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Rep:Hoss.Module3

2011-03-25T16:58:27Z

Ha508: /* Chair Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules using various computational techniques. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2Å - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS AM1 opt+freq.mol</uploadedFileContents> <text>Exo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS DFT opt+freq.mol</uploadedFileContents> <text>Exo DFT TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b) AM1 opt+freqENDO.mol</uploadedFileContents> <text>Endo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss DFTTsberny endo DA2 TS.mol</uploadedFileContents> <text>Endo DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:48:56Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules using various computational techniques. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS AM1 opt+freq.mol</uploadedFileContents> <text>Exo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS DFT opt+freq.mol</uploadedFileContents> <text>Exo DFT TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b) AM1 opt+freqENDO.mol</uploadedFileContents> <text>Endo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss DFTTsberny endo DA2 TS.mol</uploadedFileContents> <text>Endo DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:48:35Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules using various computational techniques. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS AM1 opt+freq.mol</uploadedFileContents> <text>Exo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS DFT opt+freq.mol</uploadedFileContents> <text>Exo DFT TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b) AM1 opt+freqENDO.mol</uploadedFileContents> <text>Endo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss DFTTsberny endo DA2 TS.mol</uploadedFileContents> <text>Endo DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:47:56Z

Ha508: /* Physical Module 3 */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules using various computational techniques. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS AM1 opt+freq.mol</uploadedFileContents> <text>Exo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS DFT opt+freq.mol</uploadedFileContents> <text>Exo DFT TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b) AM1 opt+freqENDO.mol</uploadedFileContents> <text>Endo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss DFTTsberny endo DA2 TS.mol</uploadedFileContents> <text>Endo DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:45:34Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS AM1 opt+freq.mol</uploadedFileContents> <text>Exo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b)TS DFT opt+freq.mol</uploadedFileContents> <text>Exo DFT TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.DA2guess(b) AM1 opt+freqENDO.mol</uploadedFileContents> <text>Endo AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss DFTTsberny endo DA2 TS.mol</uploadedFileContents> <text>Endo DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

File:Hoss DFTTsberny endo DA2 TS.mol

2011-03-25T16:44:47Z

Ha508:

File:Hoss.DA2guess(b) AM1 opt+freqENDO.mol

2011-03-25T16:43:19Z

Ha508:

File:Hoss.DA2guess(b)TS DFT opt+freq.mol

2011-03-25T16:41:54Z

Ha508:

File:Hoss.DA2guess(b)TS AM1 opt+freq.mol

2011-03-25T16:40:39Z

Ha508:

Rep:Hoss.Module3

2011-03-25T16:39:11Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) AM1 opt+freq TSberny.mol</uploadedFileContents> <text>AM1 TS</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.DA1 DFT ts.png|200px]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.guess(1) DFT opt+freq TSberny.mol</uploadedFileContents> <text>DFT TS</text> <color>white</color></jmolAppletButton> </jmol>
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

File:Hoss.guess(1) DFT opt+freq TSberny.mol

2011-03-25T16:38:03Z

Ha508:

File:Hoss.guess(1) AM1 opt+freq TSberny.mol

2011-03-25T16:37:15Z

Ha508:

Rep:Hoss.Module3

2011-03-25T16:34:42Z

Ha508: /* Intrinsic Reaction Coordinate (IRC) Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:34:25Z

Ha508: /* Intrinsic Reaction Coordinate (IRC) Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.chair ts opt+freq derivative IRC optlocmin.mol</uploadedFileContents> <text>Chair IRC Berny Opt</text> <color>white</color></jmolAppletButton> </jmol> || [[Image:Hoss.boat IRC LocMin.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents> Hoss.boat ts opt+freq IRC optlocmin.mol</uploadedFileContents> <text>Boat IRC</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

File:Hoss.chair ts opt+freq derivative IRC optlocmin.mol

2011-03-25T16:33:48Z

Ha508:

File:Hoss.boat ts opt+freq IRC optlocmin.mol

2011-03-25T16:33:28Z

Ha508:

Rep:Hoss.Module3

2011-03-25T16:26:59Z

Ha508: /* Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq.mol</uploadedFileContents> <text>TS Berny Opt</text> <color>white</color></jmolAppletButton> </jmol>
|| [[Image:Hoss.chair frozen opt.png|200px|centre ]] <jmol> <jmolAppletButton> <uploadedFileContents>Hoss.chair ts opt+freq derivative.mol</uploadedFileContents> <text>Frozen Coord Opt</text> <color>white</color></jmolAppletButton> </jmol>
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

File:Hoss.chair ts opt+freq derivative.mol

2011-03-25T16:26:07Z

Ha508:

File:Hoss.chair ts opt+freq.mol

2011-03-25T16:24:16Z

Ha508:

Rep:Hoss.Module3

2011-03-25T16:19:36Z

Ha508: /* Conclusion */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

The computational calculations carried out successfully explained the preferred mechanism by which a simple Diels Alder reaction, such as the Cope rearrangement, proceeds by. Furthermore, the results from the last section helped explain why the endo isomer is preferred in the reaction between maleic anhydride and 1,3-cyclohexadiene.

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:10:49Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:09:43Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry. R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|800px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T16:05:23Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry. R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

The LUMO+1 and LUMO+2 for both isomers clearly illustrate that there exists this secondary orbital interaction in the endo isomer. The MO's for the endo isomer demonstrates that there is significant overlap between the C=C π orbitals and the C=O π*. This interaction is absent in the MO's for the exo isomer. Overall it can be said that consideration of secondary orbital interactions helps explain why the endo isomer is the favoured product in this Diels Alder reaction.

= Conclusion =

= References =
<references/>

Rep:Hoss.Module3

2011-03-25T15:32:35Z

Ha508: /* The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|200px]]||[[Image:Hoss.diene LUMO exp DA2.png|200px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|200px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|200px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry. R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T15:32:10Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry. R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|200px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|200px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|200px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concerns the overlap between the π* orbitals on the C=O and the conjugated π system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher. This can be shown using both the LCAO method and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T15:29:45Z

Ha508: /* The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required to reach the transition state is less. The stereochemistry observed here will be investigated and explained using the computational techniques already used in this report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of the oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought that it would be better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn in a separate window in Gaussview, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation, as well as modifying the (terminal) C-C bond lengths to 2.0 Å.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control, the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry. R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show that the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T15:24:49Z

Ha508: /* Optimisation of the Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

The MO's obtained from both methods are shown below:

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T15:23:31Z

Ha508: /* The Diels Alder Cycloaddition */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4π + 2π] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state; the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new σ bonds, which result due to the interaction between the π orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of cis-Butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry perspective, this is a very poor example of the Diels Alder reaction as the ethene does not behave as a good dienophile. However, for the purposes of the computational techniques used here, this reaction is still adequate.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step in locating the transition structure involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the opposite of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to be similar in appearance to a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00 Å. It was initially set to 2.20 Å, however this resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out using the more accurate DFT/B3LYP/6-31G (d) method and basis set. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''Terminal C-C Bond Distance (Å)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a longer C-C bond and a lower (imaginary) frequency - lengthening the bond results in increasing the shallowness of the potential energy slope thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginary frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two σ bonds between the terminal carbons.

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:59:23Z

Ha508: /* Activation Energies of the Transition States */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The table below compares the electronic energies obtained from the two methods:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}

The activation energies for both transition states were calculated using the thermochemical data:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
The calculated values match reasonable well with literature indicating that the results are fairly accurate. Overall, it is evident that the activation energy for the chair transition state is lower in energy, suggesting that the preferred mechanism in the Cope Rearrangement proceeds vis this structure. To summarise this section, one can say that the anti-2 structure i.e. the reactant, undergoes a Cope Rearrangement via a chair transition state to yield the gauche 3 structure as the product.

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:32:19Z

Ha508: /* Activation Energies of the Transition States */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation were used to determine the activation energies for the reaction via both transition structures. A comparison of the energies, bond lengths and imaginary frequencies of the two optimised structures is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

Unsurprisingly, the more accurate method resulted in a more stable molecule for both the chair and the boat. The values for the terminal C-C bond length did not change much but the values for the imaginary frequency did. This value is determined by taking the second derivative of the potential energy surface and will change depending on the method used.

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:20:36Z

Ha508: /* Intrinsic Reaction Coordinate (IRC) Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Table showing the results from the IRC calculation when the force constant was calculated at every step'''
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:19:07Z

Ha508: /* Intrinsic Reaction Coordinate (IRC) Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see [[Mod:phys3#Appendix 1|Appendix 1]]), thus the chair transition state connected 1,5-hexadiene in the gauche-2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connected 1,5-hexadiene in the gauche-3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:17:10Z

Ha508: /* Intrinsic Reaction Coordinate (IRC) Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
In order to determine the mechanistic characteristics of this pericyclic reaction, we need to investigate which conformers of reactants and products each transition state connects. The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surface. It achieves this by taking small energy steps, which involve a change in the geometry of the structure under consideration, in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that resulted did not produce the most stable geometry and further calculations were required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the structure corresponding to the 43rd structure was then copied into a new window and another IRC calculation was carried out (this time the force constant was calculated at every step and the number of points to be calculated was set to 100). Finally, the last structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (this time the initial step showed that 44 runs of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:09:32Z

Ha508: /* Boat Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

The negative frequency obtained confirmed that a transition state had been located, and the vibration at -839cm-1 corresponded to the asynchronous bond formation observed in the Cope Rearrangement.
The QST2 method proved useful, however the downside is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation). This requires knowledge of the transition state structure and may not always be known.

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:07:40Z

Ha508: /* Boat Transition State */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
A different method will be employed for the optimisation of this structure, namely the QST2 method. In QST2, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state. Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time and labelled as the product. In the Cope Rearrangement, the reactant and product is identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the atom numbers so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calculation failed - this occurred because upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T14:03:46Z

Ha508: /* Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', 'Opt=NoEigen' was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:59:25Z

Ha508: /* Chair Transition State/Structure */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', Opt=NoEigen was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:59:01Z

Ha508: /* Optimisation of the Chair and Boat structures */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus on the transition state of the Cope Rearrangement in some detail. In this rearrangement process, there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration were separately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for the chair structure was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond separated by a distance of approximately 2.2 Å.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield good results if the structure suggested for the transition state is accurate. The transition state for the Cope Rearrangement has been studied before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed in the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as 'Opt+Freq', Opt=NoEigen was then typed into the additional keywords section and the force constant was set to calculate once.
The Frozen Co-ordinate method is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02 Å. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirmed the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken. This is indicative of the asynchronous bond formation found in this pericyclic reaction.
Although both methods are useful, as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:47:57Z

Ha508: /* Frequency Analysis */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Thermo-calculations on the Re-optimised Anti 2 conformer at 298.15 K (All energies are in a.u)
| Sum of Electronic and Zero-point Energies || - 234.469204
|-
| Sum of Electronic and Thermal Energies || - 234.461857
|-
| Sum of Electronic and Thermal Enthalpies || - 234.460913
|-
| Sum of Electronic and Thermal Free Energies || - 234.500776
|-
|}

The sum of the electronic and zero-point energies is the potential energy at 0, including the zero-point vibrational energy: E = Eelec + ZPE.
The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans. The sum of the electronic and thermal enthapies is the previous energy but contains an extra correction for room temperature, RT (where H = E + RT). Finally, the sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS.

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:40:08Z

Ha508: /* Frequency Analysis */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised Anti-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were observed, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be separated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:33:09Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:31:54Z

Ha508: /* Re-optimisation */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation of Anti-2 ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and the 6-31G(d) basis set, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''sp2-sp2 bond distances (Å) ''' || '''sp2-sp3 bond distances (Å) ''' || '''sp3-sp3 bond distances (Å) ''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || 1.32 || 1.51 || 1.55 || -231.69253528 ||
|-
| DFT/B3LYP/6-31G (d) || [[Image:Hoss.app2.png|200px|center]] || 1.33 || 1.51 || 1.55 || -234.6117038 ||
|-
| Literature<ref>G. Schultz, I. Hargitta, ''J. Mol. Struc.'', '''1995''', ''346'', pp. 63-69 </ref> || || 1.34 || 1.51 || 1.54 || ||
|}

The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the dihedral angle between the non-terminal carbons was 180oC on both occasions.
Visually, the geometry obtained looks identical to the one obtained earlier using the HF method; comparison of the bond lengths show that the structure has changed only slightly, and that both sets of values are reasonably well matched with the literature values (the values for DFT method were marginally closer to the literature).

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:16:05Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:13:48Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (a.u)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (a.u)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (a.u) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:12:25Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:12:00Z

Ha508: /* Optimisation of the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before it optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:11:41Z

Ha508: /* Optimising the Reactants and Products */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimisation of the Reactants and Products ==

The diene to be considered generally has two different stable structures: an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on Gaussview with an anti-periplanar structure and was then 'cleaned' before it optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before. Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified as:

- 1st anti and gauche conformations idenitified
{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the anti-periplanar and gauche conformations in order to determine the lowest energy configuration of the molecule. The overall results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || Ci || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C2
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C2h || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C1 || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C1
|}

Overall, the most stable anti-periplanar conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although Gaussview calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that overall, the most stable gauche conformation (gauche 3) is more stable than the most stable anti conformation (anti-1). One possible reason for this is due to the orbital interactions and consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}

The HOMO for the gaauche conformation clearly shows that there is a stabilising π interaction between the two double bonds. This interaction is not present in the anti-periplanar form.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T13:02:03Z

Ha508: /* The Cope Rearrangement */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a reaction involving a [3,3]-Sigmatropic Shift. Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The mechanism by which this reaction proceeds has been a controversial topic for many years, however it is now widely accepted that the reaction occurs in a concerted manner involving the migration of a group from one point to another. The concerted migration goes through a chair (shown below) or boat transition state:

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope Rearrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures; this will provide information regarding the preferred mechanism which takes place.

== Optimising the Reactants and Products ==
The diene to be considered generally has two different stable conformations, an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on GaussVIEW with an antiperiplanar structure and was then 'cleaned' before it was optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before.Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified:

- 1st anti and gauche conformations idenitified
{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the antiperi-planar and gauche conformations in order to determine the lowest energy configuration of the molecule. The results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C
|}

Overall, the most stable anti conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although GaussVIEW calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that the gauche conformation (gauche 3) is more stable than the anti (anti-1). One possible explanation can be propsoed by consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}
The HOMO for the gaauche conformation clearly shows that there is a stabilising pi-pi interaction between the two double bonds. This interaction is not present in the anti periplanar conformation.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T12:50:59Z

Ha508: /* Physical Module 3: Aims and Introduction */

= Physical Module 3 =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a [3,3]-Sigmatropic Shift involving....Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The major product in this reaction is the more thermodynamically stable regioisomer. The reaction proceeds in a concerted fashion and goes through a chair (shown below) or boat transition state.

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope re-arrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures. This will procide information regarding the preferred mechanism which takes place.

== Optimising the Reactants and Products ==
The diene to be considered generally has two different stable conformations, an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on GaussVIEW with an antiperiplanar structure and was then 'cleaned' before it was optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before.Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified:

- 1st anti and gauche conformations idenitified
{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the antiperi-planar and gauche conformations in order to determine the lowest energy configuration of the molecule. The results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C
|}

Overall, the most stable anti conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although GaussVIEW calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that the gauche conformation (gauche 3) is more stable than the anti (anti-1). One possible explanation can be propsoed by consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}
The HOMO for the gaauche conformation clearly shows that there is a stabilising pi-pi interaction between the two double bonds. This interaction is not present in the anti periplanar conformation.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]

Rep:Hoss.Module3

2011-03-25T12:50:37Z

Ha508: /* Physical Module 3: Aims and Introduction */

= Physical Module 3: Aims and Introduction =
'''''Hossay Abas'''''

The aim of this experiment will be to investigate structural and chemical characteristics for a range of molecules. The transition state for several reactions will be analysed and used to determine the preferred reaction mechanism. Different levels of theory and optimisation methods will be used throughout.

= The Cope Rearrangement =

The Cope Rearrangement is a [3,3]-Sigmatropic Shift involving....Specifically, it is the thermal isomerisation of a (substituted) 1,5-diene and results in a regioisomeric 1,5-diene. The major product in this reaction is the more thermodynamically stable regioisomer. The reaction proceeds in a concerted fashion and goes through a chair (shown below) or boat transition state.

[[Image:Hoss.cope rearranegement.gjf.png|thumb|250px|The Cope Rearrangement|center]]

The Cope re-arrangement will be investigated using 1,5-hexadiene. Initially, the lowest energy configuration of this molecule will be determined, followed by investigation of the transition state structures. This will procide information regarding the preferred mechanism which takes place.

== Optimising the Reactants and Products ==
The diene to be considered generally has two different stable conformations, an anti conformation and a gauche conformation.
1,5 Hexadiene was drawn on GaussVIEW with an antiperiplanar structure and was then 'cleaned' before it was optimising it using the HF/3-21G method and basis set. A second 1,5 Hexadiene was then drawn, this time with a gauche conformation, and was optimised as before.Using [[Mod:phys3#Appendix 1|Appendix 1]], the structures were identified:

- 1st anti and gauche conformations idenitified
{| class="wikitable" border="5" align="center"
|+ '''Energies of anti and gauche optimisation'''
| '''Conformer''' || '''Structure''' || '''Energy (RHF)''' || '''Point Group'''
|-
| Anti 1 || [[Image:Hoss.app1.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1 Opt</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C2
|-
| Gauche 3 || [[Image:Hoss.gauche3.png|250px|center]]<jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C1
|}

Some alterations were made to both the antiperi-planar and gauche conformations in order to determine the lowest energy configuration of the molecule. The results along with the respective identifications are shown below:

{| class="wikitable" border="5" align="center"
|+ '''Energies of all app and gauche optimisation'''
| '''Anti FORM''' || '''Energy (RHF)''' || '''Point Group''' || || '''Gauche FORM''' || '''Energy (RHF) || '''Point Group'''
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP1.mol</uploadedFileContents> <text>Anti 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6926 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE1.mol</uploadedFileContents> <text>Gauche 1</text> <color>white</color></jmolAppletButton> </jmol> || -231.6878 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP2.mol</uploadedFileContents> <text>Anti 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6925 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE2.mol</uploadedFileContents> <text>Gauche 2</text> <color>white</color></jmolAppletButton> </jmol> || -231.6917 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP3.mol</uploadedFileContents> <text>Anti 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6891 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE3.mol</uploadedFileContents> <text>Gauche 3</text> <color>white</color></jmolAppletButton> </jmol> || -231.6927 || C
|-
| <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE APP4.mol</uploadedFileContents> <text>Anti 4</text> <color>white</color></jmolAppletButton> </jmol> || -231.6910 || C || || <jmol> <jmolAppletButton> <uploadedFileContents>HOSS HEXADIENE GAUCHE6.mol</uploadedFileContents> <text>Gauche 6</text> <color>white</color></jmolAppletButton> </jmol> || -231.6892 || C
|}

Overall, the most stable anti conformer was found to be anti-1 and the most stable gauche conformer was found to be gauche-3. Although GaussVIEW calculates absolute geometries for each molecule drawn, it is still possible to compare the values obtained for isomers. The values above indicate that the gauche conformation (gauche 3) is more stable than the anti (anti-1). One possible explanation can be propsoed by consideration of the MO's involved:

{| align=center
|+
| [[Image:Hoss.HOMOanti1.png|thumb|275px|'''HOMO - Anti 1''']]
| [[Image:Hoss.HOMOgauche3.png|thumb|275px|'''HOMO - Gauche 3''']]
|}
The HOMO for the gaauche conformation clearly shows that there is a stabilising pi-pi interaction between the two double bonds. This interaction is not present in the anti periplanar conformation.

==Re-optimisation ==
The structure corresponding to 'anti-2' was re-optimised using the more accurate DFT/B3LYP method and 6-31G(d) method, and the energy was compared with the value obtained earlier:

{| class="wikitable" border="5" align=center style="text-align:center"
|+ '''Optimisation of the Anti 2 conformer: 2 different methods
| ''' Method''' || Structure || '''Bond distances (Å)''' || '''Dihedral angle''' || '''Energy (Atomic Units)]'''
|-
| HF/3-21G || [[Image:Hoss.app2reopt.png|200px|center]] || A____p || _____ o || -231.69253528 ||
|-
| DFT/B3LYP/6-31G* || [[Image:Hoss.app2.png|200px|center]] || A----- || .... o || -234.6117038 ||
|}

compare to lit cptha
The values above indicates that the more accurate basis set had optimised the structure to an even lower energy state (which is a decrease of approx 7____kJmol-1) - this molecule is significantly more stable than the one which was previously generated.
It should be noted that for both optimisations, the point group was found to be Ci as is expected due to the centre of inversion present in the molecule. This is further confirmed by the fact that the C1-C4 dihedral angle on both occasions is 180C.

Visually, the geometry obtained closely resembles that obtained earlier using the HF method; thus in order to compare the two results, the bond lengths and dihedral angles were analysed as shown below:

The re-optimised molecule has slightly different values, with these new values being in better agreement with that of literature.

== Frequency Analysis ==
Using the re-optimised app-2 structure, a frequency analysis was carried out using the same method (DFT-B3LYP) and basis set (6-31G(d)).
The calculated frequencies were all found to be positive and no imaginary frequencies were seen, which confirmed that a minima had been reached. The IR spectra is shown below:

[[Image:Hoss.IRapp2.png|thumb|400px|IRspectra|center]]

Thermodynamic information can be extracted from the results of the frequency analysis. Specifically, the energy of the molecule can be seperated into kinetic and potential energy components, which can then be used to determine the enthalpy and free energy. The calculation was carried out at 298.15K and the results are shown below:

- The sum of the electronic and zero-point energies is the potential energy at 0 , including the zero-poin vibrational energy: E = Eelec + ZPE: 234.469204
- The sum of the electronic and thermal energies is the energy at 298.15 K and 1 atm of pressure; it includes contributions from the translational; rotational; and vibrational energy modes: E' = E + Evib + Erot + Etrans: 234.461857
- the sum of the electronic and thermal enthapies is the previous energy but containes an extra correction for room temperature, RT (where H = E + RT): 234.460913
-The sum of electronic and thermal free energies includes the entropic contribution to the free energy: G = H - TS: 234.500776

= Optimisation of the Chair and Boat structures =

The section will focus onj the transition state of the Cope Re-arrangement in some detail. In this re-arrangement process there are two possible configurations which can be adopted, the chair and the boat:

[[Image:Hoss.chair boatTS.png|thumb|300px|TS for Chair and Boat|center]]

The transition states associated with each configuration was seperately optimised using two different techniques.

== Chair Transition State/Structure ==

The transition state for this method was first modelled and then optimised using two different methods. For both methods, an allyl fragment was initially optimised on Gaussview at the HF/3-21G level of theory. The C3H5 fragment was chosen because it is known that the transition state for the reaction resembles two allyl fragments connected by a partial bond seperated by a distance of approximately 2.2A.

=== Optimisation using the TS (Berny) method and the Frozen Co-ordinate Method===
This method will only yield accurate results of the structure suggested for the transition state is accurate. The transition state for the Cope rearrangement has been studies before and thus a reasonable structure for the transition state is already known. In this method, the force constant matrix is computed ion the first optimisation step and is then continuously recalculated during the optimisation process. 'Optimisation to a TS (Berny) was selected after setting the Job Type as Opt+Freq. Opt=NoEigen was typed into the additional keywords section and the force constant was set to calculate once.

This is a more reliable method when the guess structure for the transition state is not sufficient; the first step of this method requires the optimisation of the two allyl fragments with distance between both terminal carbon being fixed at exactly 2.2A - this was achieved by freezing the reaction coordinate. Once completed, the reaction coordinate is then unfrozen to allow for the optimisation of the rest of the molecule.
The outcomes of the two methods are shown below:

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''Simple TS (Berny) Optimisation''' || '''Frozen Coordinate Method Optimisation'''
|-
| '''Optimised Structure''' || [[Image:Hoss.chair tsberny opt.png|200px|centre]]|| [[Image:Hoss.chair frozen opt.png|200px|centre ]]
|-
|'''Energy (a.u.)''' || -231.61932241 || -231.61932239
|-
| '''Terminal C-C bond length/Å''' || 2.02 ||2.02
|-
| '''Imaginary Frequency/cm-1''' || -818 || -818
|-
| '''Vibration''' || [[Image:Hoss.chair tsberny freq.png|200px|centre]]|| [[Image:Hoss.chair frozen freq.png|200px|centre ]]
|}

It is clear that both structures look almost identical, thus suggesting that both successfully converged to the same optimised structure. This is further supported by the fact that the bond distances for both were both 2.02A. The frequency analysis for the two techniques showed that the both had an imaginary vibration at 818 cm-1 – this confirmed that the optimisation was successful as a negative (imaginary) frequency confirms the presence of a transition state. The animation for this vibration corresponds to the bond breaking and formation process which occurs in the Cope rearrangement; specifically, there is a vibration along the bond to be formed and an asymmetrical one along the bond to be broken – this is indicative of an asynchronous bond formation in the pericyclic reaction.
Both methods are useful however as in all computational methods each has flaws associated with it. The TS(Berny) method requires an accurate guess of the transition state which may not always be known, and the Frozen Coordinate Method requires that we input a specified bond distance, which again may not be known.

== Boat Transition State ==
a different methods will be employed for the optimisation of this structure, namely the QST2 method. In this method, the reactants and products will be specified and the reaction will then interpolate between the two to try and determine the transition state between them.Initially, the chk. file for the optimised anti-2 structure was copied into a new Gaussview window and was labelled as the reactant. Using the 'add to Molgroup' function, this structure was copied a second time - this time it was labelled as the product. In the cope re-arrangement, the reactant and product are identical (both 1,5 hexadiene) and thus the only way to distinguish between the two was to alter the numberings so that it corresponded to the re-arrangement reaction:

[[Image:Hoss.boatTS labelling.png|thumb|400px|center]]

Having changed the numbering of the product to correspond to the re-arrangement of the reactant, a QST2 optimisation cailulation was ran to determine the transition state. The following structure was obtained:[[Image:Hoss.boatTS failed.png|200px|center]]

The result obtained indicated that the calcilation failed - this occured because the upon interpolating between the reactant and product, rotation around the central C-C bond was not taken in to account. To correct for this, the structure was altered to resemble more of a boat like structure as shown below:[[Image:Hoss.boatTS changed.png|400px|center]]
Having done this for both the reactant and product, the QST2 optimisation was set up again and produced the following results:

- Results of final optimisation of boat including the vibration animation -

The negative frequency obtaine confirmed that a transition state had been locayed, and the vibration at -8---- corresponded to the asynchronous nond formation of the Cope-Rearrangement.
The QST2 method had proved useful, however the downside to this method is that the reactants and products need to closely resemble the transition state otherwise the calculation will fail (as was the case in the first optimisation).

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''QST2 Optimisation'''
| '''Optimised Structure''' || [[Image:Hoss.boatTS opt.png|200px|centre]]
|-
|'''Energy (a.u.)''' || -231.60280199
|-
| '''Terminal C-C bond length/Å''' || 2.14
|-
| '''Imaginary Frequency/cm-1''' || -839
|-
| '''Vibration''' || [[Image:Hoss.boatTS freq.png|200px|centre]]
|}

== Intrinsic Reaction Coordinate (IRC) Method ==
Referring back to the cope re-arrangement, in order to determine the mechanistic characteristics of this pericyclic reaction, we need toinvestigate which conformers of reactants and products each transition state connects.
The IRC method allows one to follow the minimum energy pathway from the transition structure to the local minimum on a potential energy surfcae. It achieves this by taking small energy steps which involve a change in the geometry of the structure under consideration in the direction where the gradient of the energy surface is the steepest.

Using the optimised chair structure obtained previously, an IRC calculation was set up by opting to calculate the reaction co-ordinate in the forward direction. Initially, the force constant was calculated only once and the number of points along the IRC which was to be calculated was set to 50. The first output that is produced does not usually produce the most stable geometry and further calculations are required. Opening up the chk. file for the initial IRC calculation showed that only 43 points had been calculated; the 43rd structure was then copied into a new window and another IRC calculation was carried out, this time the force constant was calculated at every step and the number of points to be calculated was set to 100. Finally, the final structure obtained form this output file was then optimised to a local minimum.
The same procedure was carried out for the boat transition structure (but this time the initial step showed that 44 points of the specified 50 had been calculated).

{| class="wikitable" border="7" align=center style="text-align:center"
|+
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure Obtained''' || [[Image:Hoss.chair IRC.png|200px|centre]] || [[Image:Hoss.boat IRC.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.67705100|| -231.69119648
|-
| '''IRC Energy Pathway''' || [[Image:Hoss.chair IRC TotalEnergy.png|500x500px]] ||[[Image:Hoss.boat IRC TotalEnergy.png|500x500px|500x500px]]
|-
| '''IRC Gradient''' || [[Image:Hoss.chair IRC Gradient.png|500x500px|500x500px]]||[[Image:Hoss.boat IRC Gradient.png|500x500px|500x500px]]
|}

{| class="wikitable" border="7" align=center style="text-align:center"
|+ '''Final IRC Calculation - Optimisation to a Local Minimum
| || '''CHAIR'''||'''BOAT'''
|-
| '''Structure''' || [[Image:Hoss.chair IRC LocMin.png|200px|centre]] || [[Image:Hoss.boat IRC LocMin.png|200px|centre]]
|-
|'''Energy/a.u.''' || -231.69166701|| -231.69266122
|-
|'''Label''' || Gauche 2 || Gauche 3
|}
The results above indicate that the final structure obtained for the chair transition state was the Gauche 2 structure (see Appendix 2), thus the chair transition state connected 1,5-hexadienes in the Gauche 2 conformation.
The boat transition state on the other hand resulted in the gauche 3 conformation and therefore the boat transition state connects 1,5-hexadienes in the Gauche3 conformation.

== Activation Energies of the Transition States ==
The HF/3-21G optimised structures were then optimised using the more accurate and complex DFT/B3LYP/6-31G(d) level of theory. The results form this optimisation was then used to determine the activation energies for the reaction via both transition structures. A comparison of the energies of the two optimised structures using the two methods is shown below:

{| class="wikitable" border="1" align=center style="text-align:center"
|+
| || ''' CHAIR - HF/3-21G ''' || ''' CHAIR - B3LYP/6-31G(d) ''' || ''' BOAT - HF/3-21G''' || ''' BOAT - B3LYP/6-31G(d)'''
|-
| '''Energy/a.u.'''|| -231.61932 || -234.61068|| -231.60280 || -234.54309
|-
| '''Terminal C-C Bond Length/Å''' || 2.02 || 1.97 || 2.14 || 2.21
|-
| '''Imaginary Frequency/cm-1''' || -818 || -545 || -839 || -530
|}

The thermochemical data is shown below:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G(d)'''
|-
| ''' '''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
| width="100" align="center" | '''Electronic energy'''
| width="100" align="center" | '''Sum of electronic and zero-point energies'''
| width="100" align="center" | '''Sum of electronic and thermal energies'''
|-
| ''' '''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| ''' '''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
|-
| '''Chair TS'''
| align="center" | -231.61932
| align="center" | -231.46671
| align="center" | -231.46135
| align="center" | -234.61068
| align="center" | -234.41492
| align="center" | -234.40900
|-
| '''Boat TS'''
| align="center" | -231.60280
| align="center" | -231.45093
| align="center" | -231.44531
| align="center" | -234.54309
| align="center" | -234.40234
| align="center" | -234.39601
|-
| '''Reactant'''
| align="center" | -231.69254
| align="center" | -231.53954
| align="center" | -231.53257
| align="center" | -234.61171
| align="center" | -234.46920
| align="center" | -234.46186
|}.............jk
The activation energies for both transition states and hence there pathways were calculated using the thermochemical date:

{| border="1" cellspacing="1" cellpadding="10" align="center"
|-
| ''' '''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''HF/3-21G'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''B3LYP/6-31G (d)'''
| width="100" align="center" | '''Expt.'''
|-
| ''' '''
| width="100" align="center" | '''at 0 K '''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
| width="100" align="center" | '''at 298 K'''
| width="100" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 39.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

= The Diels Alder Cycloaddition =

The Diels Alder reaction is a [4pi + 2pi] cycloaddition reaction which occurs between a conjugated diene and a dienophile. The reaction is known to proceed via a 6 membered aromatic transition state, the stabilisation due to this aromatic characters helps explain why the reaction works so well. The reactions involves the formation of two new sigma bonds, which result due to the interaction between the pi orbitals on the diene and dineophile. Whether the reaction is 'allowed' or 'forbidden' is dependent on the number of electrons involved in the transition state. Specifically, the reaction will be 'allowed' if the HOMO of one of the species interacts in a favourable manner with the LUMO of the other species; this depends on factors such as the relative energy as well as the symmetry of the orbitals.

== The Diels Alder Reaction of Cis-butadiene and Ethene ==

The simplest example of the Diels Alder reaction is the reaction between cis-butadiene and ethene. Speaking from a chemistry persepctive, this is a very poor reaction as the diene is not very electron......... however it is still useful to investigate the transition states involved using computational methods.

[[Image:Hoss.DA1 mech.png|thumb|250px|Mechanism for the Diels Alder Reaction|center]]

The first step involved the optimisation of the two reactants using the AM1 semi-emperical method:

[[Image:Hoss.DA1 diene+dienophile.png|400px|center]]

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"|Reactant !! width="90"|Molecular Orbital !! Theoretical MO's !! Experimental MO's !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Cis-butadiene''' || HOMO || [[Image:Hoss.diene HOMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || -0.34382 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.diene LUMO theory.png|100px]] || [[Image:Hoss.diene HOMO exp.png|100px]] || 0.01708 || Symmetric
|- align="center"
| rowspan="2"|'''Ethene''' || HOMO || [[Image:Hoss.dienophile HOMO theory.png|75px]] || [[Image:Hoss.dienophile HOMO exp.png|100px]] || -0.38777 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.dienophile LUMO theory.png|75px]] || [[Image:Hoss.dienophile LUMO exp.png|100px]] || 0.05284 || Antisymmetric
|}

The MO's show that the symmetries of the HOMO and LUMO of the butadiene are the oppositive of the ethene. Since the orbitals can only interact if they have the same symmetry (conservation of orbital symmetry), the MO's above show that the HOMO and LUMO of the butadiene is able to interact with the LUMO and HOMO of the ethene respectively.

=== Optimisation of the Transition State ===
The transition state for this cycloaddition is known to resemble an envelope type cyclohexane structure. The optimised structures from before were copied into a new Gaussian window and orientated so as to resemble this envelope shape. The fragments were modified in that the ethene hydrogens were distorted so as to resemble a tetrahedral like structure, and the terminal C-C bond distances were set to approximately 2.00A. It was initially set to 2.20 A, however thsi resulted in an error.
[[Image:Hoss.DA1 initial guess.png|thumb|250px|Guess structure for transition state|center]]
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. After, the same calculation was carried out, but this time the more accurate DFT/B3LYP/6-31G (d) method and basis set was used. The TS(Berny) method was chosen over the QST2 method as it was less time consuming.

{| class="wikitable" border="7" style="text-align:center" align=center
|+
! '''Method''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimized Structure'''|| [[Image:Hoss.DA1 AM1 ts.png|200px]] || [[Image:Hoss.DA1 DFT ts.png|200px]]
|-
| '''C-C Bond Distance (A)'''|| 2.12 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA1 AM1 ts freq.png|200px]] || [[Image:Hoss.DA1 DFT freq.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -955.80 || -524.78
|}

The energy associated with each has not been included as a different method was used. The more accurate DFT method resulted in a larger C-C bond and a lower (imaginery) frequency - lengthening the bond results in increasing the shallowness of the potential energy slop thus resulting in a lower total energy for this transition state.
The results above indicated that only one imaginery frequency was present which confirms that the transition state had been located. For both methods, this vibration corresponds to the synchronous formation of the two sigma bonds between the terminal carbons.
--sp3c and vdwa radii--

{| class="wikitable" border="7" align="center"
|+ ''' '''
! width="90"| !! width="90"|Molecular Orbital !! Image !! Energy (a.u.) !! width="90"|Symmetry
|- align="center"
| rowspan="2"|'''Semi-empirical/AM1 - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS AM1 HOMO.png|125px]] || -0.32395 || Antisymmetric
|- align="center"
|| LUMO || [[Image:Hoss.DA1 TS AM1 LUMO.png|125px]] || 0.02316 || Symmetric
|- align="center"
| rowspan="2"|'''DFT/B3LYP/6-31G(d) - TS (Berny)''' || HOMO || [[Image:Hoss.DA1 TS DFT HOMO.png|125px]] || -0.21897 || Symmetric
|- align="center"
| LUMO || [[Image:Hoss.DA1 TS DFT LUMO.png|125px]] || -0.00863 || Symmetric
|}

Focussing on the MO's obtained form the AM1 method, the antisymmetric HOMO of the transition state is formed due to the overlap between the HOMO of the diene and the LUMO of the dienophile, both of which are also antisymmetric. This again re-iterates that only effective overlap will be achieved only when the symmetry of the interacting orbitals are the same. The LUMO of the transition state also abides by this rule and can be rationalised by considering the interaction between the LUMO of the diene and the HOMO of the dienophile. The HOMO of the transition state for the DFT method has now become symmetric, this is because upon changing the calculation method, the ordering of the orbitals changes and thus the HOMO obtained from the AM1 method may not be the same HOMO obtained from the DFT method.

== The Diels Alder Reaction of Cyclohexadiene and Maleic Anhydride ==

This section focusses on the facile Diels Alder reaction which takes place between cyclohexadiene (the diene) and maleic anhydride (the dienophile). Unlike before, this reaction is stereoselective with the endo isomer dominating over the exo isomer. The reaction proceeds under kinetic control and thus one needs to consider the role that the transition state plays. Since the endo isomer is the major product, this means that the activation energy required for its formation is less and that the transition state corresponding to this isomer is lower in energy and more stable. The stereochemistry observed here will be investigated and explained using the computational technqiues already used in thsi report.

[[Image:Hoss.DA2 mech.png|thumb|500px|Mchanism for the Diels Alder Reaction|center]]

As before, the two reactants were optimised using the semi-emperical AM1 method, this was followed by a an MO analysis:

--Image of Diene and Dienophile**

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Cyclohexa-1,3-diene'''||colspan="2"|'''Maleic Anhydride'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''||[[Image:Hoss.diene HOMO exp DA2.png|250px]]||[[Image:Hoss.diene LUMO exp DA2.png|250px]]||[[Image:Hoss.dienophile HOMO exp DA2.png|250px]]||[[Image:Hoss.dienophile LUMO exp DA2.png|250px]]
|-
|'''Energy (a.u)'''|| -0.32194 ||0.01680 ||-0.44184 ||-0.05946
|-
|'''Symmetry'''|| Antisymmetric ||Symmetric ||Symmetric ||Antisymmetric
|}

In this reaction, the dienophile is very electron poor by virtue of the electron withdrawing nature of oxygens present in the anhydride. This helps explain why this reaction is facile. Taking this into account and the fact that the diene is electron rich, one would expect the HOMO of the diene to interact favourably with the LUMO of the dienophile. It is evident from above that both are antisymmetric and thus this interaction is possible.

=== Optimisation of the Transition State ===

Instead of using fragments (i.e. the individual reactants) to form an initial guess structure for the transition state, it was thought perhaps better to modify the products (i.e. the exo and endo isomers) instead. Thus, each isomer was drawn on a seperate window in Gaussian, after which it was cleaned and sent for an initial optimisation using the semi-empirical AM1 method. The optimised structures were then modified slightly so as to resemble a possible transition state. The changes which were made included changing the dihedral angles of the hydrogens attached to the carbons involved in the bond formation as well as modifying the (terminal) C-C bond lengths to 2.0 A.
An Opt+Freq calculation to a TS(Berny) was carried out on this structure using the semi-empirical/AM1 method. The exact same process was carried out for both isomers and the results are shown below:

{| class="wikitable" border="1" align="center" style="text-align:center"
|+
! rowspan="2"|-!! colspan="2"|'''Exo''' !! colspan="2"|'''Endo'''
|-
! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)''' !! '''Semi-empirical/AM1 - TS (Berny)'''!! '''DFT/B3LYP/6-31G(d) - TS (Berny)'''
|-
| '''Optimised Structure'''|| [[Image:Hoss.DA2 AM1 ts.png|200px]] || [[Image:Hoss.DA2 DFT ts.png|200px]] || [[Image:Hoss.DA2 AM1 ts ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts ENDO.png|200px]]
|-
| '''Energy (a.u)'''|| -0.05050324 || -604.80884528 || -0.05159422 || -612.68339678
|-
| '''C-C Bond Distance (A)'''|| 2.17 || 2.38 || 2.16 || 2.27
|-
| '''Vibration corresponding to TS''' || [[Image:Hoss.DA2 AM1 ts freq.png|200px]] || [[Image:Hoss.DA2 DFT ts freq.png|200px]] || [[Image:Hoss.DA2 AM1 ts freq ENDO.png|200px]] || [[Image:Hoss.DA2 DFT ts freq ENDO.png|200px]]
|-
| '''Imaginary Frequency (cm-1)'''|| -811.40 || -485.90 || -805.91 || -446.97
|}

In Diels-Alder reactions such as the one considered here, if the reaction proceeds under kinetic control then the endo product usually forms exclusively over the exo form. Woodward and Hoffmann believed that the reason was due to orbital interactions. Investigation of the transition states have shown that orbital symmetry favours the endo conformer over the exo conformer. <ref>The conservation of orbital symmetry.
R. Hoffmann, R. B. Woodward, ''Acc. Chem. Res.'' , 1968 , '''1''' , 17 - 22: {{DOI|10.1021/ar50001a003}}.</ref>
The above results show tha the endo conformer is the more stable. This is unsurprising when one takes into account the fact that there are repulsive steric interactions occuring between oxygen atoms and the bridghead group. This steric hindrance is absent for the endo case whereby the anhydride is on found on the opposite face of the bridgehead group. Thus one can conclude that the endo isomer in this case is both kinetically and thermodynamically favoured.

Investigation of secondart orbital overlap also explains why the endo form is more stable. This takes into account interactions between atoms which do not form a bond with each other. The experimentally calculated MO's are shown below:

{| class="wikitable" border="7" style="text-align:center" align=center
|+
|-
|rowspan="2"| ||colspan="2"|'''Exo'''||colspan="2"|'''Endo'''
|-
| '''HOMO'''||'''LUMO'''||'''HOMO'''||'''LUMO'''
|-
|'''Molecular Orbital'''|| [[Image:Hoss.DA2 TS DFT HOMO.png|250px]]||[[Image:Hoss.DA2 TS DFT LUMO.png|250px]]||[[Image:Hoss.DA2 TS DFT HOMO ENDO.png|225px]]||[[Image:Hoss.DA2 TS DFT LUMO ENDO.png|250px]]
|-
|'''Symmetry'''|| Antisymmetric || Antisymmetric || Antisymmetric || Antisymmetric
|}

All the MO's are antisymmetric. The HOMO for both isomers have significant orbotal overlap and thus areas of high electron density between the terminal carbons of the reactants and products. This MO thus describes the newly formed sigma bond and occurs as a result of the interaction between the HOMO of the diene and the LUMO of the dienophile. The LUMO for both cases shown lots of anti-bonding character as can be seen by the several number of nodes present.
Woodward and Hoffmann rationalised the stability of the endo isomer by showing that there existsed a stabilising secondary-orbital overlap in the endo transition state<ref>R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[http://pubs.acs.org/doi/abs/10.1021/ja00947a033]</ref>. Specifically, this concenrs the overlap between pi* orbitals on the C=O and the conjugated pi system on the diene. This stabilisation cannot occur for the exo transition state, thus explaining why the activation energy is higher.

Comparisons could aldo be made between the qualitatively determined LCAO and the experimentally determined MO's:

[[Image:Hoss.DA2 LCAO.png|300px|center]]

The stabilising interactin described here was shown above on the LCAO diagram and can be compared to the following MO found for the endo isomer:

[[Image:Hoss.LUMO+1+2 DA2.png|500px|center]]