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Module 3

Introduction

The kinetics of a reaction and the selectivity of a kinetically controlled reaction can be predicted from the energy or relative energies of the transition states. In this experiment, the several transition structures will be studied through computational modelling. Molecular orbital-based methods, through numerically solving the Schrödinger equation, will be used. The location of transition structures can be predicted based on the local shape of a potential energy surface. Reaction pathways and barrier heights can also be calculated.

The Cope Rearrangement of 1,5-Hexadiene

In this section, the Cope rearrangement of 1,5-hexadiene will be studied. The [3,3]-sigmatropic shift rearrangement has been the subject of numerous experimental and computational studies and the reaction mechanism has been a source of controversy for some time. The DFT B3LYP method has given relatively accurate results in the calculation of transition structures.


The Cope rearrangement of 1,5-hexadiene


Optimising the Reactants

A few of the possible conformations of 1,5-hexadiene were drawn in GaussView and optimised in Gaussian using the Hartree-Focke method and 3-21G basis set. Firstly an anti conformation was optimised and this was expected to be the lowest energy conformer due to the minimal steric interactions. (The numbering of the conformations is matched to Appendix 1 on the tutorial page.) Hence the first conformation drawn was the Anti-1 conformation. The corresponding gauche conformation (Gauche-4) was drawn and optimised in the same manner. The energy of this conformation was found to be ~2.8 kJ mol-1 higher in energy than the Anti-1 conformation.

However it was found in literature[1] that the gauche conformation was in fact more stable. The Gauche-3 conformation was drawn and optimised using the same methods and came to a lower energy than the Anti-1 conformation by ~0.2 kJ mol-1. This stability is due to the favourable donation of electron density from the π orbital of the C=C bond to the σ* orbital of the C-H bond to the vinyl proton.[2]. The results summaries of the optimisations are shown below.


Jmol
Calculation Type FOPT FOPT FOPT
Calculation Method RHF RHF RHF
Basis Set 3-21G 3-21G 3-21G
Charge 0 0 0
Spin Singlet Singlet Singlet
Total Energy (RHF) -231.69260 a.u. -231.69153 a.u. -231.69266 a.u.
Appendix 1 energies -231.69260 a.u. -231.69153 a.u. -231.69266 a.u.
RMS Gradient Norm 0.00002 a.u. 0.00002 a.u. 0.00002 a.u.
Dipole Moment 0.20 Debye 0.13 Debye 0.34 Debye
Point Group C2 C2 C1


The energies for the conformations are identical to those found in Appendix 1 to 5 decimal places indicating that the correct conformation was optimised to in each case.

Optimising the Anti-2 conformer

The Ci Anti-2 conformation from Appendix 1 was drawn and optimised using at the HF/3-21G level and then optimised again using a DFT method at the higher B3LYP/6-31G* level.


Jmol
Calculation Type FOPT FOPT
Calculation Method RHF RB3LYP
Basis Set 3-21G 6-31G(d)
Charge 0 0
Spin Singlet Singlet
Total Energy -231.69254 a.u. (RHF) -234.61171 a.u. (B3LYP)
Appendix 1 energies -231.69154 a.u. N/A
RMS Gradient Norm 0.00000 a.u. 0.00001 a.u.
Dipole Moment 0.00 Debye 0.00 Debye
Point Group Ci Ci


The two optimised geometries appear to look very similar at first sight. The difference in total energies cannot be compared as they were measured using different methods. However specific bond lengths, bond angles and dihedral angles were compared and these are shown below.


Comparison of geometries from optimisations using HF/3-21G and B3LYP/6-31G* methods
Parameter Bond length/ Å
Bond angle/ °
HF/3-21G B3LYP/6-31G* Literature[3]
C=C 1.316 1.334 1.340±0.003
=C-C 1.509 1.504 1.508±0.012
C-C 1.553 1.548 1.538±0.027
C-C-C 111.3 112.7 111.5±0.9
C-C=C 124.8 125.3 124.6±1.0


The optimised geometries are relatively similar to the results found in literature. The two geometries found through the different optimisation methods do contain small differences between them. In terms of bond lengths, the B3LYP/6-31G* level of optimisation gives results closer to the literature values but in terms of bond angles it seems the HF/3-21G level gives results closer to the literature.

The total energies given in the output represent the energy of the molecule on the bare potential energy surface and not experimentally measurable results. In order to compare the data with literature a frequency analysis needs to be performed. The frequency analysis can also be used to confirm that the geometry found is a minimum. If all vibrational frequencies are real and positive then the critical point is a minimum in energy, if one of the vibrational frequencies is negative then it is a transition state, and if more than one of the frequencies are negative then the optimisation has failed.

Frequency Analysis of the Anti-2 conformer
IR spectrum of Anti-2 conformer

A frequency analysis of the B3LYP/6-31G* optimised structure was carried out at the same level of theory. The low frequencies (zero frequencies) found in the .log file range from -9 to 13 indicating the level of accuracy in the optimisation is relatively low. However there were no negative frequencies indicating that the geometry found is indeed a minimum in energy. The .log file for the frequency analysis can be found here. The IR spectrum was also simulated and is shown on the left.

The .log file also contained thermochemical data. The sum of electronic and zero-point energies is the potential energy at 0K including the zero-pont vibrational energy (E1 = Eelec + ZPE). The sum of electronic and thermal energies is the energy at 298.15K and 1atm which includes contributions from the translational, rotational and vibrational energy modes at this temperature (E2 = E1 + Evib + Erot + Etrans). The sum of electronic and thermal enthalpies contains an additional correction for RT (H = E +RT). Finally the sum of electronic and thermal free energies includes the entropic contribution to the free energy (G = H - TS).


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 234.500777


Optimising the Transition Structures

The Cope rearrangement is believed to occur through two possible transition structures[4], the chair (C2h) and the boat (C2v) (a third possible transition structure involves a biradical chair transition structure). Both transition structures consist of two C3H5 allyl fragments positioned approximately 2.2 Å apart. This section involves the optimisation of these transition states through several methods.

Optimising the Chair Transition Structure

The C3H5 allyl fragment was optimised using the HF/3-21G level of theory. Two of these fragments were then combined in an approximate chair transition state as seen in Appendix 2 with the distance between the terminal ends of the allyl fragments at around 2.2 Å. The transition state was then optimised using two different methods.

  1. Optimising by computing force constants at the beginning of the calculation

The optimisation was set up as a Opt+Freq job type with Optimisation to a TS Berny. Force constants were calculated once and Opt=NoEigen was added to the job type. The job was run and the resultant transition structure looked identical to that in Appendix 2. The transition structure contains an imaginary frequency at -818 cm-1. This vibrational mode is animated to the left.

  1. Optimising using the frozen coordinate method

The distance between the terminal carbon atoms of the allyl fragments was fixed using the Redundant Coordinate Editor at 2.2 Å and the conformation of the two allyl fragments was optimised to a minimum. After this had completed, the structure was resubmitted for optimisation to a Transition State (Berny) to give the final transition structure. The two transition structures are compared in the table below.

Comparison of transition structures from the two methods
Optimised by Transition State (Berny) Optimised by frozen coordinate method
Jmol
Imaginary Frequency/ cm-1 -818 -818
Animation
Vibration
Vibration
Bond forming/ bond breaking length/ Å 2.02 2.02
Energy/ a.u. -231.61932 -231.61932
.log file TS (Berny) Frozen Coordinate

The two chair transition structures optimised by the two different methods yield identical results to the accuracy that the results are given suggesting that both methods are reliable and relatively accurate in locating this transition structure. The imaginary frequency vibrational mode shows the formation of the new bond and the breaking of the other bond, suggesting the reaction occurs concerted, asynchronous fashion.

Optimising the Boat Transition Structure

The boat transition structure would be optimised using the QST2 method. In this method, after specifying the reactants and products, the calculation will interpolate between them to find the transition state between them.

The optimised Anti-2 .chk file was opened and the structure was pasted into both the reactant and product windows. The carbon and hydrogen atoms were then relabelled to match up as they would be after the Cope rearrangement. The Opt+Freq job type was run with the QST2 method but the job failed to run to completion. The output transition state looked similar to the chair transition state but with more dissociated fragments. The calculation simply translated the top allyl fragment and id not consider the possibility of rotation around the central C-C bond.

Using this reactant and product, the calculation cannot find the boat transition structure and the reactant and product will have to be adapted to be more similar to the boat transition structure. The dihedral angle in the reactant C2-C3-C4-C5 was changed to 0° and the inside C2-C3-C4 and C3-C4-C5 angles were reduced to 100°. The reactant and product geometries are shown below. This was optimised using the same method as before and gave the boat transition structure.


Jmol
Imaginary frequency/ cm-1 -841
Animation
Vibration
Bond breaking/ bond forming length 2.14
Energy/ a.u. -231.60280
.log file Boat

The energy of the boat transition structure can be seen to be ~43 kJ mol-1 higher in energy than the chair transition structure shown earlier as expected. The imaginary frequency vibrational mode is of a higher frequency in the boat transition structure. This suggests that the energy released in the formation of the new bond and breaking of the old bond is larger in the boat transition structure as it is of higher energy. The bond breaking/ bond forming length in the boat transition structure is also larger. The destabilisation of the boat transition structure is due to the steric clashes of the eclipsing hydrogens which is reduced in the chair transition structure.

Intrinsic Reaction Coordinate (IRC)

Predicting the conformer of the product of the Cope rearrangement is very difficult to do visually. However the Intrinsic Reaction Coordinate calculation follows the lowest energy pathway from the transition structure to a local energy minimum, thereby giving the conformation of the product. The IRC calculation was used on the chair transition state optimised by the Berny transition state method to find the closest conformation. The job was set up calculating force constants once and in the forward direction only, due to the symmetrical nature of the reaction coordinate. The IRC calculation would also follow 50 points along the line.

Unfortunately after 50 points along the IRC, the transition structure had not yet reached a minimum in energy and the output gave an intermediary geometry between the transition state and the final product. There were three possible solutions to this outcome:

  1. Take the output and optimise the geometry to find the energy minimum. This method is fast but if the geometry is far from the local minimum then the wrong minimum may be obtained.
  2. Carry out the IRC with a larger number of points, so it can finish optimising to the product.
  3. Carry out the IRC calculating the force constants at each step.

The first and third solution were followed and they both resulted in identifying the local minimum as the gauche-2 conformation. The IRC pathways are shown below for the two IRC runs.

Initial IRC calculation Revised IRC calculation

The .log file for the first IRC run and the subsequent optimisation. The .log file for the IRC calculating force constants at each step can be found on D-space. DOI:10042/to-6282

Activation Energies

Before calculating the activation energies to the transition structures, the transition structures were optimised to a higher level of accuracy using the B3LYP/6-31G* method and frequency calculations were carried out.


HF/ 3-21G B3LYP/ 6-31G*
Electronic Energy Sum of electronic and
zero point energies (0K) 		Sum of electronic and
thermal energies (298.15K) 		Electronic Energy Sum of electronic and
zero point energies (0K) 		Sum of electronic and
thermal energies (298.15K) 		.log file
Chair TS -231.61932 -231.46670 -231.46134 -234.55698 -234.41493 -234.40901 Chair TS
Boat TS -231.60280 -231.45092 -231.44529 -234.54309 -234.40234 -234.39601 Boat TS
Anti-2 -231.69254 N/A N/A -234.61171 -234.46920 -234.46186 N/A


The activation energies from the Anti-2 conformer to the chair and boat transition states were then calculated by taking the differences in electronic energy. The differences in sum of electronic and zero-point energies were used to calculate the activation energy at 0K and the sum of electronic and thermal energies were subtracted to give the activation energies at 298.15K. However since the thermochemical data for the Anti-2 conformer using the HF/ 3-21G level of theory had not been taken, only the calculated activation energy using the B3LYP/ 6-31G* level of theory could be calculated.

B3LYP/ 6-31G* Experimental
0K /a.u. (kcal mol-1) 298.15K/ a.u. (kcal mol-1) 0K/ kcal mol-1
ΔE(chair) 0.05427 (34.05) 0.05285 (33.16) 33.5±05
ΔE(boat) 0.06686 (41.96) 0.06585 (41.32) 44.7±2.0


The data shows that the activation energy to the chair transition structure is lower than that to the boat transition structure as would be expected. The activation energy to the chair transition structure matches the experimental results well, within the error limits of the data. However the activation energy calculated to the boat transition structure is just outside the error limits of the experimental data. The activation energies to both transition structures decreases at higher temperatures.

The Diels Alder Cycloaddition

The Diels Alder reaction is a type of well known pericylclic reaction in which the $pi; orbitals of the dienophile are used to form new σ bonds with the π orbitals of the diene in a concerted fashion. However the reaction is only allowed if there is significant overlap between the HOMO of one reactant and the LUMO of the other reactant. The HOMO-LUMO pair must have the same symmetry. The following computations were carried out at the semi-empirical AM1 level of theory unless otherwise stated.

Prototypical Diels Alder reaction of cis-butadiene and ethylene

The Molecular Orbitals of the reagents

In the reaction of cis-butadiene with ethylene to form hexene, the symmetric HOMO of ethylene and LUMO of cis butadiene interact and the antisymmetric LUMO of ethylene and HOMO of cis butadiene also interact. These molecular orbitals can be seen below.

cis-butadiene ethylene
HOMO
Energy -0.34381 -0.38776
Symmetry antisymmetric (a) symmetric (s)
LUMO
Energy 0.01707 0.05284
Symmetry symmetric (s) antisymmetric (a)


Optimising the Transition Structure

In order to find the transition structure the frozen coordinate method was used. The TS (Berny) method was used initially due to its ease of use but it failed to find the transition structure. The QST(2) method was found to be time consuming due to the relabelling of all the atoms. As a rough approximation to the transition state, the structure of [2,2,2]-bicylcooctane was used as the starting point. The extra CH2-CH2 were removed and double bonds were placed in the appropriate places. A dashed bond was also used to represent the bonds to be formed. The bond forming lengths were frozen at 2.2 Å in the first stage. The AM1 optimised transition structure was then reoptimised at the B3LYP/ 6-31G*level calculating all force constants. The results are shown and compared below.


Semi-empirical/ AM1 B3LYP/ 6-31G*
Jmol
Imaginary frequency/ cm-1 -956 -525
Animation
Vibration
Vibration
Bond forming length 2.12 Å 2.27 Å
Energy/ a.u. 0.11163 -234.54390
.log file AM1 DOI:10042/to-6286


The bond forming length in the B3LYP optimised structure is considerably longer than that in the AM1 optimised structure. The imaginary frequencies are also significantly different suggesting that the methods give considerably different transition structures. The typical sp3 bond length is 1.54 Å and a typical sp2 C-c is 1.34 Å. The van der Waals radius of a carbon atom is 1.70 Å[5]. The bond lengths of 2.12 Å and 2.27 Å are both longer than that of a C-C bond but smaller than the sum of two van der Waal's radii, suggesting that there is a bonding interaction occurring in the transition structures. The imaginary frequency vibrational mode shows that bond formation occurs in a concerted synchronous fashion.


HOMO
Energy -0.32395
Symmetry antisymmetric (a)
LUMO
Energy 0.02317
Symmetry symmetric (s)


Examining the molecular orbitals shown it can be seen that the asymmetric HOMO of the transition structure is formed from the interaction of the asymmetric HOMO of cis-butadiene with the asymmetruc LUMO of ethylene and the symmetric LUMO of the transition structure is formed from the interaction of the symmetic HOMO of the ethylene and the symmetric LUMO of cis-butadiene. This reaction is allowed as the symmetry of the interacting molecular orbitals is the same.

Regioselectivity in the reaction of Cyclohexa-1,3-diene with Maleic Anhydride

The reaction of cyclohexa-1,3-diene with maleic anhydride is known to occur with stereoselectivity for the endo product[6]. The reasons for the endo product being the kinetic product will be investigated in this section.

Optimisation of the Transition Structures

The endo and exo transition structures were optimised using the frozen coordinate method as before with a frozen bond forming length of 2.2 Å. The B3LYP/ 6-31G* level was also used to reoptimise the transition structures. The results are shown below.


Endo product Exo product
Semi-empirical/ AM1 B3LYP/ 6-31G* Semi-empirical/ AM1 B3LYP/ 6-31G*
Jmol
Imaginary frequency/ cm-1 -805 -477 -810 -448
Animation
Vibration
Vibration
Bond forming length/ Å 2.16 2.27 2.17 2.29
Energy/ a.u. -0.05159 -612.68340 -0.05050 -612.67931
.log file AM1 endo DOI:10042/to-6285 AM1 exo DOI:10042/to-6283


The imaginary frequency vibrational mode shows that the reaction occurs in a concerted synchronous fashion as before. The bond forming lengths are slightly longer in the exo product although the difference is small. As before, the B3LYP/ 6-31G* optimisation gives considerably different results to the AM1 optimisation.

endo geometry
exo geometry

The diagram above shows the C-C bond lengths and the distance from the anhydride structure to the rest of the system. It can be seen that the steric strain will be less in the exo transition structure due to the slightly longer spacial distance of 3.03 Å between the anhydride and the opposite carbon atom. However the endo transition state is stabilised by the secondary orbital overlap effect[7]. A diagrammatic example of secondary orbital ovrlap is shown below. A stabilising interaction between the p orbitals anhydride carbonyl carbons and the p orbitals of the diene present in the endo transition structure but not in the exo structure cause the endo transition structure to be lower in energy. The steric strain and secondary orbital overlap work against each other but in this case the secondary orbital overlap is more significant and so the endo product is the kinetic product.


endo product exo product
HOMO
Energy -0.34508 -0.34270
Symmetry antisymmetric (a) antisymmetric (a)
LUMO
Energy -0.03568 -0.4048
Symmetry antisymmetric (a) antisymmetric (a)

Both the HOMO and LUMO of both transition structures are antisymmetric.

Conclusion

Computational methods can be used to predict the geometries and relative energies of transition structures in a reaction and therefore predict the kinetic product to a reasonable accuracy. The quantum mechanical methods also allow us to predict whether reactions are allowed or disallowed on the basis of molecular orbital symmetry.

References

  1. ↑ B. W. Gung, Z. Zhu and R. A. Fouch, J. Am. Chem. Soc., 1995, 117, 1783-1788 DOI:10.1021/ja00111a016
  2. ↑ M. Nishio and M. Hirota, Tetrahedron, 1989, 45, 7201-7245 DOI:10.1016/S0040-4020(01)89185-7
  3. ↑ G. Schultz and I. Hargittai, J. of Molecular Structure, 1995, 346, 63-69 DOI:10.1016/0022-2860(94)09007-C
  4. ↑ K. Morokuma, W. T. Borden and D. A. Hrovat, J. Am. Chem. Soc., 1988, 110, 4474-4475 DOI:10.1021/ja00221a092
  5. ↑ S. S. Batsanov, J. of Molecular Structure, 1999, 468, 151-159 DOI:10.1016/S0166-1280(99)00077-9
  6. ↑ K. Nagai and H. Yonezawa, Journal of Polymer Science: Polymer Letters Edition, 2003, 21, 115-121 DOI:10.1002/pol.1983.130210208
  7. ↑ M. A. Fox, R. Cordona and N. J. Kiwiet, J. Org. Chem., 1987, 52, 1469-1474 DOI:10.1021/jo00384a016
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