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:

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 Appendix 1, the structures were identified as:

**Energies of anti and gauche optimisation**
Conformer	Structure	Energy (a.u)	Point Group
Anti 1		-231.6926	C₂
Gauche 3		-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:

**Energies of all app and gauche optimisation**
Anti FORM	Energy (a.u)	Point Group	Gauche FORM	Energy (a.u)	Point Group
	-231.6926	C₂		-231.6878	C₂
	-231.6925	C_i		-231.6917	C₂
	-231.6891	C_2h		-231.6927	C₁
	-231.6910	C₁		-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:


HOMO - Anti 1	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:

**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		1.32	1.51	1.55	-231.69253528
DFT/B3LYP/6-31G (d)		1.33	1.51	1.55	-234.6117038
Literature^[1]		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 C_i 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 180^oC 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:

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:

**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:

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:


	Simple TS (Berny) Optimisation	Frozen Coordinate Method Optimisation
Optimised Structure
Energy (a.u.)	-231.61932241	-231.61932239
Terminal C-C bond length/Å	2.02	2.02
Imaginary Frequency/cm^-1	-818	-818
Vibration

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:

Having changed the numbers, a QST2 optimisation calculation was ran to determine the transition state and the following structure was obtained:

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:

Having done this for both the reactant and the product, the QST2 optimisation was set up again and produced the following results:

**QST2 Optimisation**
Optimised Structure
Energy (a.u.)	-231.60280199
Terminal C-C bond length/Å	2.14
Imaginary Frequency/cm^-1	-839
Vibration

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 43^rd 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).

**Table showing the results from the IRC calculation when the force constant was calculated at every step**
	CHAIR	BOAT
Structure Obtained
Energy/a.u.	-231.67705100	-231.69119648
IRC Energy Pathway
IRC Gradient

**Final IRC Calculation - Optimisation to a Local Minimum**
	CHAIR	BOAT
Structure
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 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:


	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:

	HF/3-21G			B3LYP/6-31G(d)
	Electronic energy	Sum of electronic and zero-point energies	Sum of electronic and thermal energies	Electronic energy	Sum of electronic and zero-point energies	Sum of electronic and thermal energies
		at 0 K	at 298 K		at 0 K	at 298 K
Chair TS	-231.61932	-231.46671	-231.46135	-234.61068	-234.41492	-234.40900
Boat TS	-231.60280	-231.45093	-231.44531	-234.54309	-234.40234	-234.39601
Reactant	-231.69254	-231.53954	-231.53257	-234.61171	-234.46920	-234.46186

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

	HF/3-21G	HF/3-21G	B3LYP/6-31G (d)	B3LYP/6-31G (d)	Expt.
	at 0 K	at 298 K	at 0 K	at 298 K	at 0 K
ΔE (Chair)	45.71	44.70	39.04	33.15	33.5 ± 0.5
ΔE (Boat)	55.60	54.76	41.94	41.32	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.

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


Reactant	Molecular Orbital	Energy (a.u.)	Symmetry
Cis-butadiene	HOMO	-0.34382	Antisymmetric
Cis-butadiene	LUMO	0.01708	Symmetric
Ethene	HOMO	-0.38777	Symmetric
Ethene	LUMO	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.

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.


Method	Semi-empirical/AM1 - TS (Berny)	DFT/B3LYP/6-31G(d) - TS (Berny)
Optimized Structure
Terminal C-C Bond Distance (Å)	2.12	2.27
Vibration corresponding to TS
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:


	Molecular Orbital	Energy (a.u.)	Symmetry
Semi-empirical/AM1 - TS (Berny)	HOMO	-0.32395	Antisymmetric
Semi-empirical/AM1 - TS (Berny)	LUMO	0.02316	Symmetric
DFT/B3LYP/6-31G(d) - TS (Berny)	HOMO	-0.21897	Symmetric
DFT/B3LYP/6-31G(d) - TS (Berny)	LUMO	-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.

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


	Cyclohexa-1,3-diene		Maleic Anhydride
	HOMO	LUMO	HOMO	LUMO
Molecular Orbital
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:


-	Exo		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
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
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. ^[2] 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:


	Exo		Endo
	HOMO	LUMO	HOMO	LUMO
Molecular Orbital
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^[3]. 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:

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

↑ G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69
↑ R. Hoffmann, R. B. Woodward, Acc. Chem. Res. , 1968 , 1 , 17 - 22: DOI:10.1021/ar50001a003 .
↑ R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[1]

[1] G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69

[2] R. Hoffmann, R. B. Woodward, Acc. Chem. Res. , 1968 , 1 , 17 - 22: DOI:10.1021/ar50001a003 .

[3] R. Hoffmann, R.B. Woodward, J. Am. Chem. Soc., 1965, 87, 4388[1]

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