Investigating Transition Structures - Daniel Spencer, 00736964
In this computational experiment, the aim was to investigate the transition structures for the Cope rearrangement and the Diels-Alder cycloaddition reaction. For the Cope rearrangement of 1,5-hexadiene, chair and boat transition structures were identified - the activation energy via the chair conformation was found to be 34.34 kcal mol-1, whilst that via the boat conformation was 43.05 kcal mol-1. Both of these calculated activation energies were found to be in agreement with the experimentally observed values. For the prototype Diels-Alder reaction of cis-butadiene and ethylene, a consideration of the molecular orbitals involved in the reactants and the transition structure provided evidence for the symmetry-allowed nature of the reaction. The activation energy was calculated to be 18.52 kcal mol-1 using DFT or 23.01 kcal mol-1 using AM1, which was observed to be closer to literature. For the Diels-Alder cycloaddition of cyclohexa-1,3-diene and maleic anhydride, it was computationally observed that the 'endo' transition structure is lower in energy than the 'exo' structure - this was reinforced through the activation energy being lower for the 'endo' pathway (15.74 kcal mol-1 for 'endo' versus 18.30 kcal mol-1 for 'exo').
Introduction
In this investigation, transition structures and their potential energy surfaces will be examined through a number of different means, focusing on the Cope rearrangement and the Diels-Alder reaction. For this purpose, GaussView 5.0 will be utilised in order to analyse the Gaussian quantum mechanical calculations performed on the molecules under consideration, using multiple levels of theory (dependent upon the situation) to compare the results obtained. Brief introductions to the two primary systems under consideration, the Cope rearrangement and the Diels-Alder reaction, are given below.
The Cope Rearrangement
Figure 1: The Cope rearrangement
The Cope rearrangement is an example of a thermally-induced pericyclic reaction - a [3,3] sigmatropic shift.[1] A reaction scheme for the Cope rearrangement using 1,5-hexadiene is presented in Figure 1.[2] In a computational study of this rearrangement for 1,5-hexadiene, it is important to consider the possible conformations of the molecule due to the possibility for significant differences in energy. Previous research into this system has revealed that of a possible 27 conformations for the molecule, there are 10 unique conformations (as a consequence of the symmetry of the system).[3] This work identified different minimum energy conformations depending upon the method used and hence, this will be one of the starting points for this investigation. However, a key conclusion of the research was that, as a consequence of the extremely small energy differences between the 10 1,5-hexadiene conformations, the conformations can be considered to be degenerate for most purposes.[3] For the overall study of the Cope rearrangement in this computational investigation, the aim will be to model the transition structures (chair and boat structures) for the system presented, with the intention of finding activation energies and comparing these to literature.
The Diels-Alder Cycloaddition
Figure 2: The prototype Diels-Alder reactionFigure 3: The Diels-Alder reaction of cyclohexa-1,3-diene and maleic anhydrideFigure 4: exo productFigure 5: endo product
The Diels-Alder cycloaddition is an extremely popular pericyclic reaction used and studied in many areas of organic chemistry, as evidenced by extensive reviews into the reaction's use in synthesis.[4] The cycloaddition is a π4 + π2 process, which, given that it is thermally driven, means that the reaction proceeds via a transition state of Hückel topology.[5][6] This reaction is therefore formally classified as a π4s + π2s process. The prototype reaction to be first studied in this investigation is that of ethylene and cis-butadiene reacting to give cyclohexene, as shown in Figure 2. The focus here will be on characterising the transition structure and examining the HOMO and LUMO orbital interactions that occur during the reaction.
Whilst the above defined prototype reaction is important to begin an investigation into the Diels-Alder cycloaddition, it does not reveal much about the selectivity of the reaction - the regioselectivity will be studied computationally through the use of the reaction between cyclohexa-1,3-diene and maleic anhydride, shown in Figure 3. Through this method, the relative energies of the 'exo' and 'endo' transition structures (for which the resulting products are shown in Figures 4 & 5 respectively) will be considered, allowing for a computational justification of the observation that the 'endo' products of Diels-Alder reactions are the kinetically preferred products (i.e. they have the lower activation barrier).[7]
Quantum Mechanical Calculations and Gaussian
In this computational investigation, numerous computational methods will be used in order to conduct quantum mechanical calculations through the use of Gaussian. The principal methods to be used are those of Hartree-Fock (HF), Density Functional Theory (DFT), Quadratic Synchronous Transit (QST), Intrinsic Reaction Coordinate (IRC) and AM1 (semi-empirical).
Hartree-Fock
The Hartree-Fock method, used at the HF/3-21G (where 3-21G defines the basis set) level of theory in this computational experiment, is an iterative method of solving the Schrödinger equation to find the ground-state electronic wavefunctions for a given molecule.[8][9] Significantly, the method assumes the mean field approximation, such that electrons are treated as an electric field, as opposed to individual entities.[10] Hence, the method lacks a full accounting of electron correlation (electron interactions in the system) - a full consideration of electronic effects is not performed by the calculation and therefore, some possible stabilising electronic orbital interactions cannot be invoked.[3] Whilst some electron correlation is accounted for by Hartree-Fock using the mean field approximation, it does not factor in individual electron-electron repulsions and as a consequence of this, it is likely that the conformational energies found here will be somewhat erroneous when compared to those computed at higher levels of theory.[11]
Density Functional Theory (DFT)
DFT is a computational method that uses functionals of electron density in order to compute the properties, primarily ground-state energy, of a molecular system.[12] In this computational experiment, DFT will be implemented as B3LYP/6-31G(d) (also referred to as B3LYP/6-31G*), where B3LYP indicates the functionals to be used, in this case Becke's three-parameter exchange and Lee-Yang-Parr's correlation functionals.[13][14][15] The basis set used here, 6-31G(d), is considered to be an improvement compared to the 3-21G basis set used for Hartree-Fock in this investigation.[16] In contrast to the Hartree-Fock method, DFT takes a more comprehensive account of the electron correlation, leading to results that may be considered to be more accurate.
Quadratic Synchronous Transit (QST)
In this investigation, the Quadratic Synchronous Transit (QST) method will be employed as a means of finding the boat transition state for the Cope rearrangement.[17][18] The QST2 method will be used, which requires the reactant and product structures to be input in order to compute the transition structure. More advanced methods (QST3) can be employed generate more accurate results.
Intrinsic Reaction Coordinate (IRC)
Intrinsic Reaction Coordinate (IRC) calculations are made using the Hartree-Fock HF/3-21G level of theory in this investigation. These calculations are run from the transition state in an attempt to locate local minima in the potential energy surface for the reaction - these minima will be the reactants or products of the reaction. The IRC calculations can be run with differing numbers of points along the potential energy surface considered, resulting in the identification of different minima depending upon the settings used.
AM1 (Semi-empirical)
AM1 (Austin Model 1) is a semi-empirical computational method designed to improve on existing semi-empirical methods, by implementing aspects missing from the likes of the Modified Neglect of Diatomic Overlap (MNDO) method whilst not extending computational time.[19] In this computational experiment, the AM1 method will be used in order to investigate the Diels-Alder cycloaddition reaction, with a focus on the molecular orbitals involved in the reaction.
(This computational introduction is mostly ok. It should only be clear that Hartree-Fock, DFT and AM1 are different methods and approximations to calculate quantum mechanical energy of the electrons for a given nuclear configuration. The electronic energy as a function of the nuclear coordinates is the PES, and QST2, IRC, etc. use first and eventually second derivative of this energy to obtain nuclear configurations/molecular geometries that correspond to interesting features of the PES and consequently of the chemical reactionJoão (talk) 01:17, 24 December 2014 (UTC))
The Cope Rearrangement
Optimising the Reactants and Products
Table 1: Anti and gauche conformations of 1,5-hexadiene
Conformation
Figure 6: Anti conformation of 1,5-hexadiene (anti1)
Figure 7: Gauche conformation of 1,5-hexadiene (gauche2)
Energy, hartree
-231.69260234
-231.69166697
Point Group
C2
C2
To investigate the conformations of 1,5-hexadiene, an antiperiplanar conformation of the central four carbon atoms was first drawn, then optimised using Hartree-Fock HF/3-21G and symmetrised (The structure should not be further modified after optimization, in particular it should not be symmetrised, as this will move the geometry away from the minimum. The point group editor can nevertheless be used to evaluate the point group of the moleculeJoão (talk) 01:17, 24 December 2014 (UTC)), giving the conformation shown in Figure 6, with a point group of C2. This process was repeated, using a gauche conformation of the same central four carbon atoms, giving the conformation presented in Figure 7, again with a point group of C2. A comparison of these two observed conformations reveals that the 'anti' conformation is the lower in energy - this is as would usually be expected from steric considerations.
In general, antiperiplanar conformations can often be predicted to be the lowest energy conformations of a system, due to the maximisation of the distance between sterically bulky groups; however, gauche conformations can be stabilised by electronic effects (such as the donation of electron density from a filled σ donor bond to an empty σ* acceptor bond). In order to identify the lowest energy conformation of 1,5-hexadiene, further possibilities for gauche and anti conformations can be considered as before. Some of these are tabulated in Table 2, where it can clearly be observed that one of the located gauche conformations, of energy -231.69266120 hartree, is the minimum energy conformer of 1,5-hexadiene - it has the most negative energy. This conformation can be determined to be the gauche3 conformation. The further conformations of 1,5-hexadiene presented have also been assigned this notation, which is used to identify conformations.
Table 2: Additional conformations of 1,5-hexadiene
Conformation
Figure 8: gauche3
Figure 9: anti2
Figure 10: anti3
Figure 11: anti4
Energy, hartree
-231.69266120
-231.69253529
-231.68907061
-231.69097055
Point Group
C1
Ci
C2h
C1
Figure 12: DFT optimisation of anti2
Figure 9 presents the anti2 conformation, which has an energy of -231.69253529 hartree and a point group of Ci. This energy can be compared to that provided, -231.69254 hartree, which indicates that this conformation has been assigned correctly to anti2. Further study can be conducted on this conformation making using of a DFT B3LYP/6-31G(d) optimisation. The result of this optimisation is presented in Figure 12, with an energy of -234.61170280 hartree, retaining the Ci point group. Comparing the results of this optimisation with that from the Hartree-Fock HF/3-21G optimisation, it is clear that DFT arrives at a lower energy - this is a consequence of the use of an alternative method (using differing reference energies).
(And how do the two optimized geometries compare to each other?João (talk) 01:17, 24 December 2014 (UTC))
Table 3: Thermochemistry output for DFT optimised anti2
Energy, hartree, at 298.15 K
Energy, hartree, at 0.01 K
Sum of electronic and zero-point energies
-234.469212
-234.469212
Sum of electronic and thermal energies
-234.461856
-234.469212
Sum of electronic and thermal enthalpies
-234.460912
-234.469212
Sum of electronic and thermal free energies
-234.500821
-234.469211
Using this DFT optimised model, a frequency calculation can be run in order to investigate the vibrational frequency modes of the molecule. Here, it was found that for this conformation, all of the frequencies calculated were real (with no imaginary modes), indicating that the conformation is indeed a minimum. Using the Thermochemistry section of the output file produced using Gaussian, the energy values presented in Table 3 were found (recorded at 298.15 K). Here, the sum of the electronic and zero-point energies represents Eelec + ZPE (the potential energy at 0 K including the zero-point vibrational energy); the sum of the electronic and thermal energies is the energy at 2918.15 K and 1 atm, E + Evib + Erot + Etrans; the sum of the electronic and thermal enthalpies contains a correction for enthalpy, H = E + RT; the sum of electronic and thermal free energies also accounts for the entropic contribution, G = H - TS.
These data can be collected at a temperature approaching 0 K, here taking 0.01 K. The results are again presented in Table 3, as for those recorded at 298.15 K. By subtracting the sum of electronic and thermal energies from the sum of electronic and zero-point energies, the zero-point energy of the molecule can, to the accuracy of the computation, be found to be 0 hartree (This is not correct. The zero point energy is included in the vibrational energy.João (talk) 01:17, 24 December 2014 (UTC)); however, were more significant figures accounted for, it is likely that a non-zero energy would be found. Alternatively, a temperature closer to 0 K could be used in an attempt to obtain a more accurate zero-point energy.
Optimising the chair and boat transition structures
Figure 13: Approximation of chair transition structure
Figure 14: Animation showing the frequency corresponding to the Cope rearrangement
Figure 15: Animation showing the frequency corresponding to the Cope rearrangement (using frozen coordinates)
Figure 16: Numbering system for QST2 method
Figure 17: Result of QST2 optimisation
Figure 18: Animation showing the frequency corresponding to the failed boat conformation for the Cope rearrangement
Figure 19: Adjusted reactant for the boat transition structure
Figure 20: Boat transition structure obtained through QST2 optimisation
Figure 21: Animation showing the frequency corresponding to the boat transition state for the Cope rearrangement
Figure 22: Result of IRC first run for chair transition structure
Figure 23: Result of IRC first run for boat transition structure
Table 4: Transition structure and reactant energies
Energy, hartree
Chair transition structure
-234.55698291
Boat transition structure
-234.54309304
Reactant (anti2 1,5-hexadiene)
-234.61170280
Table 5: Transition structure and reactant electronic and thermal energies (298.15 K)
Sum of electronic and thermal energies, hartree
Chair transition structure
-234.409005
Boat transition structure
-234.396004
Reactant (anti2 1,5-hexadiene)
-234.461856
In order to investigate the chair and boat transition structures of the Cope rearrangement, an allyl fragment (CH2CHCH2) was first optimised (using the Hartree-Fock HF/3-21G level of theory). This optimised fragment was then duplicated and arranged with respect to the original fragment in order to approximate a chair transition structure, with a distance of 2.2 Å between the termini of the allyl fragments. This approximated arrangement is shown in Figure 13 (note that this is not shown as a delocalised system here).
Performing an Opt + Freq computation on this approximated transition structure (using HF/3-21G, optimising to a TS (Berny) and calculating force constants once) gave an imaginary frequency of -818.02 cm-1 - this is visualised in Figure 14, where it can be observed to correspond to the Cope rearrangement, through the breaking and formation of the expected bonds.
This system can now be considered using the frozen reaction coordinate method. This was achieved through the use of the Redundant Coordinated Editor to freeze the bond distances between the termini of the allyl fragments. Performing this optimisation to a minimum, as before, gives a structure similar to that visible in the animation in Figure 14. This structure can then be further optimised with regards to the bond forming/breaking distances that were previously frozen. Here, the bonds are set to Derivative, with the transition structure optimisation not calculating force constants (The force cosntants are calculated but with respect to the forming bonds only, instead of all motions in the molecule. This represents a computational advantage.João (talk) 01:17, 24 December 2014 (UTC)). The imaginary vibrational mode of the transition structure obtained through these optimisations (this time using the DFT B3LYP/6-31G(d) level of theory) is presented in Figure 15. Here, it should be noted that the imaginary frequency calculated was -565.39 cm-1, compared to that from the previous optimisation to a TS (Berny) of -818.02 cm-1 - the difference here is likely to be primarily due to the different methods used. This is reinforced as a consideration of the animation of the vibrational mode indicates that this frequency still corresponds to the Cope rearrangement. Clearly, there is little difference in structure between the two transition structures obtained via different optimisations.
For further investigation, a boat transition structure can be optimised using a QST2 method (at the HF/3-21G level of theory). This was set-up by the use of the anti2 conformation of 1,5-hexadiene as both the reactant and the product molecules, where the atomic numbering was adjusted in order to account for the changes observed during the Cope rearrangement - this numbering system is highlighted in Figure 16. An Opt + Freq calculation was then performed, optimising to a TS(QST2). Here, it was observed that the calculation did not optimise to the desired boat transition structure - this can be considered through an examination of the checkpoint file produced, where it can be observed that the structure produced bears significant similarities to the chair transition structure found previously - this is shown in Figure 17. Whilst the numbering of the atoms is not visible here, the two allyl fragments are C1-C2-C3 and C4-C5-C6, with C1 above C4, C3 above C6 (this numbering can be observed in the animation, presented in Figure 18). Notably, the dashed bonds shown in Figure 17 are between C1-C6 and C3-C4. Performing a frequency calculation allowed for visualisation of the vibrational mode associated with the reaction, presented in Figure 18. Here, it can be observed that it is the bonds between C1-C4 and C3-C6 which are observed to be forming and breaking, not the expected C1-C6 and C3-C4 bonds - thus, it is clear that this computation has not converged on the desired transition structure.
(It is a good thing that you did a frequency calculation to convince yourself it was a normal chair transition state. The dashed lines bare no meaning as in a quantum mechanical calculation what counts is the inter-nuclear distances.João (talk) 01:17, 24 December 2014 (UTC))
In order to make an enhanced attempt at optimsing the boat transition structure for the Cope rearrangement, the bond angles for the reactant and product molecules were altered to better approximate this transition structure. The new set-up is presented for the reactant in Figure 19. It should be noted that these adjustments were also made for the product, consistent with the numbering system previously established. The QST2 optimisation was then performed on these adjusted conformations, given the boat transition structure shown in Figure 20. Through the use of a frequency calculation, the single negative vibrational frequency obtained was -839.94 cm-1, this is visualised in Figure 21. A consideration of the transition structure and animation produced using this optimisation reveals that with these adjusted reactant/product conformations, the desired boat transition structure was indeed obtained. This is also made evident through the vibration animation, which shows that the expected bond breaking and formation is observed (breaking of the C3-C4 bond, formation of the C1-C6 bond), as was not the case for the previous QST2 optimisation.
Whilst these calculations allowed for optimisation of chair and boat transition structures, it can be difficult to establish which conformation of 1,5-hexadiene will be formed from a particular transition state. In order to further investigate the transition states, the Intrinsic Reaction Coordinate (IRC) was used to examine the energy paths from the transition structure to local minima on the potential energy surface for the system. Firstly, the chair transition structure obtained through the use of the frozen coordinate method was considered using the IRC method, using the Hartree-Fock HF/3-21G level of theory - the IRC was run in the forward direction only, calculating the force constants always, with 50 points along the IRC. The result of this calculation is presented in Figure 22. It should be noted here that the distance between the terminal carbon atoms, between which a bond is expected to form, is 1.55219 Å, which is close to the bond length expected. In order to reach a minimum energy, the IRC was re-run on this structure, instead using 100 points along the IRC - this calculation gave a structure with an energy of -231.69161 hartree, close to that of the gauche2 conformation. Hence, it was decided that, as the structure was very close to a local minimum, an optimisation to a minimum would be the best method of reaching this conformation with minimal computing time. This optimisation gave the gauche2 structure, as evidenced through its energy of -231.69166701 Hartree, which matches that previously found for this gauche2 conformation. By performing the IRC calculating force constants at every point, it is likely that the global minimum, the gauche3 conformation, would be obtained as the product.
(If/since there is a barrier between conformers gauche2 and gauche3, are you still confident the IRC would give you the path from one conformer to the other?João (talk) 01:17, 24 December 2014 (UTC))
This IRC method can also be used to investigate the energy paths from the optimised boat transition structure. Using the same initial settings as for the chair transition structure, the result of the initial calculation was that presented in Figure 23. It can immediately be observed to resemble the gauche3 conformation of 1,5-hexadiene - this structure was, therefore, optimised to a minimum, as for the chair structure. After optimisation using HF/3-21G, the energy was found to be -231.69266121 hartree, which corresponds to that of the gauche3 conformation, the energy minimum for 1,5-hexadiene, as shown previously in Figure 8. In this case, it is clear that the IRC method has found the minimum energy path from the boat transition structure to the global minimum for the 1,5-hexadiene molecule. Altering the IRC settings is likely to allow different local minima to be obtained as the products of the Cope rearrangement.
Calculating Activation Energies
Activation energies can be calculated for the chair and boat transition structures, using the previously optimised structures for each. Here, the structures are re-optimised using the DFT B3LYP/6-31G(d) level of theory - the energies of these structures are presented in Table 4(Did you check if you still had an imaginary frequency after re-optimization?João (talk) 01:17, 24 December 2014 (UTC)), alongside the energy for the reactant molecule. The activation energy can be obtained from these energies through subtraction of the reactant energy from the transition state energy.
Using these energy values, the activation energy for the chair transition structure can be found to be 0.05471989 hartree, or 34.33722345 kcal mol-1. This can be compared to the experimental activation energy of 33.5 ± 0.5 kcal mol-1, indicating a good degree of agreement between computational and experimental results. For the boat transition structure, the activation energy can similarly be found to be 0.06860976 hartree, or 43.05324189 kcal mol-1. Here, the experimental value for the activation energy is 44.7 ± 2.0 kcal mol-1, hence the value calculated computationally is within the error range of the experimental value. For both transition structures, it is clear that there is a good agreement between the energies calculated computationally and those measured experimentally. It should be noted that the activation energies calculated here are at 0 K - the activation energies can also be investigated at 298.15 K through the thermal correction present (the sum of electronic and thermal energies) in the output file of the computations performed. These data are presented in Table 5. Using these data, the activation energy for the reaction via the chair transition structure can be calculated to be 33.16437246 kcal mol-1, whilst that via the boat transition structure was found to be 41.32259096 kcal mol-1. These values are both lower than those calculated at 0 K, as expected due to the presence of thermal energy (Are thermal corrections the same for reactant and transition state structures?João (talk) 01:17, 24 December 2014 (UTC)); however, literature values could not be located with which to compare those computed in this aspect of the investigation.
The Diels-Alder Cycloaddition
Prototype Diels-Alder Reaction
Ethylene
Figure 24: HOMO of ethylene
Figure 25: LUMO of ethylene
Figure 26: Defined plane for symmetry considerations
In order to begin an investigation into the transition structure of the Diels-Alder cycloaddition of cis-butadiene and ethylene, ethylene was first constructed and optimised (using the semi-empirical AM1 method, optimising to a minimum). The HOMO and LUMO of the molecule generated using this method are presented in Figures 24 & 25 respectively. Defining a plane of symmetry as in Figure 26, the molecular orbitals produced via these calculations can be characterised as symmetric or antisymmetric, with respect to this plane. In Figure 26, this plane is that which bisects the C-C single bond of cis-butadiene and the C=C double bond of ethylene. With respect to this defined plane, the HOMO of ethylene is clearly symmetric, whilst the LUMO is antisymmetric.
Figure 27: HOMO of cis-Butadiene
Figure 28: LUMO of cis-Butadiene
For later calculation of the activation energy, it should be noted that the energy of the optimised ethylene molecule was calculated using DFT B3LYP/6-31G(d) to be -78.58745744 hartree. Using semi-empirical AM1 optimisation, the energy was found to be 0.02619027 hartree.
cis-Butadiene
As for ethylene, the HOMO and LUMO orbitals generated using this method are presented in Figures 27 & 28 respectively. Here, it is evident that the HOMO of cis-butadiene is antisymmetric with respect to the previously defined plane, whilst the LUMO is symmetric. The energy of cis-Butadiene was found to be -155.98594956 hartree, using DFT B3LYP/6-31G(d). Using semi-empirical AM1 optimisation, the energy was calculated to be 0.04879719 hartree.
Transition State Geometry of the Prototype Reaction
Figure 29: Bicyclic precursor to guess transition structure
Figure 30: Guess transition structure
Figure 31: Animation showing the frequency corresponding to the transition structure for the prototype Diels-Alder cycloaddition
In order to compute the transition structure for the prototype Diels-Alder cycloaddition shown in Figure 2, a 'guess' structure was first produced - the bicyclic structure shown in Figure 29 shows the precursor to this guess, prior to optimisation. This structure was then optimised via the Hartree-Fock HF/3-21G level of theory, followed by removal of one -CH2-CH2- unit. This gives the guess structure shown in Figure 30, where the distance between the terminal C atoms (with dashed bonds) has been set to 2.20 Å. This structure was then optimised using the semi-empirical AM1 molecular orbital method to a TS(Berny), resulting in an imaginary vibration of -956.26 cm-1, which is visualised in Figure 31. Symmetrisation of the transition structure showed that its point group was Cs. The geometry of the transition structure is presented in Figure 32
Using this optimised transition structure, the HOMO and LUMO orbitals can be calculated to be those shown in Figures 33 & 34. A consideration of the symmetry these molecular orbitals reveals that the HOMO of the transition structure is antisymmetric with respect to the previously defined plane, whilst the LUMO is symmetric. The overall geometry of the transition structure is visible in the animation, where it should be noted that the dashed bond distances (the partially formed C-C σ bonds) is 2.11925 Å for both bonds. This is clearly significantly larger than the C-C single bond length (sp3 C-C bond length is, on average, 1.53 Å)[20], but is closer than twice the van der Waal's radius of carbon (1.70 Å)[21] and this therefore supports the formation of the two new C-C σ bonds. This is also demonstrated through the vibrational mode which is visualised in Figure 31, where it is clear that the new bonds are formed in a synchronous manner.
Figure 33: HOMO of prototype reaction transition structure
Figure 34: LUMO of prototype reaction transition structure
Considering these molecular orbitals alongside those produced for ethylene and cis-butadiene individually, it is evident that it is combinations of these individual molecular orbitals which form those for the transition structure. For the HOMO of the transition structure, it is the HOMO of cis-butadiene and the LUMO of ethylene which form the molecular orbital; whilst for the LUMO of the transition structure, the LUMO of cis-butadiene and the HOMO of ethylene form the molecular orbital. In each case, the HOMO-LUMO pairs interact according to symmetry - the reaction is allowed as the orbitals interact with others of the same symmetry. (Is that the only condition according to Woodward-Hoffman rules?João (talk) 01:17, 24 December 2014 (UTC))
The activation energy for this prototype Diels-Alder reaction can be calculated by subtracting the total reactant energy from the energy of the transition structure found. The transition state energy was found to be 0.11165465 hartree (using AM1), with the total reactant energy being 0.07498746 hartree - this gave an activation energy of 23.0089184 kcal mol-1. Alternatively, using DFT, the transition state energy was -234.5438974 hartree and the total reactant energy was -234.573407 hartree - this gives an activation energy of 18.51746174 kcal mol-1. Notably, these two values are not the same, indicating a differing degree of accuracy in the transition state optimisation achieved by the two methods. Comparison to an experimental literature value for the activation energy of this Diels-Alder reaction, that of 25.1 kcal mol-1 at 0 K, indicates that the activation energy calculated using the AM1 method is closer to that experimentally observed than that found using DFT.[22][23] Considering the large difference between the DFT-calculated value and the experimental literature value, some of the difference will be a consequence of approximations made when using Density Functional Theory; however, evidence has been found to suggest that this is not the cause of the whole difference - namely through the location of a different minimum on the potential energy surface of the reaction, found using a Nudged Elastic Band (NEB) method.[24]
Investigating the Regioselectivity of a Diels-Alder Reaction
Figure 35: 'exo' transition structure
Figure 36: Animation showing the frequency corresponding to the exo transition structure
In order to investigate the regioselectivity of the Diels-Alder cycloaddition, the reaction specified in Figure 3, between cyclohexa-1,3-diene and maleic anhydride, was considered. Here, both the 'exo' and 'endo' products presented in the Introduction are possible (via 'exo' and 'endo' transition structures respectively), where the 'endo' structure is the kinetic product of the reaction.[7]
Figure 37: HOMO of exo transition structure
Figure 38: LUMO of exo transition structure
To find the transition structure of the 'exo' product, this product was first drawn, then edited, with a 'new bond' distance of 2.20 Å. This was then optimised at the AM1 level of theory to a minimum, giving a structure with an energy of -0.05041968 hartree, where the partially formed bond lengths are both 2.17053 Å. The structure was observed to have an imaginary vibrational mode of -811.96 cm-1, which, when visualised, corresponded with the synchronous formation of the two new C-C σ bonds. For the 'endo' transition structure, the same procedure was used, giving an energy of -0.05150480 hartree, along with a partially formed bond distance of 2.16239 Å. Here, the imaginary vibrational mode was at a frequency of -806.05 cm-1, which again corresponded to synchronous bond formation. The structures resulting from these calculations are not presented here, due to their similarities with those calculated at the DFT B3LYP/6-31G(d) level of theory (presented below). It should be noted that the differing energies are simply a consequence of the different methods (using different references and different approximations). (Yes, that is why it is much better to report energy differences instead of absolute energies.João (talk) 01:17, 24 December 2014 (UTC))
For the DFT calculations, the frozen coordinate method, above applied to the Cope rearrangement, was used in order to determine the transition structure from the approximate geometry predicted using the method applied for AM1 calculations. In this manner, the transition structure calculated for the 'exo' product is shown in Figure 35. The energy of this structure was found to be -612.67931096 hartree, with a partially formed bond distance of 2.29058 Å (notably larger than that set as the incoming distance). The imaginary vibrational frequency was found to be -448.49 cm-1 - this is visualised in Figure 36, where the animation can be observed to correspond to the synchronous bond formation expected for the Diels-Alder cycloaddition. Figures 37 & 38 give the HOMO and LUMO of the transition structure respectively.
Figure 39: 'endo' transition structure
Figure 40: Animation showing the frequency corresponding to the endo transition structure
Figure 41: HOMO of endo transition structure
Figure 42: LUMO of endo transition structure
Figure 43: HOMO-1 of endo transition structure
For the 'endo' transition structure, the structure calculated was that presented in Figure 39. The energy of this structure was found to be -612.68339677 hartree, with the partially formed bond distances being 2.26836 Å and 2.26835 Å. Other bond lengths in this transition structure are also significant, when considering a C-C sp3 length to be 1.54 Å (on average) and a C-C sp2 bond length to be 1.34 Å (on average). In the cyclohexa-1,3-diene fragment of the transition structure, the bond lengths are 1.39123 Å, 1.40308 Å and 1.39123 Å, where the larger value is for the C-C single bond separating the two double bonds in the reactant molecule. From these lengths, with a comparison to the sp2 and sp3 C-C bond lengths, it is clear that the bonds are all between single and double bonds - this is as would be expected for the delocalised transition state. This is also found for the bond of the maleic anhydride fragment (that which is a C-C double bond in the reactant), which has a length of 1.39394 Å. Given that the bonds of length ~1.39 Å are closer to a double bond length than that at 1.40 Å, it is clear that the system has an early transition state - the transition structure more closely represents the reactants than the product. These observations were also made for the 'exo' transition structure. For this 'endo' transition structure, the imaginary vibrational mode was found to be observed at a frequency of -446.95 cm-1, visualised in Figure 40. The HOMO and LUMO for this structure are presented in Figures 41 & 42 respectively; here, the HOMO-1 is also included, Figure 43, for a consideration of secondary orbital overlap. Geometrically, the two transition structures are different as a consequence of their molecular alignment - for the 'endo' structure, the incoming maleic anhydride fragment is aligned with the π-system (allowing for an interaction), whilst for the 'exo' structure, the maleic anhydride fragment is orientated away from this π-system, preventing a favourable interaction.
When comparing the 'exo' and 'endo' transition structures found in this investigation, it is also important to consider some through-space interactions which may be significant. For the 'exo' structure, the C-C through-space distances of the carbonyl carbons and the -CH2-CH2- carbons is important for steric considerations - these distances are 3.02794 Å. Given that twice the van der Waals radius of carbon is 3.40 Å, it is evident that these atoms are within a repulsive distance (Why is that evident? Shouldn't you be comparing the distance to the hydrogen atoms instead?João (talk) 01:17, 24 December 2014 (UTC))- this indicates that there are steric interactions which will increase the energy of the structure. For the 'endo' transition structure, the interactions to consider are those between these carbonyl carbons and the carbon atoms of the new double bond formed (that which appears in the product molecule). These distances are both 2.99010 Å. Whilst this distance may be considered repulsive as for the 'exo' case, there is an additional factor to consider here - the secondary orbital interaction that occurs between this double bond π-system and the maleic anhydride fragment, as can be observed in Figure 43 (the HOMO-1 for the 'endo' transition structure). Here, this secondary orbital interaction can be seen to be between the π-system and the p-like orbitals of oxygen - this interaction will act to stabilise the 'endo' transition state. This secondary orbital interaction must be significant, as it results in the transition state energy of the 'endo' structure being lower than that of the 'exo' structure.
Activation Energies
In order to calculate an activation energy for the Diels-Alder reaction of cyclohexa-1,3-diene and maleic anhydride, these reactants were optimised using DFT B3LYP/6-31G(d) - the energies were recorded as -233.41893610 hartree for cyclohexa-1,3-diene and -379.28954469 hartree for maleic anhydride. Thus, the total reactant energy was -612.7084808 hartree. By subtracting this from the transition state energies for the 'exo' and 'endo' products, the activation energy for each case can be found. This gives an activation energy of 18.30433085 kcal mol-1 for the 'exo' case and 15.74044831 kcal mol-1 for the 'endo' case. These results are clearly in accordance with the expected product of the kinetically-controlled reaction being the 'endo' product, as this has the lower activation energy of the two possibilities. A literature value for the activation energy of this reaction is 11.4 kcal mol-1, which indicates that the activation energy calculated for the 'endo' transition state is close to that experimentally observed.[25] Some of the difference between the computational and experimental values is likely derived from the use of a solvent - experimentally, dichloromethane was used, whilst the computational calculations did not account for any solvation. Additionally, the approximations made using the DFT calculations are also likely to have contributed to this difference.
Conclusion and Evaluation
In the investigation into the Cope rearrangement, computational techniques were used in order to identify and optimise the different conformers of 1,5-hexadiene, which were then used for a consideration of the transition states for the [3,3]-sigmatropic rearrangement.[1] Here, a 'guess' chair transition structure was used to find an optimised structure, where the imaginary vibrational mode was found to correspond to the bond breaking/formation expected to be observed in the Cope rearrangement - this transition structure was also obtained using a frozen coordinate method, where little change in geometry was observed when comparing the two optimisations. Additionally, the boat transition structure was obtained through the use of the QST2 method - this was achieved after an alteration to the input geometries in order to ensure that the correct C-C bonds were formed/broken. For this Cope rearrangement, the activation energies were found to be 34.33722345 kcal mol-1, for the chair transition state, and 43.05324189 kcal mol-1, for the boat transition state. Comparison to experimental literature values revealed a good agreement with those computed in this investigation.
With regards to the prototype Diels-Alder reaction (that of cis-butadiene and ethylene) considered, the results obtained clearly demonstrated that the HOMO-LUMO pairs of the two reactants were combined, according to symmetry, in order to produce the HOMO and LUMO obtained for the transition structure, justifying the symmetry-allowed nature of the reaction. Using DFT B3LYP/6-31G(d) optimisations, the activation energy of the reaction was found to be 18.52 kcal mol-1, whilst using AM1 calculations, the activation energy was 23.01 kcal mol-1. Comparison of these data with literature indicated that, in this case, the semi-empirical AM1 method produced the more accurate results.
For investigating the regioselectivity of the Diels-Alder cycloaddition, the reaction between cyclohexa-1,3-diene and maleic anhydride was studied. This revealed that the transition state energy was lower for the 'endo' product than the 'exo' product. The predominance of the 'endo' product was also indicated through the activation energies calculated, 15.74 kcal mol-1 for 'endo' versus 18.30 kcal mol-1 for 'exo'. This was justified through a consideration of steric interactions and secondary orbital interactions, where it was concluded that whilst both suffer from steric repulsions between the maleic anhydride fragment carbonyl carbons and carbon atoms from the cyclehexa-1,3-diene fragment, only the 'endo' product experiences stabilising secondary orbital interactions. These interactions were observed in the HOMO-1 between the oxygen p-orbitals and the π-system of cyclohexa-1,3-diene.
In order to improve the results of the computations conducted to find the boat transition structure for the Cope rearrangement of 1,5-hexadiene, a QST3 method could be implemented, requiring the additional input of a 'guess' transition structure.[17] This method is likely to result in an energetically more accurate transition structure than that computed using QST2 - this increase in accuracy would be expected to be reflected in the activation energy for the rearrangement via the boat transition state, noted through comparison to literature. (Assuming you had obtained a transition state structure using the QST3 method, in what way would you assess that structure to be "better" than the QST2 one? Would they be different?João (talk) 01:17, 24 December 2014 (UTC))
With the transition structures computed for the Diels-Alder cycloadditions presented here, it is evident that several effects have been emitted in the calculations. One of the most obvious of these is solvation - this was noted through comparison of the activation energy calculated for the reaction of cyclohexa-1,3-diene and maleic anhydride to a literature value, which used dichloromethane as a solvent whilst the computations neglected solvation effects. Clearly this is likely to influence the activation energy for the reaction and hence, further computations could be made, accounting for these effects, in order to better calculate the transition states and their energies for these reactions.
Given that butadiene is known to readily undergo dimerisation to give 4-vinylcyclohexene[22], further study could be conducted into this competing reaction, allowing a consideration of the activation barrier in comparison to that for the reaction with ethylene. Consequently, a computational study into the competition could be undertaken.
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