Introduction

In a reaction, the transition state is the point along the reaction co-ordinate at which there is the highest potential energy. Due to it's very nature it is not possible to "trap" the transition structure chemically - it is very unstable - and so instead we can only observe such states of the reaction computationally, which is what is observed throughout this experiment.

This can be achieved through the use of Gaussview, software that works out the potential energy surfaces of molecules and their reaction geometries. There are a number of different calculations that can be applied to the molecules being studied; optimisation forms the lowest energy conformation of a molecule, whilst TS(Berny) and TS(QST2) methods both produced results for the transition structure potential energy surfaces.

Within these methods, there are further settings that are changed within Gaussview. For this experiment we will be using two sets of methods; HF/3-21G and B3LYP/6-31G*, the latter of which is a higher level of theory. HF otherwise known as Hartree-Fock which is a method of calculation that takes the minimum energy for a single determinant, through the use of approximating the spin orbitals of the molecule. B3LYP is a sub setting of DFT or Density Functional Theory, which uses functions of functions to find the electronic density of the ground state of the molecule. 3-21G and 6-31G* are both basis sets which are used by Gaussview to linearly combine basis functions in order to determine the structure's molecular orbitals.

Semi-empirical methods are used for large molecules, in cases where the cost of using the HF method would be too expensive; in terms of both time and computer power. These methods make a number of approximations and use empirical data in order to reduce the number of calculations being made. Within this setting, there are further methods that can be examined. For this experiment we will be using the Austin Model 1 or AM1 setting, which neglects certain integrals.

Nf710 (talk) 13:06, 24 February 2016 (UTC) Fairly good basic understanding of the methods.

The Cope Rearrangement Tutorial

Background

The [3,3] sigmatropic rearrangement of carbon atoms is known as the Cope Rearrangement, which is a type of pericyclic reaction. This requires for the Woodward-Hoffman rules to be obeyed.

For the Cope Rearrangement to be thermally allowed then the number of suprafacial components ((4q+2)_s) plus antarafacial ((4r)_a) must be odd. If the sum of these components is even, then whilst the reaction is thermally disallowed, it will be photochemically allowed.

For this section of the experiment, we will be examining different transition structures of the Cope Rearrangement of 1,5-hexadiene. This reaction in itself is not chemically useful (as the reactant is the same as the product!), but for larger molecules the Cope Rearrangement can be a useful synthetic tool. An example of this is in the synthesis of citral, which has a [3,3] sigmatropic rearrangement as an intermediate step, as is discussed on page 949 of Organic Chemistry by Clayden.¹

Nf710 (talk) 13:08, 24 February 2016 (UTC) Good mechanistic understanding

Optimising the Reactants and Products

Optimising the "Anti" conformation of 1,5-hexadiene

1,5-hexadiene was drawn in Gaussview with an anti linkage about the central atoms (dihedral angle of 180^o). This molecule was then optimised using the Hartree-Fock(HF)/3-21G parameters in order to give the positions of the nuclei in their lowest energies. The .chk file was opened and the energy found to be -231.69260235 a.u., corresponding well with the figure given in Appendix 1 to be the anti1 linkage. Once the "Symmetrize" function was applied, a point group of C₂ was shown, which also matches the results given in Appendix 1 for anti1.

anti1 structure

Optimising the "Gauche" conformation of 1,5-hexadiene

1,5-hexadiene was drawn in Gaussview with a gauche linkage about the central atoms (dihedral angle of 60^o). It is expected that this structure to have a higher energy than the previous structure, due to the conformational analysis energies often placing the gauche conformer higher in energy than the antiperiplanar conformer.

This molecule was again optimised using the Hartree-Fock(HF)/3-21G parameters in order to give the positions of the nuclei in their lowest energies. The .chk file was opened and the energy found to be -231.69199701 a.u.. This is in fact a lower energy structure than the "anti" structure, with ΔE = -0.00093534a.u..

The point group was found to be C₂ after running the "Symmetrize" function, suggesting that the structure optimised was gauche2.

gauche2 structure

Lowest Energy conformers of 1,5-hexadiene

For the "anti" conformer, it is expected that the anti1 structure is that that is the most stable. This is due to σ-conjugation of the C-H bonding orbital into the π* orbital of the C=C bond, as well as the low amount of steric clashing by the alternation of the position of the C-H bonds about the 4 central carbons. It is also expected that all of the anti structures will have a lower energy than the gauche structures due to conformational analysis as previously discussed.

Gauche3 is the lowest energy type of the gauche conformer of 1,5-hexadiene. It has a dihedral angle of 60^o, and a C₁ point group. Upon optimisation it was found that this conformer has an energy of -231.69266120 a.u. which corresponds to the lowest energy conformation of the 1,5-hexadiene. This suggests that for the molecule, the gauche conformation is actually lower in energy than the anti conformation.

The reason for this is likely two-fold. The structure appears to have a lower amount of steric clashing between the hydrogens, due to the twisted nature of the central chain. There may also be some long range interactions from the two double bonds as they are parallel to one another (and hence the π and π* orbitals can interact with one another allowing for stabilisation, and hence a lower energy conformer).

Nf710 (talk) 13:18, 24 February 2016 (UTC) Correct conclusion you could have shown this with the orbitals from the .chk file.

gauche3 structure

Optimising the "anti2" conformer of 1,5-hexadiene

When optimised at the HF/3-21G level of theory, the anti2 structure, which has a C_i point group, has an energy of -231.69253528 a.u., which is slightly lower than the figure given in Appendix 1 (however this could be due to rounding), and is comparatively higher in energy than the anti1 structure.

Comparison of HF/3-21G and B3LYP/6-31G* optimisations

The best comparison of the two theory levels is to look at the bond lengths and angles that they are optimised to. As we are analysing the anti2, C_i conformation of the 1,5-hexadiene, it is possible to look at the bond lengths of the C=C and C-C linkages, and to compare the two theory levels with literature.

B3LYP optimised anti2 1,5-hexadiene

HF optimised anti2 1,5-hexadiene

Bond Lengths (Å)

Dihedral Angle (^o)

Bond

HF

B3LYP

Bond

HF

B3LYP

C=C

1.316

1.333

C=C-C-C

114.69

118.57

C-C (adjacent to C=C)

1.509

1.504

=C-C-C-C=

180.00

C-C (central linkage)

1.553

1.548

As can be seen in the literature, the bond lengths of the anti2 structure are closer to the literature when optimised to the B3LYP level of theory, than they are when optimised to the HF theory level.

For the dihedral angles, we would expect the C=C-C-C linkage to be 120, which once again makes the B3LYP theory more suitable.

However, as can be seen for the C-C bonds, and the saturated dihedral angle, the HF correlates fairly well with the experimental results, suggesting that the use of B3LYP is only justified if the compound is unsaturated.

Vibrational modes and energy modes for "anti2" 1,5-hexadiene

anti2-1,5-hexadiene vibration

This higher level of optimisation allows for an accurate descriptor of the key vibrations of the bonds in the 1,5-hexadiene (anti2 structure). Analysis shows that there are no negative vibrations, and hence no imaginary vibrations. If there had been an imaginary vibration, it would have corresponded to the transition state of the molecule. This is due to the fact that the quantum harmonic oscillator has a force constant, k. If this is negative, then there will be an imaginary vibration will occur. As the bonds are at their maxima at the transition state, and hence their weakest, this is the point at which the quantum harmonic oscillator will have a negative value for k.

Nf710 (talk) 13:35, 24 February 2016 (UTC) No weakest so to speak. The second derivative of the PES gives the force constant and hence when that is negative the PES is at a maximum and hence a TS

Further examination of the B3LYP/6-31G* .log output file shows fundemental energies of the anti2-1,5-hexadiene.

The sum of electronic and zero point energies:
- 234.469215a.u.

The sum of electronic and thermal energies
- 234.461866a.u.

The sum of electronic and thermal enthalpies
- 234.460922a.u.

The sum of electronic and thermal free energies
- 234.500803a.u.

Optimising the "Chair" and "Boat" Transition Structures

Allyl fragment optimsation

Prior to the examination of the Chair and boat transition structures, the acyl fragment was optimised to the HF/3-21G theory level.

Optimised allyl fragment

Chair Transition Structure

Imaginary vibrational mode of chair TS

Reoptimised chair TS

The two allyl structures were drawn in Gaussview, and set in the Chair transition structure, parallel to one another (yet flipped), with the distance between the two fragments being 2.1194Å.

A calculation on Gaussview was then run, with the settings being; Opt+Freq, with optimisation being to TS (Berny), and the force constants being calculated once. Opt=NoEigen was the additional keyword. The calculation provided a C₁ point group, with an energy of -231.61932246a.u. The vibrational modes found following the calculations were real vibrations for the main part, with one imaginary vibration at -817.94cm^-1 corresponding to the concerted reaction of the two allyl fragments in the Cope rearrangement reaction. The reason this appears as a vibration (even though it is imaginary) is due to the fact that for the bonds to break the bonds must overcome the force constant, k, of the quantum harmonic oscillator. Nf710 (talk) 13:27, 24 February 2016 (UTC) you must get out of the local minimum for that degree of freedom The guess structure was then pasted into a new window, and the co-ordinates between the bonds that will be formed/broken in the reaction were set to freeze, and the optimisation set to a minimum. The settings appeared as Freeze bond (1,14) and Freeze bond (6,9). Using the Redundant Coordinate Editor from the edit menu, the Hessian guess method was used in order to determine the distance between the bonds during the bond forming and breaking. This was found to be 2.02032Å.

Boat Transition Structure

The chair transition structure is not the only possible intermediate, with the boat conformation also being plausible. The atoms on the 1,5-hexadiene were numbered in order to see how the carbons "move" between the reactants and products.

Using the TS(QST2) method of calculation, the reactant and products were numbered as shown below, and optimised. However, this method failed, with an impossible geometry of the transition structure being formed as show.

Numbering of reactant and product for Cope rearrangement	Transition structure of failed QST2 calculation	Summary of failed QST2 calculation

This issue was resolved by changing the central dihedral angle, C₂-C₃-C₄-C₅ for the reactant, C₂-C₁-C₆-C₅ to 0^o and changing the internal angles to 110^o. This allowed for the transition structure to converge to the boat conformation.

Imaginary vibrational mode of QST2 optimised boat TS

As can be seen from the Jmol file above, there is an imaginary vibration at -840.23cm^-1. There are also no disallowed structures, suggesting that this is an appropriate guess for the transition structure.

Intrinsic Reaction Co-ordinate

In order to determine the minimum of the potential energy surface of this reaction, calculations were run using the Intrinsic Reaction Coordinate method, for 50 jobs. This did not create a completely minimised reaction surface, and so three separate methods were employed:

i) The final state (of 44) of the reaction energy surface was minimised under the standard optimisation settings of HF/3-21G in order to give a quick and easy minimisation.

ii) The IRC calculation was rerun using 100 steps rather than 50 steps. This took a little longer.

iii) The most accurate, yet time consuming method, was to switch the Calculate Always setting to Calculate Once and this calculation switched from every 3rd step, to every step over all 50 iterations.

Intrinsic Reaction Co-ordinate of the Cope Rearrangement of 1,5-hexadiene

The total energy plot shows how the transition state tends to a lower energy as the reaction co-ordinate increases. The RMS Gradient Norm graph shows the derivative of the total energy plot, and when this value is zero, then the product has been produced.

Nf710 (talk) 13:37, 24 February 2016 (UTC) you could have found out the connecting conformer by optimising down the final IRC conformer

Higher Optimisation Levels

So far, the chair and boat transition structures have only been optimised to the HF/3-21G level of theory. However, it can be optimised to a higher level, i.e. B3LYP/6-31G* theory level. This was achieved by taking the optimised output files for the lower levels of theory and optimising to the higher level.

For the chair transition structure, the output energy was -234.55698303a.u. and the point group was C₁.

For the boat transition structure, the output energy was -234.54309307a.u. and the point group was C₁.

These are slightly different energies, whilst the geometries appear to be relatively similar. The only difference is that down the plane of the two molecules (looking from a side on point of view), the chair is a mirror plane, whilst the boat has the two molecules superimposed onto one another.

Using the figures from the script, it is possible to determine the activation energies from the transition state energies, and that of the reactant in the anti2, as well as the conversion of 1a.u.(hartree)=627.508 kcalmol^-1. This gives the following activation energies (product E - reactant E):

Transition State	Activation Energy (kcalmol^-1)	Within Experimental Range?
Chair	627.509((-234.55698303)-(-234.611710))=34.34	Outside of 33.5 ± 0.5 kcalmol^-1 range
Boat	627.509((-234.54309307)-(-234.611710))=43.06	Inside of 44.7 ± 2.0 kcalmol^-1 range

Nf710 (talk) 13:42, 24 February 2016 (UTC) You have not accounted for zero point energy!

This suggests that the chair transition state has a lower activation energy (ΔE) than the boat transition state, and correlates well with the figures given in the script, but not perfectly.

The above results correspond to the reaction at 0K, which does not happen often. Instead, reactions tend to take place at a higher temperature, such as room temperature, which is 298.15K. These figures are also calculated in the process, with the corresponding results.

Transition State 298.15K	Activation Energy (kcalmol^-1)
Chair	627.509((-234.443814)-(-234.461856))=11.32
Boat	627.509((-234.431751)-(-234.461856))=18.89

Nf710 (talk) 13:46, 24 February 2016 (UTC) In general you folowed the script but you have made a few mistakes and missed out a few things. Also your knowledge of the theory could have been much better.

The Diels-Alder Cycloaddition

Background

The Diels-Alder cycloaddition is arguably the most famous pericyclic reaction. It is a [4+2] cycloaddition and requires a diene and a dienophile, the latter of which is often an alkene with an electron withdrawing group attached (although not always).

There are usually two potential products of the Diels-Alder cycloaddition; the endo and the exo product, which are dependent upon the transition state of the reaction, and which is formed will be the focal point for this section of the reaction.

As with all pericyclic reactions, this [4+2] cycloaddition is dependent upon the Woodward-Hoffman rules which were described in the Background section of the Cope Rearrangement.

Cis Butadiene Optimisation

Using the AM1 semi-empirical method for the calculations, it was possible to visualise the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the cis-butadiene.

The HOMO of the cis-butadiene had a relative stabilisation energy of -0.34381 whilst the LUMO had a relative energy of 0.01707, with a positive value being destabilising (antibonding). The HOMO was anti-symmetric whilst the LUMO was symmetric. The JMol versions of the molecular orbitals for this reaction can be found by following the link on the image description.

HOMO of cis-butadiene - anti-symmetric	LUMO of cis-butadiene - symmetric	summary of cis-butadiene AM1 optimisation

(These are the HOMO and LUMO of butadiene+ethene Tam10 (talk) 11:55, 23 February 2016 (UTC))

Ethene Optimisation

Using the AM1 semi-empirical calculation, the ethene was optimised.

HOMO of Ethene - symmetric	LUMO of Ethene - antisymmetric	summary of optimisation

The HOMO of ethene is symmetric, whilst the LUMO is antisymmetric.

Cis Butadiene + Ethylene

Once again using the AM1 semi empirical method, the transition structure (set to TS(Berny)) was optimised in order to show how the Diels-Alder cycloaddition progresses. The HOMO and LUMO were once again anti-symmetric and symmetric respectively, with relative energies of -0.32393 and 0.02314.

The ethylene dienophile is attacking from above the plane of the planar diene, with the double bond lengths being 1.38187Å and the single bond length being 1.39745Å. The distance of the C-C bonds for the newly formed bonds was 2.11931Å suggesting that the transition state is early. The Van der Waals radius of carbon is 1.7Å which also hints towards the idea that the transition state is early on in the reaction co-ordinate.

By checking the imaginary vibration at -955.99cm^-1 then it is possible to visualise the transition state as being synchronous (or concerted). This is the opposite to the lowest positive frequency, 147.16cm^-1, which appears to have only one bond forming or breaking at a time.

(There's no bond forming/breaking in the lowest positive frequency. The positive frequency suggests the geometry is at a minimum wrt that normal mode ie stable. In addition, the C-C distances barely change during that vibration Tam10 (talk) 11:55, 23 February 2016 (UTC))

Imaginary vibrational mode of chair TS

First positive vibrational mode of chair TS

(These diagrams seem to conflict with your Jmols as the Jmols aren't using the optimised geometry. Also, which MOs of butadiene and ethene form the HOMO and LUMO of the TS and why is the reaction allowed (from a purely symmetry point of view)? Tam10 (talk) 11:55, 23 February 2016 (UTC))

Intrinsic Reaction Co-ordinate

An IRC calculation was run from the .chk file that was produced following the optimisation of the transition structure. When run with 50 steps, and at the HF/3-21G theory level, the Diels-Alder addition of ethene to butadiene to form cyclohexene is visualised.

The IRC seems to show that the reaction stabilises the two reactants (as would be expected from a thermodynamically favourable reaction), and that the final product is reached by the end of the animation. The peak on the RMS Gradient Norm against IRC graph suggests that at 1.5 on the IRC, the reaction reaches a "point-of-no-return", from which the product of cyclohexene will be formed and the molecules will not move away from one another. As can be seen on the RMS graph, the RMS is tending to zero as the IRC progresses, suggesting that the product is formed.

(RC=1.5 shows where the gradient is at its steepest. The "point-of-no-return" as it were would be just after the TS is crossed Tam10 (talk) 11:55, 23 February 2016 (UTC))

The IRC of the DA cycloaddition of ethene and butadiene

Cyclohexa-1,3-diene + Maleic Anhydride

During the Diels-Alder reaction there are two possible modes of attack for the reactants. One mode is the exo, the other is the endo - with the first being thermodynamically favourable and the latter being kinetically favourable.

By drawing the products of the reaction in gaussview and then setting the "newly formed" bond lengths to 2Å, and the π-bond length to 1.44Å, the transition state was optimised under TS(Berny), with the AM1 semi-empirical method for both the endo and exo transition states.

Endo Transition State

Exo Transition State

Once the optimisation had been run the following results were achieved (All MO Jmol files have been posted on a separate wiki page that have been linked to in the anti-symmetric/symmetric designations):

TS type	HOMO	LUMO	Activation Energy (a.u.)	Activation Energy (kcalmol^-1
Endo	Anti-symmetric	Anti-symmetric	-0.05150479	-32.3197
Exo	Anti-symmetric	Anti-symmetric	-0.05041985	-31.6389

(These are absolute energies of the calculation, not the activation energy. The activation energy can be calculated as the energy of the TS minus the sum of the energies of the reactants Tam10 (talk) 11:55, 23 February 2016 (UTC))

As can be seen, the Exo conformer of the transition structure has a higher activation energy (more positive) than the Endo conformer, making the latter the kinetic product of the Diels-Alder reaction. In fact, the energy difference between the two conformers is ΔE=0.6808kcalmol^-1.

The Secondary Orbital Overlap Effect allows for the transition state to be stabilised by π-orbital overlap between the HOMO on the diene and the LUMO on the dienophile, that allows for the endo to be more stable a transition state than the exo transition state.

As can be seen by the HOMO of the endo transition state, there is very good overlap of the HOMO and LUMO of the two reactants, exemplified by the large MO between the two in the transition state. The polarity of the MO has switched for the exo, but the MO between the two does not appear to be as large, possibly pertaining to worse secondary orbital overlap effects.

(Again, you're not using the correct geometries for the MOs here. Tam10 (talk) 11:55, 23 February 2016 (UTC))

However, the afformentioned secondary orbital overlap effect, results in the endo product having lower energy due to the π orbitals in the diene and the π bonds of the C=O interacting. This results in the HOMO having an attractive interaction between these two compounds. However the calculation does not take this into effect.

This is exemplified by the fact that the output of the calculation for both HOMO's the endo and exo have a node about the O=C-CH₂C=O part of the maleic anhydride, which suggests it does not get heavily involved in the reaction. This does not agree with the literature².

Examining the bond lengths of the transition structures shows the potential steric effects that these compounds are undergoing.

Bond	Bond Length	Endo TS of DA cycloaddition numbered
C7-C9	1.40848
C7-C20	1.48921
C20-O19	1.40897
O19-C22	1.40896
C22-C7	1.48923
C4=C1	1.39730
C3-C4	1.39304
C3-C13	1.49054
C13-C16	1.52297
C16-C2	1.49052
C2-C1	1.39304

Bond	Bond Length	Exo TS of DA cycloaddition numbered
C1-C4	1.41011
C4-C17	1.48820
C17-O21	1.40963
O21-C18	1.40963
C18-C1	1.48820
C15=C16	1.39676
C3-C15	1.39438
C3-C10	1.48976
C10-C7	1.52208
C7-C2	1.48976
C2-C16	1.39438

The σ-bond being formed in the exo transition structure is longer than the endo form.

(What is the length of this σ-bond? Tam10 (talk) 11:55, 23 February 2016 (UTC))

On top of this, the exo form is more strained due to the fact that the C-C bonds that are formed in the exo structure are in the equatorial position of the cyclohexene ring that is formed, whilst the endo product is in more an axial position, anti to the ethane bridge across the ring, and so the exo product is also more susceptible to steric clashing with the ethane bridge than the endo product is.

As discussed in the Introduction, AM1 semi-empirical method takes empirical results and inputs them rather than calculating them. This results in some effects potentially not being taken into account as the programme runs, and so the energies calculated may vary from the experimental results for the transition state.

Intrinsic Reaction Co-ordinates

An IRC calculation was run for both the endo and the exo forms of the Diels-Alder cycloaddition of maleic anhydride and cyclohexadiene.

Endo

The Endo IRC had the features as shown.

Intrinsic Reaction Coordinate graphs of the endo DA cycloaddition of maleic anhydride and cyclohexadiene

Exo

The Exo IRC had the features as shown.

IRC of the DA cycloaddition of the maleic anhydride + cyclohexadiene exo form

For both of these RMS graphs it can be seen that the product is being formed as the RMS tends to zero.

Conclusion

For this experiment, two key reactions were examined; the Cope Rearrangement and Diels-Alder cycloaddition, through the use of the software Gaussian.

1,5-hexadiene was used to examine it's Cope Rearrangement with both the chair and boat transition structures. These were modeled using both the HF/3-21G and B3LYP/6-31G* levels of theory. Both of these transition states had one negative frequency vibration, or imaginary frequency, which corresponds to a negative k, and the mechanism of the reaction. Examining the different levels of theory further seemed to suggest that the B3LYP/6-31G* level of theory, whilst more time consuming and computationally expensive, is a better model for the true structure if the molecule is unsaturated than the HF/3-21G theory level.

Two different Diels-Alder reactions were examined throughout this study; that of ethene + cis-butadiene, and cylohexa-1,3-diene + maleic anhydride. For these calculations, the semi-empirical AM1 level of theory was used.

In the ethene + cis-butadiene reaction, it was found that the reaction was relatively simple, running in accordance of the FMO theory. The reaction is allowed thermally with regards to the Woodward-Hoffmann rules. The cylohexa-1,3-diene + maleic anhydride did not follow FMO theory, and instead the kinetic, endo product was formed due to it's lower activation energy barrier. The exo had a higher activation energy, as well as a higher level of steric hinderance. The downfall, however, for these calculations was the lack of the secondary orbital overlap effect in the measurements, which plays a major part in stabilising the endo transition state.

If this experiment were to be done again, then it would be best to run the Diels-Alder calculations at a higher level of theory, perhaps through a supercomputer if possible, in order to provide more accurate results. It would also be interesting to examine a chemically useful Cope Rearrangement (i.e. use an asymmetric molecule), as well as to examine reactions that are not thermally allowed, in accordance with the Woodward-Hoffmann rules, to see if the output of the calculations recognises this issue.

References

1. Clayden. Organic Chemistry, 1st Edition. Oxford University Press.

2. Fleming, I (1976). Frontier Orbitals and Organic Chemical Reactions. Wiley-Interscience: John Wiley & Sons Ltd.

Further references have been directly linked to in the text.