible overlap would be 0.457 (a.u.)

This would show that the first combination where the Cyclohexa-1,3-diene acting as the HOMO would have a smaller energy gap which would allow a more effective orbital overlap and thus would result in greater orbital splittings. This is more favoured as it would bring about a greater degree of stabilisation. Thus it can be concluded that the system would prefer to have the HOMO via the Cyclohexa-1,3-diene as anticipated by theory where Cyclohexa-1,3-diene is the electron rich diene.

=== Transition State Analysis ===

==== Vibrational Analysis ====

From the MO analysis carried out, it is seen that the HOMO of the Cyclohexa-1,3-diene will have an effective orbital overlap with the LUMO of maleic anhydride as it would have the smaller orbital energy gap. To go with a tried and tested theory, the same method of optimisation for the Diels-Alder reaction between ethylene and butadiene was used. A guessed structure of the transition state was drawn for both the endo and the exo transition states. The C-C bond that was expected to be formed via the cycloaddition was 'frozen' at 2.10 (Å).

Following this step the '''Redundant''' option was selected and an optimisation via the Ts(Berny) mode via the HF/3-21G method and basis set was carried out. The same process was then carried out with a further optimisation with the DFT/B3LYP/6-32G(d)method and basis set.

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 25 - Summarising the Exo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_exo.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41898.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_pl1208_exo_b3l.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41897.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_exo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_exo_b3ly.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-647 ||-449
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.31;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41898_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Vib_start.JPG|thumb|200px|Showing vibration start]] | [[Image:Vib_end.JPG|thumb|200px|Showing vibration end ]]>
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |DFT/6-31G(d)]]
|}

{| class="wikitable" border="1" style="text-align:center" align=center
|+ ''' Table 26 - Summarising the Endo-Transition state via two different methods/basis sets'''
! '''Method''' !! '''Ts (Berny) via HF/3-21G'''!! '''Ts (Berny) via B3LYP/6-31G(d)'''
|-
| '''Optimized Structure'''|| [[Image:Diagram_pl1208_endo_hf.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41899.mol</uploadedFileContents><text>Ts Berny (HF)</text><color>black</color></jmolAppletButton></jmol>
| [[Image:Diagram_endo_pl1208_dft.JPG|150x150px]]
<jmol><jmolAppletButton><uploadedFileContents>Checkpoint_41896.mol</uploadedFileContents><text>Ts Berny (B3LYP)</text><color>black</color></jmolAppletButton></jmol>
|-
| '''Energy/a.u.'''|| [[Image:Summary_pl1208_endo_hf_summary.JPG|thumb|200px|Total energy = -606 a.u.]] || [[Image:Summary_pl1208_endo_dft.JPG|thumb|200 px|Total energy = -613 a.u.]]
|-
| '''Imaginary Frequency/cm<sup>-1</sup>'''||-643 ||-446
|-
| '''Animation of Imaginary Frequency''' || <jmol><jmolApplet>
<title>Vibration</title><color>black</color><size>250</size>
<script>zoom 100;frame 1.27;vectors 4;vectors scale 2.0;color vectors yellow;vibration 3;
</script><uploadedFileContents>Log_41899_2.txt</uploadedFileContents></jmolApplet></jmol>
|| [[Image:Strart_vib.JPG|thumb|200px| start of vibrational stretch ]] | [[Image:End_vib.JPG|thumb|200px|end vibrational stretch ]]
|-
| D-Space Links || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 |HF/3-21G]] ||
[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |DFT/6-31G(d)]]
|}

From the table it is observed that the RMS gradient is well below the present threshold and so convergence is observed and there being only a single negative frequency (imaginary) would be indicative of successful optimisation to the endo transition state (table 25). As different method and basis sets are used there is some difference in the total energies to be expected. The successful formation of the optimisation to the transition state is further observed by the animation of the vibrations.

It would show the synchronous movement of the 'middle' two carbons closer to the end of the anhydride. This is indicative of bond formation via a concerted reaction. And when this happens the apex C-C bond in the cyclohexadiene becomes smaller which is indicative of the double bond formation due to the [4+2] cycloaddition. Similary the lengthening of the C=C bonds in the diene are indicative of the π bond being broken. Although different magnitudes of the imaginary frequencies are observed, both the vibrational motions are equivalent. (As the log file for the 6-31G frequency is very large, only screen shots were uploaded)

Suprisingly the exo optimisation would also result in relatively similar energy values and the vibrations both point towards effective optimisation towards the formation of the transition state. As there is no change seen in the rounded up total energy, more quantitative analysis is to be carried out to determine which is the more stable state (Table 26).

==== Geometric Analysis ====

{| class="wikitable" border="1" style="text-align:center" align=center
| '''Type of Method - B3lyp/6-31G(d) || Endo Ts || Exo Ts'''
|-
|'''Diagram with labelling''' ||colspan="2"| [[Image:Numbering_PL1208_TS_endoexo.JPG|thumb|500px| Labelled Carbons for both Ts ]]
|-
|Interfragment Distance C-C (9-6 and 12-5) (Å)||2.26933 (2.27) || 2.29012 (2.29)
|-
|Sigma Bond C-C (7-6 and 4-5) (Å)||1.47943 (1.48) || 1.47952 (1.48)
|-
|Sigma Bond C-C (C5-C6)/ Former Double bond (Å)||1.39379 (1.39) || 1.39793 (1.40)
|-
|C=C Bond Product (8-C13 and 10-11)/ Former Single bond (Å)||1.40320 (1.40) || 1.40344 (1.40)
|-
|Sigma Bond C-C (9-C8 and 12-13) (9-C10 and 12-C11)/ Former Double bond (Å)||1.39105 (1.39) || 1.39145 (1.39)
|-
|Dihedral Angle btw C=C in Cyclohexa and anhydirde plane (deg)||-64.811 C11-10-9-6 || 67.738 C13-8-9-6
|-
|Through Space Distance (C=C)-(C-C) (Å)||2.87336 (2.87) ||2.97342 (2.97)
|-
|Typical C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.54
|-
|Typical C=C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å)||colspan="2"|1.35
|-
|Typical sp2-sp3 C-C bond length<ref> G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69</ref> (Å) ||colspan="2"|1.50
|}

For the exo transition state, the 'spear-end' O=C-O-C=O of the Maleic anhydride is significantly further from the C=C bond than in the endo transition state. This would explain why the C-C forming sigma bonds (Interfragment) are longer in the exo transition state than in the endo. The C=C bonds which were former single bonds are much longer than the typical C=C bond which is due to the calculations being made in a transition state. In the transition state the pi bond are not fully broken or formed and thus there is the deviation from the typical bond lengths.

The Length of the C=C bonds being formed and being broken are between 1.40 and 1.39 Å while is as expected since it would be indicative of the concerted formation and breaking of bonds. When comparing the space distance of the C=C bond group from the dienophile, the endo Ts is seen to have a smaller distance of 2.87 Å as compared to the exo while had a further, sterically more favourable distance of 2.97 Å. However, due to the secondary orbital stabilisations the increased steric repulsions are overcomed by the favourable orbital overlaps to make the endo more stable.

==== Energetic Analysis of Transition States ====

This section aims to energetically determine which is the major product of the cycloaddition reaction. From the previous sections, it has been shown that the Endo product is the energetically favourable one due to the additional secondary orbital interactions it has. Thus it is anticipated that the Endo transition would be of a lower energy level so it is more readily attained and also the endo product should be energetically lower in energy.

There are three different groups that have to be analysed and the energies attained from them. To set a common ground of comparison, all the different molecules and transition states will be optimised by the B3LYP/6-31G(d) method and basis set.

i) Specific Energies for the Reactants

ii) Specific Energies for the two different Transition states

iii) Specific energies for the two types of products.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table Comparing the Specific Energies for the Reactants of the reaction'''
! '''Specific Energy (a.u.) !! cyclohexa-1,3-diene !! maleic anhydride''' !! Total sum of Reactants'''
|-
| Electronic Energy || -233.41891079|| -379.28954470 || -612.7084554
|-
| Sum of electronic and zero-point Energies at 0 K || -233.296099 || -379.233657 || -612.6525677
|-
| Sum of electronic and thermal energies 298.15 K || -233.290921|| -379.228473 || -612.519394
|-
| Source || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Cyclo_hexa.out |Log File]] || [[https://wiki.ch.ic.ac.uk/wiki/index.php?title=Image:Maleic.out |Log File]] || -
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ '''Table Comparing the Specific Energies for the Exo and Endo Transition States'''
! '''Specific Energy (a.u.) !! Exo Transition State !! Endo Transition State'''
|-
| Electronic Energy || -612.67931093|| -612.8339606
|-
| Sum of electronic and zero-point Energies at 0 K || -612.498014 || -612.502133
|-
| Sum of electronic and thermal energies 298.15 K || -612.487663 || -612.491779
|-
| Source || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8031 |Log File]] || [[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8029 |Log File]]
|}

There are three sets of Activation Energies for the Transition states for each of the transition states Which is attained by taking the transition state specific energy and determine the difference relative to the total energies of the reactants. The results are summarised in the two tables below.

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ Activation energies for the Exo Transition State
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0291445 || 18.28843 (18.3)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1545537 || 96.9838377 (97.0)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.031731 || 19.91148808 (19.9)
|}

{| class="wikitable" border="1" align="center" style="text-align:center"
|+ Activation energies for the Endo Transition State
! Specific Activation Energy !! (a.u) !! Kcal mol <sup>-1</sup>
|-
| Activation Energy (Electronic Energy) || 0.0255052 || 16.00474255 (16.0)
|-
| Activation Energy (Sum of electronic and zero-point Energies at 0 K) || 0.1504347 || 94.39912816 (94.4)
|-
| Activation Energy (Sum of electronic and thermal energies 298.15 K) || 0.027615 || 17.32866104 (17.3)
|}

[[Image:PL1208_GRAH_COMPARE_TS_ENDOEXO.bmp|thumb|500px|Diagram comparing the relative energies for Endo Against Exo |centre ]]

From the above graph it can be seen that comparing all the different activation energies would show that the endo-transition state would have the lower activation energy barrier. This would mean it is more energetically accessible as compared to the exo transition despite the endo having a greater steric hindrance as compared to the exo. This can be rationalised as being due to the secondary orbital interactions present in the endo and not the exo that allows the steric hindrance experiences to be overpowered by the stabilisation brought about by the overlapping secondary orbitals.

== Theoretical Explaination Compared Against Computational Results <ref>Alder, K.; Stein, G. Justus Liebigs Ann Chem 1934, 514, 1;</ref> <ref> Garcı´a, J. I.; Mayoral, J. A.; Salvatella, L. Acc Chem Res 2000, 33, 658.</ref> ==

[[Image:Secondary_orbital_overlap_PL1208_DIELS.bmp|thumb|500px|Diagram '''Primary and Secondary Orbital interactions''' |centre ]]

The primary orbital interactions are defined as interactions that result in the formation of formal bonds such as sigma bonds and pi bonds which are the primary head-head and side-side orbital overlap respectively. In contrast, the secondary orbital overlaps make reference to the orbital interactions that

{| class="wikitable" border="1" style="text-align:center" align=center
|+ '''Table 27 - HOMO-LUMO comparison for the Endo and Exo Ts'''
|-
|rowspan="2"| ||colspan="2"|'''Exo-Transition State'''||colspan="2"|'''Endo Transition state'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Homo_exo_pl1208_ts.JPG|200px]]||[[Image:lumo_exo_pl1208_ts.JPG|200px]]||[[Image:Homo_endo_pl1208_ts.JPG|200px]]||[[Image:lumo_endo_pl1208_ts.JPG|200px]]
|-
| '''Energy of MO (a.u.)'''||-0.323 ||0.058 ||-0.324 ||0.073
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric|| Antisymmetric||Antisymmetric
|-
| '''Source File'''|| colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8030| D-Space]] ||colspan="2"|[[https://spectradspace.lib.imperial.ac.uk:8443/dspace/handle/10042/to-8028 | D-Space]]
|}

== Conclusion ==