File:Dielsalderscheme2.png

2014-12-05T02:19:17Z

Ipm12:

Rep:Mod:iulianamaria

2014-12-05T02:18:53Z

Ipm12: /* Introduction */

== Third Year Computational Lab: Physical module. Transition states and reactivity ==

== Introduction ==

The structures and energy of the species involved in a chemical reaction offer valuable insight into the mechanism and general features of the process. Due to the advances in optimization algorithms, information about the most stable geometry, vibrational frequencies and in principle any observable physical quantity can be computed with a high degree of accuracy. Synthetic sequences often involve two or more reaction pathways with similar activation energies and thus the location of the transition structures corresponding to each pathway on the potential energy surface is of great importance in predicting the major products.

[[File:copescheme.png|right|300x300px|thumb|'''Figure 1.''' Cope rearrangement reaction]]
The Cope rearrangement reaction of 1,5-hexadiene is an archetypal [3,3] sigmatropic shift which has been the subject of numerous theoretical and experimental studies. Computational evidence has been provided to show that the reaction can proceed via either an aromatic transition state consisting of two partially bonded three-carbon units or via a cyclohexane- 1,4-diyl-like structure. <ref name="chairTS"/>. This study employs computations at the HF/3-21G and B3LYP/6-31G* levels of theory to investigate the lowest energy conformation of the reactant molecule and to locate the two possible, chair and boat, transition structures. The results are subsequently employed to rationalize the preferred mechanism and a comparison with previous research is also provided.

[[File:dielsalderscheme2.png|right|300x300px|thumb|'''Figure 2.''' Diels-Alder cycloaddition reaction]]
The [4+2] Diels-Alder reaction between ethylene and butadiene is another example of a process which has been extensively investigated from a theoretical and experimental point of view, but which is still not completely elucidated. Possible mechanisms that have been proposed for this reaction include a synchronous concerted approach via a cyclcic aromatic transition state, a two-stage asynchronous concerted mechanism and a stepwise process, which occurs via a stable diradical intermediate. <ref name="intro"/> This study uses AM1 semi-empirical molecular orbital method to locate and investigate the geometrical features of the transition state, followed by an analysis of the possible interaction between the frontier orbitals involved in the reaction.Computations at the AM1 level of theory are also employed to probe the Diels-Alder endo rule and study the possible pathways of the cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride. Both ''endo'' and ''exo'' adducts shown in Figure 2 are considered and their relative energies and nodal properties of the molecular oribitals are compared in order to determine the product that is preferentially formed.

==Results and Computational Methods==

 

=== Exercise 1. Optimizing the Reactants and Products of the Cope Rearrangement===

Due to the presence of three C-C bonds and three rotational minima around each of these bonds <ref name="pi interaction" />, 1,5-hexadiene is expected to have 27 conformations in total and ten of them to be energetically distinct. The conformations shown in Figure 3 are optimized at the B3LYP/6-31G* and HF/3-21G levels of theory and their electronic energies are compared, in order to determine the lowest energy conformation of 1,6-hexadiene. Frequency calculations performed on the optimized structures at the same level of theory yielded only real and positive frequencies, confirming that a minimum was obtained. The electronic energies and symmetries, along with the sum of electronic and zero-point energies and the sum of electronic and thermal energies of the optimized structures are summarized in Table 1.

 

'''Table 1. Summary of energies (hartrees) of the conformations under investigation'''

{| class="wikitable"
! rowspan="2" style="font-weight: bold;" | Conformer
! rowspan="2" style="font-weight: bold;" | Point Group
! colspan="3" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
|-
| style="font-style: italic;" | Anti1
| C2
| -231.692600
| -231.539601
| -231.532644
| -234.611800
| -234.469286
| -234.461965
|-
| style="font-style: italic;" | Anti2
| Ci
| -231.692535
| -231.539539
| -231.532565
| -234.611702
| -234.469212
| -234.461856
|-
| style="font-style: italic;" | Gauche3
| C1
| -231.692661
| -231.539486
| -231.532646
| -234.611326
| -234.468718
| -234.461477
|}

 
[[File:Conformers.jpg|200px|thumb|upright=2|'''Figure 3.''' Anti1 (top), Gauche3 (middle) and Anti2 (bottom) conformations of 1,5-hexadiene]]

On the basis of conformational analysis principles, one might expect the anti1 conformation to be the lowest energy conformation of 1,5-hexadiene. However, an optimization at the HF/3-21G level of theory of the gauche3 conformation, shown in Figure 3, yields an electronic energy of -231.69266 hartrees (c.f. -231.69260 hartrees for the anti1 conformation). This suggests that, in fact, the gauche3 conformation is the global minimum.

This observation is not in accordance with the general principle that a C-C-C-C anti-periplanar arrangement enables efficient overlap between σ C-C and σ* C-C orbitals, leading to a stabilization of the filled NBO. Each C-C-C-C gauche unit has been reported to contribute 0.8-1 kcal mol-1 to the total electronic energy of saturated aliphatic hydrocarbons such as n-butane, due to close contact between hydrogen atoms and increased bond-bond Pauli repulsion energy. <ref name="gauche" />. Nonetheless, the anomalous behaviour of 1,5-hexadiene can be attributed to a favourable attractive interaction between the π electrons of the C=C bonds and the nearby vinyl hydrogen atom, referred to as the CH-π interaction. <ref name="pi interaction" /> However, the energy difference between the anti1 and gauche3 conformations of 0.037 kcal mol -1 is a tenfold smaller difference <ref name="pi interaction" /> than in the case of the gauche and anti conformations of n-butane. Thus, since the Hartree-Fock Method describes only non-interacting electrons under the influence of a mean field potential and does not the include the electron correlation effects, the CH-π interaction may not be properly considered with the Hartree-Fock Method. The density functional theory optimization at the B3LYP/6-31G* level gives a higher energy difference of 0.297 kcal mol -1 between the gauche3 and anti1 conformations, but in this case, the gauche3 is the least stable one. The density functional theory includes the exchange-correlation energy term and thus might be expected to consider electronic effects with greater accuracy. However, the results suggest that the exchange-correlation functional used in these computations is not appropriate for the treatment of CH-π interactions. A post-Hartree-Fock Method such as the Møller–Plesset perturbation theory might have been more accurate in determining the lowest energy conformation.

The slight destabilisation of the ''anti2'' conformation compared with the ''anti1'' conformation can be attributed to the closer contact between one of the allylic and the terminal vinyl protons in the '' anti2'' structure, with a distance of 2.52251 Å ('' c.f.'' 2.52266 Å in the case of the ''anti1'' conformer). The ''gauche3'' conformation shows a distance of 2.61312 Å between these protons and thus reduced repulsive steric interactions, providing further evidence to support the observation that ''gauche3'' is the lowest energy conformation of 1,5-hexadiene.

The density functional theory optimization at the B3LYP/6-31G* level of the ''anti2'' conformation yielded a significantly lower electronic energy than in the case of the Hartree-Fock computations. Since DFT includes an approximate treatment of electron correlation, it is expected to provide more accurate results. However, in the absence of experimental data this assumption cannot be confirmed. It is worth noting that even if the overall symmetry of the optimized structure was Ci for both methods, the lengths of the C=C, C-H and C-C bonds were slightly higher when employing a DFT optimization method, as shown in Figure 4. The destabilizing interaction between occupied σ orbitals is thus diminished, resulting in a lower electronic energy. A lower electronic energy was also obtained for the DFT optimization of the other two conformations as shown in Table 1.

[[File:bondsHFandDFT.jpg|center|800px|thumb|upright=4|'''Figure 4.''' Bond distances in optimized ''anti2'' at the HF/3-21G (left) and B3LYP/6-31G* (right) levels of theory ]]

=== Exercise 2. Optimizing the "Chair" and "Boat" Transition Structures of the Cope Rearrangement===

[[Image:ChairTSipm.gif|frame| right| 234 px|'''Animation 1.''' Normal mode with the single imaginary frequency in the chair transition structure ]]
Two allylic fragments optimized separately at the Hartree-Fock level with a basis set of 3-21G were oriented in a chair-like conformation, with a distance between the allyl fragments of approximately 2.2 Å.The optimization to a Berny transition state of the chair transition state was performed initially at the HF/3-21G level of theory by computing the force constants at the beginning of the calculation. A vibrational analysis revealed convergence to a geometry with two imaginary vibrational frequencies and a normal mode motion which did not correspond to the desired reaction coordinate. Thus, the proposed structure was not close enough to the transition state geometry. After a basic manipulation of the position of the two allylic fragments in order to obtain a more symmetrical guess structure, the modified arrangement was re-optimized using the previous method. A single imaginary frequency of magnitude 817.94 cm-1 and the expected normal mode motion shown in Animation 1 were obtained, confirming the successful outcome of the computation. 

The same guess structure was optimized to a minimum at the HF/3-21G level of theory with the frozen coordinates method, by fixing the distance between opposite terminal allylic carbon atoms to 2.2 Å. A structure with decreased C-C bond lengths (1.381 Å c.f. 1.389 Å for the previous method) and an electronic energy of -231.619185 hartrees was obtained. The two bond/forming breaking distances were subsequently optimized using a Hessian modified to include information about the two coordinates along which the differentiation was performed. The computation gave a geometry almost identical to the one obtained by computing force constants at the beginning of the reaction, with only negligible variations of less than 0.00005 Å in both C-C and C-H bond lengths. The imaginary frequency of magnitude 817.88 was described by the same normal mode motion as for the previous method (Animation 1). 

The distance between opposite terminal allylic carbon atoms and the C-C bond length were approximately 2.021 Å and 1.389 Å, respectively, for both computational methods. The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

 
''' Table2.''' Summary of computed energies (hartrees) for the chair and boat transition states *0 K **298.15 K 
{| class="wikitable"
! rowspan="3" style="font-weight: bold;" | Transition state
! colspan="9" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| colspan="3" style="text-align: center; font-weight: bold;" | Force constant computation
| colspan="3" style="text-align: center; font-weight: bold;" | Redundant coordinate method
| colspan="3" style="text-align: center; font-weight: bold;" | QST2
| rowspan="2" | Electronic energy
| rowspan="2" | Sum of electronic and zero-point energies*
| rowspan="2" | Sum of electronic and thermal energies**
|-
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energy*
| Sym of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
|-
| Chair (C2h)
| -231.619322
| -231.466697
| -231.461338
| -231.619322
| -231.466705
| -231.461345
| -
| -
| -
| -234.556932
| -234.414879
| -234.408950
|-
| Boat (C2v)
| -
| -
| -
| -
| -
| -
| -231.602802
| -231.450926
| -231.445300
| -234.543093
| -234.402342
| -234.396008
|}
[[File:Boatinputfile.png|436x234 px|thumb|'''Figure 5.''' Input reactant and product geometries for the boat transition stucture computation]]
[[Image:BoatTSipm.gif|frame| right| 244 px|'''Animation 2.''' Normal mode with the single imaginary frequency in the boat transition structure ]]
The boat transition structure was located using a QST2 approach at the HF/3-21G level, by searching for a maximum along a parabola connecting the reactant and the product, and for a minimum in all directions perpendicular to this parabola. An initial calculation was performed using the ''anti2'' structure that was previously optimized to a minimum at the HF/3-21G level, as the reactant and the corresponding rearranged molecule. However, the calculation starting from these reactant and product structures yielded a chair transition structure with C2h symmetry, with a distance between the central allylic carbon atoms of 2.021 Å. The geometry was thus identical to the chair structure that was previously obtained by optimization to a transition state with the Berny algorithm. (Animation 1). This shows a limitation of the QST2 method, which is not capable of locating the desired first order saddle point if the reactant and product geometries are not similar to the geometry of the boat transition structure. Therefore, the geometry of the optimized ''anti2'' conformation was adjusted by changing the central dihedral C-C-C-C angle to 0 о and the angles formed by the central sp3 carbon atoms with each terminal allylic carbon atom to 100 о for both the reactant and the product as shown in Figure 5. Another QST2 calculation performed on the modified reactant and product structures gave a boat transition structure with an imaginary frequency of magnitude 839.87 (Animation 2). The distance between opposite terminal allylic carbon atoms was found to be approximately 2.140 Å and the C-C bond length was 1.381 Å.The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

Both optimized chair and boat transition structures at the HF/3-21G level were re-optimized using the B3LYP/6-31G* level of theory. Vibrational analysis at the same level of theory confirmed that each of the determined stationary points is a transition structure with a single negative eigenvalue at 569.16 and 530.35 cm-1 for the chair and boat structures, respectively. The optimized chair transition structure illustrated in <jmolFile text="Figure 6">ChairDFT.LOG</jmolFile> shows a distance of 1.967 Å between opposite terminal allylic carbon atoms and a C-C bond length of 1.408 Å. The optimized boat transition structure shown in <jmolFile text="Figure 7">BoatDFT.LOG</jmolFile> has interallylic distance of 2.207 Å and a C-C bond length of 1.393 Å. Thus, both optimized transition structures have similar interallylic distances and C-C bond length as the structures that were previously located using the Hartee-Fock method. As noted in the previous discussion, the DFT computations yield significantly lower energies than Hartree-Fock, as a result of the inclusion of electron correlation. The breaking/forming C-C bond length for the boat transition structure is 0.240 Å longer than for the chair transition structure, which is comparable with the value of 0.230 Å found by MCSCF calculations. <ref name="MCSCF" />. The energy difference at 298.15 K between the boat and chair TS was found to be 8.12 kcal/mol, which is slightly lower than the experimental value of 11.2 ± 2.5 kcal/mol <ref name="energy difference" />. However, these calculations at the B3LYP/6-31G* gave more accurate results than the MM2 method, which has been reported to yield an energy difference of 6.4 kcal/mol <ref name="MM2"/> 

The length of the C-C bond which are not involved in the Cope rearrangement varies between 1.381 and 1.408 Å, being comparable with a benzene C-C bond (1.40 Å). The present results suggest that the reaction proceeds via an aromatic transition state consisting
of two partially bonded three-carbon units. However, this observation is not in agreement with an AM1 study which found the biradicaloid transition structure to be lower in energy than an aromatic transition state.<ref name="experimental"/>. Thus, the preferred mechanism of the Cope rearrangement of 1,5-hexadiene is still not completely elucidated and further calculations are required to predict it.
[[Image:step44.gif|frame| right | 244 px|'''Animation 3.'''Reaction coordinate in the forward direction ]]
All calculations converged to transition structures with lengths of both C-C bonds that are forming or breaking during the rearrangement greater than a typical sp3 C-C bond, varying between 1.967 Å and 2.207 Å. Since all the pericyclic reactions that have been theoretically studied and predicted to be synchronous exhibit forming/breaking bond lengths higher than 2 Å <ref name="experimental"/>, it is reasonable to state that computations performed at HF/3-21G and B3LYP/6-31G* levels are consistent with a synchronous mechanism for the Cope rearrangement. 

IRC computations at the HF/3-21G level provide further evidence to support this affirmation. Calculations for the rearrangement via the chair transition state converged to the structure shown in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> after 44 steps and led to a reaction profile of the type indicated in Figure 9. Since the reaction coordinate is symmetrical, computations were performed only in the forward direction. It can be observed that there are no discrete intermediates characteristic for stepwise processes and thus the reaction is concerted. The 1,5-hexadiene conformation to which the calculations converged with an electronic energy of -231.691579 hartrees was further optimized at the HF/3-21G level of theory. The C2 symmetric gauche2 structure shown in <jmolFile text="Figure 10">OptimizedIRC.mol</jmolFile> was located as a minimum on the potential energy surface, with an electronic energy of -231.691667 hartrees. A further attempt was made to perform the IRC calculations in 100 steps, but the same configuration as the one illustrated in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> was obtained after 44 steps. Even if it would have been useful to verify that the conformation optimized at the HF/3-21G level is the correct minimum, the number of steps was not further increased due to time constraints.
[[File:IRC 50 steps.png|center|536x334 px|thumb|'''Figure 9.''' Reaction profile in the forward direction]] 
[[File:OrbitalsTS.png|right|294x294 px|thumb|'''Figure 10.''' Frontier orbitals in the Cope rearrangement ]]

Activation energies were calculated relative to the Ci symmetric configuration of 1,5-hexadiene (anti2) for both chair and boat reaction pathways at 0 K and 298.15 K. As shown in Table 3, inclusion of electron correlation terms enhances the accuracy of the results, the computations performed at the B3LYP/6-31G* level being in reasonable agreement with the experimental values <ref name="experimental"/>. The significantly lower activation energies for the chair pathway supports the experimental evidence that the transition structure adopts a chair-like conformation with C2h symmetry. <ref name="chairTS"/>. A possible explanation for the preference of the chair conformation is the presence of an antibonding secondary interaction in the boat form. <ref name="secondary"/> as shown in Figure 10. By dividing the 1,5-hexadiene into an ethylene moiety and a butadiene equivalent and considering the interaction between their HOMO and LUMO, it can be noticed that in the chair transition structure, the orbitals are too far apart to exhibit significant out-of-phase overlap.

''' Table 3. Summary of activation energies (kcal/mol) '''

{| class="wikitable"
| rowspan="2" width="125" align="center" | '''Activation energy'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Experimental'''
|-
| width="125" align="center" | '''0 K '''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
|-
| '''ΔE (chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.09
| align="center" | 33.20
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (boat)'''
| align="center" | 55.61
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

=== Exercise 3. The Diels-Alder Cycloaddition===
 
====Cycloaddition reaction between ethene and butadiene====
 

The AM1 semi-empirical molecular orbital method was employed to optimize to a minimum a butadiene and an ethylene fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 11. 

<gallery heights="185" widths="160" caption=" Figure 11. Frontier orbitals of ethene and butadiene" >
File: Planeofsymmetry.png|Plane of symmetry
File: HOMOethene.png|HOMO of ethene: symmetric
File: LUMOethene.png|LUMO of ethene: anti-symmetric
File: HOMObutadiene.png|HOMO of butadiene: anti-symmetric
File: LUMObutadiene.png|LUMO of butadiene: symmetric
</gallery>

The envelope-like guess transition structure for the Diels-Alder reaction was constructed starting from a bicyclo[2.2.2]-2-octene molecule which was adjusted by removing a -CH2-CH2- fragment. Optimization to a transition state using the Berny algorithm at the AM1 level of theory gave a structure with an imaginary frequency of magnitude 623 cm-1 and a normal mode which did not correspond to the desired reaction path. The initial bicyclo[2.2.2]-2-octene molecule was symmetrized and another optimization using the same method was performed. A critical point of Cs symmetry was located on the potential energy surface and vibrational analysis revealed a single imaginary vibrational frequency of magnitude 956.25 cm-1 with the desired normal mode motion (Animation 4). The relevant optimized geometrical parameters of the envelope type transition structure are reported in Figure 12. 

[[File:Dielsaldertransitionstategeometry2.png|300px|thumb|center|upright=2| '''Figure 12.''' Geometrical features of the optimized transition state for the Diels-Alder reaction (Å)]]

[[Image:Transitionstateanimation.gif|frame| right| 234 px|'''Animation 4.''' Normal mode with the single imaginary frequency in the transition structure ]]

The length of the σ C-C bonds which are not involved in forming/breaking processes indicates that the reaction proceeds via a cyclic aromatic transition state. The length of the σ C-C bonds that are expected to form (2.119 Å) is greater than a typical sp3 (~ 1.54 Å) and sp2 (~ 1.34 Å) bond length, but significantly lower than the sum of the corresponding van der Waals radii (~ 3.4 Å). Thus, there is a bonding interaction between the butadiene and ethylene fragments and the new σ C-C bonds are partly formed as expected for a transition structure. It has been generally reported that allowed pericyclic reactions are invariably synchronous if there are no factors which might strongly militate against synchronicity.<ref name="dielsalder" /> Since concerted approaches involve cyclic aromatic transition states, the transition structures are expected to be highly symmetric based on the analogy to the structure of benzene (equal C-C bond lengths of approximately 1.40 Å). <ref name="dielsalder" />. Therefore, the equal length of both C-C bonds formed in the cycloaddition reaction suggests that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cis-butadiene and ethylene. Further evidence is provided by the normal mode motion at -956.25 cm-1 corresponding to the reaction path, which depicts a simultaneous formation of the two σ C-C bonds. The vibration with the lowest positive frequency occurs at 147.30 cm-1 describes a nonsynchronous approach, but the motion is simply a twisting of both fragments rather than a bond forming process equivalent to the reaction path. Even if AM1 agrees with the ab initio calculation <ref name="abinitio"/> in predicting the reaction to be synchronous, MIND0/3 calculations are in favour of the process involving a stable biradicaloid intermediate, which collapses to cyclohexene in the rate-determining step. <ref name="rzepa"/>. The nature of the mechanism is thus still open to debate. 

The AM1 computations can be further employed to confirm theoretical observations such as the fact that the Diels-Alder reaction is an allowed process. The Woodward-Hoffmann rule for thermal pericyclic reactions states that a ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a <ref name="pericyclic"/> components is odd. "S" and "a" refer to suprafacial and antarafacial components, respectively and p and r are integers. Since the investigated Diels-Alder reaction involves only suprafacial bond formation between a diene with 4 π electrons (r=1) and a dienophile with 2 π electrons (q=0), the process can be described as having only one (4q+2)s component and no (4r)a components. The total number of components is thus 1 and the reaction between ethene and cis-butadiene is expected to be allowed.
This is confirmed by the nodal properties of the HOMO and LUMO that were previously computed for ethene and butadiene. It is clear from Figure 13 that the overlap between HOMO of one reactant and the LUMO of the other reactant is bonding at both sites, if only suprafacial components are considered. Overlap between the HOMO of butadiene and the LUMO of ethylene gives the HOMO of the Cs transition structure, whereas interaction between the LUMO of butadiene and the HOMO of ethylene gives the LUMO of the transition structure. As shown in Figure 13, the HOMO and LUMO corresponding to the transition structure are anti-symmetric and symmetric, respectively, with respect to the plane of symmetry that was previously defined.
 
<gallery heights="195" widths="170" caption=" Figure 13. Frontier orbitals of the TS and bonding interactions" >
File: bondingHOMOetheneLUMObutadiene.png| Bonding interaction between HOMO of ethene and LUMO of butadiene
File: bondingLUMOetheneLUMObutadiene.png|Bonding interaction between LUMO of ethene and HOMO of butadiene
File: HOMOtransitionstate.png|HOMO of the transition structure: anti-symmetric
File: LUMOtransitionstate.png|LUMO of the transition structure: symmetric
</gallery>
[[File:HOMOhartreefock.png|436x234 px|thumb|'''Figure 14.'''HOMO of the TS obtained at the HF/3-21G level]]
 Note: The representation of the bonding interactions is only schematic and the relative size of the butadiene and ethene frontier orbitals is not accurate. The lines identify the bonding overlap that can develop as the reaction proceeds.

An attempt was made to perform all the above computations at the HF/3-21G level of theory. However, a transition state with the symmetric HOMO shown in Figure 14 was located, suggesting that the level of theory is not appropriate for the characterisation of the frontier orbitals involved in the Diels-Alder reaction. Based on the remarks that were previously made for the Cope rearrangement, calculations at the B3LYP/6-31G* level might have given more accurate results. Even if the shape and the coefficients of the molecular orbitals should not have been affected, the DFT method might have given electronic and thermally corrected energies which are in better agreement with the experimental values.
 
 

==== Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride ====
 
The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a cyclohexa-1,3-diene and a maleic anhydride fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane shown are illustrated in Figure 15. The overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride or vice versa is bonding at both sites, if only suprafacial components are considered. The cycloaddition reaction is thus allowed, which is consistent with the predictions based on the Woodward-Hoffmann rule.
 

<gallery heights="185" widths="160" caption=" Figure 15. Frontier orbitals of cyclohexa-1,3-diene and maleic anhydride " >
File: Planeofsymmetryii.png|Plane of symmetry
File: HOMOcyclohexadiene.png|HOMO of cyclohexa-1,3-diene: anti-symmetric
File: LUMOcyclohexadiene.png|LUMO of cyclohexa-1,3-diene: symmetric
File: HOMOanhydride.png|HOMO of maleic anhydride: symmetric
File: LUMOanhydride.png|LUMO of maleic anhydride: anti-symmetric
</gallery>

As mentioned in the introduction section, the Diels-Alder between cyclohexa-1,3-diene and maleic anhydride can give an ''endo'' or an ''exo'' adduct and thus it was necessary to construct a guess transition structure for both pathways. The guess transition structures were constructed by joining together a bicyclo[2.2.2]-2-octene molecule, adjusted as for the previous part by removing a -CH2-CH2- fragment, and a five-membered cyclic anhydride fragmen, at a distance of approximately 2 Å from each other. Optimization to a transition state using the Berny algorithm at the AM1 level of theory followed by vibrational analysis gave a structure with an imaginary vibrational frequency of magnitude 806.38 and 812.02 cm-1 for the <jmolFile text=" endo">endoTSipm12.mol</jmolFile> and <jmolFile text=" exo">DIELSEXO.LOG</jmolFile> adducts, respectively. The desired normal mode motion (Animation 5 and 6) corresponding to the Diels-Alder reaction path was obtained in both cases. The relevant optimized geometrical parameters of both transition structures are reported in Figure 16 and 17.

{|
![[Image:endoTSipm12r.gif|frame| center| Animation 4. Normal mode with the single imaginary frequency in the ''endo'' transition structure ]]
![[Image:exoTS.gif|frame| center| Animation 5. Normal mode with the single imaginary frequency in the ''exo'' transition structure ]]
! [[File:geometricalfeaturesendo.png| frame| center| 208x208 px|Figure 16. Geometrical features of the endo transition structure ]]
! [[File:geometricalfeaturesexo.png| frame| center| 150x150 px|Figure 17. Geometrical features of the exo transition structure ]]
|}

The length of the σ C-C bonds which are not involved in forming/breaking processes is comparable with the length of aromatic C-C σ bonds, indicating that the reaction proceeds via a cyclic aromatic transition state. Analogous to the Diels-Alder reaction between ethene and ''cis''-butadiene, the highly symmetric transition structures suggest that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cyclohexa-1,3-diene and maleic anhydride via both ''endo'' and ''exo'' adducts.<ref name='concerted'/>. Further evidence is provided by the normal mode motion at the imaginary frequency, which depicts a simultaneous formation of the two σ C-C bonds for both transition structures. In addition, IRC calculations at the AM1 level of theory, performed in both directions in 20 steps, are consistent with the proposed synchronous mechanism. The reaction profile does not show any discrete intermediates characteristic for stepwise processes, excluding thus the possibility of a mechanism proceeding via biradical or zwitterionic intermediates. It should be noted that the in the case of the IRC computation via the ''endo'' transition structure, the first step on the reaction path corresponds to the reactants and the last step corresponds to the cycloaddition product, whereas in the case of the ''exo'' transition structure the opposite situation was observed. Thus the IRC computations performed at the AM1 level confirm the exothermic nature of the Diels-Alder cycloaddition reaction.
[[File:IRCendo20steps.png| frame| center| 300x300 px|'''Figure 18.''' Reaction profile via the endo transition structure ]]
[[File:IRCexo20steps.png| frame| center| 300x300 px|'''Figure 19.''' Reaction profile via the exo transition structure ]]

The computed electronic and thermally corrected energies of the ''endo'' and ''exo'' transition structures are reported in Table 4 below. The calculations suggest that the activation energy of the reaction pathway which proceeds via the ''endo'' transition structure is lower than the activation energy of the process involving the ''exo'' transition structure by 0.87 kcal/mol (0.0013185 hartrees) at 0 K and 0.75 kcal/mol (0.001197 hartrees) at 298.15 K. Therefore, the AM1 predictions are in agreement with the experimental observation that the Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene gives the ''endo'' product faster than it gives the ''exo'' adduct, even though the latter is thermodynamically more stable. 
 

Table 4. Summary of computed energies (hartrees) for the endo and exo transition states

{| class="wikitable"
! style="font-weight: bold;" | Optimized molecule
! style="font-weight: bold;" | Electronic energy
! style="font-weight: bold;" | Sum of electronic and zero-point energies
! style="font-weight: bold;" | Sum of electronic and thermal energies
|-
| style="font-weight: bold;" | Cyclohexa-1,3-diene
| 0.027711
| 0.152502
| 0.157726
|-
| style="font-weight: bold;" | Maleic anhydride
| -0.121824
| -0.063346
| -0.058192
|-
| style="font-weight: bold;" | Endo TS
| -0.051505
| 0.133494
| 0.143683
|-
| style="font-weight: bold;" | Exo TS
| -0.050419
| 0.134879
| 0.144880
|}

[[File:endoorbitalinteractions.png| frame| right| 208x208 px|'''Figure 20.''' Secondary orbital interaction in the ''endo'' transition structure ]]
In order to account for the formation of the ''endo'' adduct under kinetically controlled conditions, the nodal properties of the frontier orbitals of the reactants and both transition structures were examined. The interaction between the frontier orbitals can be simplified in a diagram involving only atomic orbitals (Figure 20). It can be clearly observed that the ''endo'' transition structure is characterized by an in-phase secondary overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride. Since this stabilising secondary orbital interaction is not possible for an ''exo'' adduct, the ''endo'' transition state is expected to be lowered in energy relative to the ''exo'' one. The stabilisation of the ''endo'' transition state is further reinforced by the increased C-C through space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the carbon atoms of cyclohexa-1,3-diene which are not involved in the reaction (3.897 Å c.f. 2.945 Å in the ''exo'' adduct), which indicates diminished steric repulsion.
{|
! [[File:HOMOendoTS.png| frame| 300x300 px|Figure 21. HOMO of the endo transition structure: anti-symmetric]]
! [[File:LUMOendoTS.png| frame| 300x300px|Figure 22. LUMO of the endo transition structure: anti-symmetric ]]
! [[File:HOMOexoTS.png| frame| 300x300 px|Figure 23. HOMO of the exo transition structure: anti-symmetric ]]
! [[File:LUMOexoTS.png| frame| 300x300 px|Figure 24. LUMO of the exo transition structure: anti-symmetric ]]
|}

The computations performed for the cycloaddtion reaction between butadiene and ethene demonstrated that the AM1 method provides more accurate results than Hartree-Fock, since it implicitly takes into account electronic correlation through the use parameters derived from experimental data. However, as any other semi-empirical method, it is expected to give very reliable results only for systems which are related to those from which their parameters have been fitted. A further study might involve the use of MCSCF calculations, which have been reported to give a more a balanced description of closed-shell species. <ref name="MCSCF2"/>

==Conclusions==
==References==
<references>
<ref name="gauche"> K.S. Pitzerk, Chem. Rev., 1940, 27, 39-57 {{DOI|10.1021/cr60086a003 }} </ref>
<ref name="intro"> F. Bernardi, A. Bottoni, M.J. Field, M. F. Guest, I. H. Hillier, M. A. Robb, A. Venturini, J. Am. Chem. Soc., 1988, '''110''', 3050-3055
{{DOI|10.1021/ja00218a009 }} </ref>
<ref name="pi interaction"> B.G. Rocque, J.M. Gonzales, H.F. Schaffer, Mol. Phys., 2002, 100, 441-446 {{DOI|10.1080/00268970110081412 }} </ref>
<ref name="MCSCF" > K. Morokuma, W. T. Borden, D.A. Hvorat, J. Am. Chem. Soc., 1988, '''110''', 4474. {{DOI|10.1021/ja00221a092 }} </ref>
<ref name="MM2"> F. Jensen, ''J. Am. Chem. Soc''. 1992, '''114''', 1596-1603 {{DOI|10.1021/ja00031a009}} </ref>
<ref name="energy difference"> M. J. Goldstein, M. S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''',7147–7149 {{DOI| 10.1021/ja00775a046}}</ref>
<ref name="experimental"> M.J.S. Dewar, C. Jie, ''J. Am. Chem. Soc.'', 1987, '''109''', 5897 {{DOI|10.1021/ja00254a001}} </ref>
<ref name="chairTS"> K. N. Houk, S. M. Gustafson, K. A. Black, J. Am. Chem. Soc., 1992, 114, 8565-8572 {{DOI|10.1021/ja00048a032}} </ref>
<ref name="secondary"> D. Nisipuri, Stereochemistry of Organic Compounds: Principles and Applications , New Age Publications, 2nd edn., 1996 </ref>
<ref name="dielsalder"> M. J. S. Dewar, S. Olivella,, J. P. Stewartt, J. Am. Chem. Soc., 1986, 108, 5771-5779 {{DOI |10.1021/ja00279a018}} </ref>
<ref name="abinitio"> M. Ortega, A. Oliva, J. M. Lluch, J. Bertran, ''Chem. Phys. Letr.'', 1983, '''102''', 317 {{DOI|10.1016/0009-2614(83)87048-1}} </ref>
<ref name="rzepa">M.J.S. Dewar, S. Olivella, H. Rzepa, ''J. Am. Chem. Soc'', 1978, '''100''', 5650- 5659 {{DOI|10.1021/ja00486a013}}</ref>
<ref name="pericyclic"> I. Fleming, ''Pericyclic Reactions'', Oxford Science Publications, 2002, 40 </ref>
<ref name="concerted"> R. N. Buckle, P. Liu, E. W.D Roberts, D. J. Burnell , ''Tetrahedron'', 1999, '''55''', 11455–11464 {{DOI|10.1016/S0040-4020(99)00667-5}}</ref>
<ref name="MCSCF2"> K. N. Houk, Y. Lin, F.K. Brown, ''J. Am. Chem. Soc'', 1986, '''108''', 554 {{DOI|10.1021/ja00263a059}}
</ref>
</references>

Rep:Mod:iulianamaria

2014-12-05T02:06:08Z

Ipm12:

== Third Year Computational Lab: Physical module. Transition states and reactivity ==

== Introduction ==

The structures and energy of the species involved in a chemical reaction offer valuable insight into the mechanism and general features of the process. Due to the advances in optimization algorithms, information about the most stable geometry, vibrational frequencies and in principle any observable physical quantity can be computed with a high degree of accuracy. Synthetic sequences often involve two or more reaction pathways with similar activation energies and thus the location of the transition structures corresponding to each pathway on the potential energy surface is of great importance in predicting the major products.

The Cope rearrangement reaction of 1,5-hexadiene is an archetypal [3,3] sigmatropic shift which has been the subject of numerous theoretical and experimental studies. Computational evidence has been provided to show that the reaction can proceed via either an aromatic transition state consisting of two partially bonded three-carbon units or via a cyclohexane- 1,4-diyl-like structure as shown in Figure 1. <ref name="chairTS"/>. This study employs computations at the HF/3-21G and B3LYP/6-31G* levels of theory to investigate the lowest energy conformation of the reactant molecule and to locate the two possible, chair and boat, transition structures. The results are subsequently employed to rationalize the preferred mechanism and a comparison with previous research is also provided.

[[File:dielsalderscheme.png|right|300x300px|thumb|'''Figure 2.''' Diels-Alder cycloaddtion reaction]]
The [4+2] Diels-Alder reaction between ethylene and butadiene is another example of a process which has been extensively investigated from a theoretical and experimental point of view, but which is still not completely elucidated. Possible mechanisms that have been proposed for this reaction include a synchronous concerted approach via a cyclcic aromatic transition state, a two-stage asynchronous concerted mechanism and a stepwise process, which occurs via a stable diradical intermediate. <ref name="intro"/> This study uses AM1 semi-empirical molecular orbital method to locate and investigate the geometrical features of the transition state, followed by an analysis of the possible interaction between the frontier orbitals involved in the reaction.Computations at the AM1 level of theory are also employed to probe the Diels-Alder endo rule and study the possible pathways of the cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride. Both ''endo'' and ''exo'' adducts shown in Figure 2 are considered and their relative energies and nodal properties of the molecular oribitals are compared in order to determine the product that is preferentially formed.

==Results and Computational Methods==

 

=== Exercise 1. Optimizing the Reactants and Products of the Cope Rearrangement===

Due to the presence of three C-C bonds and three rotational minima around each of these bonds <ref name="pi interaction" />, 1,5-hexadiene is expected to have 27 conformations in total and ten of them to be energetically distinct. The conformations shown in Figure 3 are optimized at the B3LYP/6-31G* and HF/3-21G levels of theory and their electronic energies are compared, in order to determine the lowest energy conformation of 1,6-hexadiene. Frequency calculations performed on the optimized structures at the same level of theory yielded only real and positive frequencies, confirming that a minimum was obtained. The electronic energies and symmetries, along with the sum of electronic and zero-point energies and the sum of electronic and thermal energies of the optimized structures are summarized in Table 1.

 

'''Table 1. Summary of energies (hartrees) of the conformations under investigation'''

{| class="wikitable"
! rowspan="2" style="font-weight: bold;" | Conformer
! rowspan="2" style="font-weight: bold;" | Point Group
! colspan="3" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
|-
| style="font-style: italic;" | Anti1
| C2
| -231.692600
| -231.539601
| -231.532644
| -234.611800
| -234.469286
| -234.461965
|-
| style="font-style: italic;" | Anti2
| Ci
| -231.692535
| -231.539539
| -231.532565
| -234.611702
| -234.469212
| -234.461856
|-
| style="font-style: italic;" | Gauche3
| C1
| -231.692661
| -231.539486
| -231.532646
| -234.611326
| -234.468718
| -234.461477
|}

 
[[File:Conformers.jpg|200px|thumb|upright=2|'''Figure 3.''' Anti1 (top), Gauche3 (middle) and Anti2 (bottom) conformations of 1,5-hexadiene]]

On the basis of conformational analysis principles, one might expect the anti1 conformation to be the lowest energy conformation of 1,5-hexadiene. However, an optimization at the HF/3-21G level of theory of the gauche3 conformation, shown in Figure 3, yields an electronic energy of -231.69266 hartrees (c.f. -231.69260 hartrees for the anti1 conformation). This suggests that, in fact, the gauche3 conformation is the global minimum.

This observation is not in accordance with the general principle that a C-C-C-C anti-periplanar arrangement enables efficient overlap between σ C-C and σ* C-C orbitals, leading to a stabilization of the filled NBO. Each C-C-C-C gauche unit has been reported to contribute 0.8-1 kcal mol-1 to the total electronic energy of saturated aliphatic hydrocarbons such as n-butane, due to close contact between hydrogen atoms and increased bond-bond Pauli repulsion energy. <ref name="gauche" />. Nonetheless, the anomalous behaviour of 1,5-hexadiene can be attributed to a favourable attractive interaction between the π electrons of the C=C bonds and the nearby vinyl hydrogen atom, referred to as the CH-π interaction. <ref name="pi interaction" /> However, the energy difference between the anti1 and gauche3 conformations of 0.037 kcal mol -1 is a tenfold smaller difference <ref name="pi interaction" /> than in the case of the gauche and anti conformations of n-butane. Thus, since the Hartree-Fock Method describes only non-interacting electrons under the influence of a mean field potential and does not the include the electron correlation effects, the CH-π interaction may not be properly considered with the Hartree-Fock Method. The density functional theory optimization at the B3LYP/6-31G* level gives a higher energy difference of 0.297 kcal mol -1 between the gauche3 and anti1 conformations, but in this case, the gauche3 is the least stable one. The density functional theory includes the exchange-correlation energy term and thus might be expected to consider electronic effects with greater accuracy. However, the results suggest that the exchange-correlation functional used in these computations is not appropriate for the treatment of CH-π interactions. A post-Hartree-Fock Method such as the Møller–Plesset perturbation theory might have been more accurate in determining the lowest energy conformation.

The slight destabilisation of the ''anti2'' conformation compared with the ''anti1'' conformation can be attributed to the closer contact between one of the allylic and the terminal vinyl protons in the '' anti2'' structure, with a distance of 2.52251 Å ('' c.f.'' 2.52266 Å in the case of the ''anti1'' conformer). The ''gauche3'' conformation shows a distance of 2.61312 Å between these protons and thus reduced repulsive steric interactions, providing further evidence to support the observation that ''gauche3'' is the lowest energy conformation of 1,5-hexadiene.

The density functional theory optimization at the B3LYP/6-31G* level of the ''anti2'' conformation yielded a significantly lower electronic energy than in the case of the Hartree-Fock computations. Since DFT includes an approximate treatment of electron correlation, it is expected to provide more accurate results. However, in the absence of experimental data this assumption cannot be confirmed. It is worth noting that even if the overall symmetry of the optimized structure was Ci for both methods, the lengths of the C=C, C-H and C-C bonds were slightly higher when employing a DFT optimization method, as shown in Figure 4. The destabilizing interaction between occupied σ orbitals is thus diminished, resulting in a lower electronic energy. A lower electronic energy was also obtained for the DFT optimization of the other two conformations as shown in Table 1.

[[File:bondsHFandDFT.jpg|center|800px|thumb|upright=4|'''Figure 4.''' Bond distances in optimized ''anti2'' at the HF/3-21G (left) and B3LYP/6-31G* (right) levels of theory ]]

=== Exercise 2. Optimizing the "Chair" and "Boat" Transition Structures of the Cope Rearrangement===

[[Image:ChairTSipm.gif|frame| right| 234 px|'''Animation 1.''' Normal mode with the single imaginary frequency in the chair transition structure ]]
Two allylic fragments optimized separately at the Hartree-Fock level with a basis set of 3-21G were oriented in a chair-like conformation, with a distance between the allyl fragments of approximately 2.2 Å.The optimization to a Berny transition state of the chair transition state was performed initially at the HF/3-21G level of theory by computing the force constants at the beginning of the calculation. A vibrational analysis revealed convergence to a geometry with two imaginary vibrational frequencies and a normal mode motion which did not correspond to the desired reaction coordinate. Thus, the proposed structure was not close enough to the transition state geometry. After a basic manipulation of the position of the two allylic fragments in order to obtain a more symmetrical guess structure, the modified arrangement was re-optimized using the previous method. A single imaginary frequency of magnitude 817.94 cm-1 and the expected normal mode motion shown in Animation 1 were obtained, confirming the successful outcome of the computation. 

The same guess structure was optimized to a minimum at the HF/3-21G level of theory with the frozen coordinates method, by fixing the distance between opposite terminal allylic carbon atoms to 2.2 Å. A structure with decreased C-C bond lengths (1.381 Å c.f. 1.389 Å for the previous method) and an electronic energy of -231.619185 hartrees was obtained. The two bond/forming breaking distances were subsequently optimized using a Hessian modified to include information about the two coordinates along which the differentiation was performed. The computation gave a geometry almost identical to the one obtained by computing force constants at the beginning of the reaction, with only negligible variations of less than 0.00005 Å in both C-C and C-H bond lengths. The imaginary frequency of magnitude 817.88 was described by the same normal mode motion as for the previous method (Animation 1). 

The distance between opposite terminal allylic carbon atoms and the C-C bond length were approximately 2.021 Å and 1.389 Å, respectively, for both computational methods. The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

 
''' Table2.''' Summary of computed energies (hartrees) for the chair and boat transition states *0 K **298.15 K 
{| class="wikitable"
! rowspan="3" style="font-weight: bold;" | Transition state
! colspan="9" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| colspan="3" style="text-align: center; font-weight: bold;" | Force constant computation
| colspan="3" style="text-align: center; font-weight: bold;" | Redundant coordinate method
| colspan="3" style="text-align: center; font-weight: bold;" | QST2
| rowspan="2" | Electronic energy
| rowspan="2" | Sum of electronic and zero-point energies*
| rowspan="2" | Sum of electronic and thermal energies**
|-
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energy*
| Sym of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
|-
| Chair (C2h)
| -231.619322
| -231.466697
| -231.461338
| -231.619322
| -231.466705
| -231.461345
| -
| -
| -
| -234.556932
| -234.414879
| -234.408950
|-
| Boat (C2v)
| -
| -
| -
| -
| -
| -
| -231.602802
| -231.450926
| -231.445300
| -234.543093
| -234.402342
| -234.396008
|}
[[File:Boatinputfile.png|436x234 px|thumb|'''Figure 5.''' Input reactant and product geometries for the boat transition stucture computation]]
[[Image:BoatTSipm.gif|frame| right| 244 px|'''Animation 2.''' Normal mode with the single imaginary frequency in the boat transition structure ]]
The boat transition structure was located using a QST2 approach at the HF/3-21G level, by searching for a maximum along a parabola connecting the reactant and the product, and for a minimum in all directions perpendicular to this parabola. An initial calculation was performed using the ''anti2'' structure that was previously optimized to a minimum at the HF/3-21G level, as the reactant and the corresponding rearranged molecule. However, the calculation starting from these reactant and product structures yielded a chair transition structure with C2h symmetry, with a distance between the central allylic carbon atoms of 2.021 Å. The geometry was thus identical to the chair structure that was previously obtained by optimization to a transition state with the Berny algorithm. (Animation 1). This shows a limitation of the QST2 method, which is not capable of locating the desired first order saddle point if the reactant and product geometries are not similar to the geometry of the boat transition structure. Therefore, the geometry of the optimized ''anti2'' conformation was adjusted by changing the central dihedral C-C-C-C angle to 0 о and the angles formed by the central sp3 carbon atoms with each terminal allylic carbon atom to 100 о for both the reactant and the product as shown in Figure 5. Another QST2 calculation performed on the modified reactant and product structures gave a boat transition structure with an imaginary frequency of magnitude 839.87 (Animation 2). The distance between opposite terminal allylic carbon atoms was found to be approximately 2.140 Å and the C-C bond length was 1.381 Å.The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

Both optimized chair and boat transition structures at the HF/3-21G level were re-optimized using the B3LYP/6-31G* level of theory. Vibrational analysis at the same level of theory confirmed that each of the determined stationary points is a transition structure with a single negative eigenvalue at 569.16 and 530.35 cm-1 for the chair and boat structures, respectively. The optimized chair transition structure illustrated in <jmolFile text="Figure 6">ChairDFT.LOG</jmolFile> shows a distance of 1.967 Å between opposite terminal allylic carbon atoms and a C-C bond length of 1.408 Å. The optimized boat transition structure shown in <jmolFile text="Figure 7">BoatDFT.LOG</jmolFile> has interallylic distance of 2.207 Å and a C-C bond length of 1.393 Å. Thus, both optimized transition structures have similar interallylic distances and C-C bond length as the structures that were previously located using the Hartee-Fock method. As noted in the previous discussion, the DFT computations yield significantly lower energies than Hartree-Fock, as a result of the inclusion of electron correlation. The breaking/forming C-C bond length for the boat transition structure is 0.240 Å longer than for the chair transition structure, which is comparable with the value of 0.230 Å found by MCSCF calculations. <ref name="MCSCF" />. The energy difference at 298.15 K between the boat and chair TS was found to be 8.12 kcal/mol, which is slightly lower than the experimental value of 11.2 ± 2.5 kcal/mol <ref name="energy difference" />. However, these calculations at the B3LYP/6-31G* gave more accurate results than the MM2 method, which has been reported to yield an energy difference of 6.4 kcal/mol <ref name="MM2"/> 

The length of the C-C bond which are not involved in the Cope rearrangement varies between 1.381 and 1.408 Å, being comparable with a benzene C-C bond (1.40 Å). The present results suggest that the reaction proceeds via an aromatic transition state consisting
of two partially bonded three-carbon units. However, this observation is not in agreement with an AM1 study which found the biradicaloid transition structure to be lower in energy than an aromatic transition state.<ref name="experimental"/>. Thus, the preferred mechanism of the Cope rearrangement of 1,5-hexadiene is still not completely elucidated and further calculations are required to predict it.
[[Image:step44.gif|frame| right | 244 px|'''Animation 3.'''Reaction coordinate in the forward direction ]]
All calculations converged to transition structures with lengths of both C-C bonds that are forming or breaking during the rearrangement greater than a typical sp3 C-C bond, varying between 1.967 Å and 2.207 Å. Since all the pericyclic reactions that have been theoretically studied and predicted to be synchronous exhibit forming/breaking bond lengths higher than 2 Å <ref name="experimental"/>, it is reasonable to state that computations performed at HF/3-21G and B3LYP/6-31G* levels are consistent with a synchronous mechanism for the Cope rearrangement. 

IRC computations at the HF/3-21G level provide further evidence to support this affirmation. Calculations for the rearrangement via the chair transition state converged to the structure shown in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> after 44 steps and led to a reaction profile of the type indicated in Figure 9. Since the reaction coordinate is symmetrical, computations were performed only in the forward direction. It can be observed that there are no discrete intermediates characteristic for stepwise processes and thus the reaction is concerted. The 1,5-hexadiene conformation to which the calculations converged with an electronic energy of -231.691579 hartrees was further optimized at the HF/3-21G level of theory. The C2 symmetric gauche2 structure shown in <jmolFile text="Figure 10">OptimizedIRC.mol</jmolFile> was located as a minimum on the potential energy surface, with an electronic energy of -231.691667 hartrees. A further attempt was made to perform the IRC calculations in 100 steps, but the same configuration as the one illustrated in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> was obtained after 44 steps. Even if it would have been useful to verify that the conformation optimized at the HF/3-21G level is the correct minimum, the number of steps was not further increased due to time constraints.
[[File:IRC 50 steps.png|center|536x334 px|thumb|'''Figure 9.''' Reaction profile in the forward direction]] 
[[File:OrbitalsTS.png|right|294x294 px|thumb|'''Figure 10.''' Frontier orbitals in the Cope rearrangement ]]

Activation energies were calculated relative to the Ci symmetric configuration of 1,5-hexadiene (anti2) for both chair and boat reaction pathways at 0 K and 298.15 K. As shown in Table 3, inclusion of electron correlation terms enhances the accuracy of the results, the computations performed at the B3LYP/6-31G* level being in reasonable agreement with the experimental values <ref name="experimental"/>. The significantly lower activation energies for the chair pathway supports the experimental evidence that the transition structure adopts a chair-like conformation with C2h symmetry. <ref name="chairTS"/>. A possible explanation for the preference of the chair conformation is the presence of an antibonding secondary interaction in the boat form. <ref name="secondary"/> as shown in Figure 10. By dividing the 1,5-hexadiene into an ethylene moiety and a butadiene equivalent and considering the interaction between their HOMO and LUMO, it can be noticed that in the chair transition structure, the orbitals are too far apart to exhibit significant out-of-phase overlap.

''' Table 3. Summary of activation energies (kcal/mol) '''

{| class="wikitable"
| rowspan="2" width="125" align="center" | '''Activation energy'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Experimental'''
|-
| width="125" align="center" | '''0 K '''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
|-
| '''ΔE (chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.09
| align="center" | 33.20
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (boat)'''
| align="center" | 55.61
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

=== Exercise 3. The Diels-Alder Cycloaddition===
 
====Cycloaddition reaction between ethene and butadiene====
 

The AM1 semi-empirical molecular orbital method was employed to optimize to a minimum a butadiene and an ethylene fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 11. 

<gallery heights="185" widths="160" caption=" Figure 11. Frontier orbitals of ethene and butadiene" >
File: Planeofsymmetry.png|Plane of symmetry
File: HOMOethene.png|HOMO of ethene: symmetric
File: LUMOethene.png|LUMO of ethene: anti-symmetric
File: HOMObutadiene.png|HOMO of butadiene: anti-symmetric
File: LUMObutadiene.png|LUMO of butadiene: symmetric
</gallery>

The envelope-like guess transition structure for the Diels-Alder reaction was constructed starting from a bicyclo[2.2.2]-2-octene molecule which was adjusted by removing a -CH2-CH2- fragment. Optimization to a transition state using the Berny algorithm at the AM1 level of theory gave a structure with an imaginary frequency of magnitude 623 cm-1 and a normal mode which did not correspond to the desired reaction path. The initial bicyclo[2.2.2]-2-octene molecule was symmetrized and another optimization using the same method was performed. A critical point of Cs symmetry was located on the potential energy surface and vibrational analysis revealed a single imaginary vibrational frequency of magnitude 956.25 cm-1 with the desired normal mode motion (Animation 4). The relevant optimized geometrical parameters of the envelope type transition structure are reported in Figure 12. 

[[File:Dielsaldertransitionstategeometry2.png|300px|thumb|center|upright=2| '''Figure 12.''' Geometrical features of the optimized transition state for the Diels-Alder reaction (Å)]]

[[Image:Transitionstateanimation.gif|frame| right| 234 px|'''Animation 4.''' Normal mode with the single imaginary frequency in the transition structure ]]

The length of the σ C-C bonds which are not involved in forming/breaking processes indicates that the reaction proceeds via a cyclic aromatic transition state. The length of the σ C-C bonds that are expected to form (2.119 Å) is greater than a typical sp3 (~ 1.54 Å) and sp2 (~ 1.34 Å) bond length, but significantly lower than the sum of the corresponding van der Waals radii (~ 3.4 Å). Thus, there is a bonding interaction between the butadiene and ethylene fragments and the new σ C-C bonds are partly formed as expected for a transition structure. It has been generally reported that allowed pericyclic reactions are invariably synchronous if there are no factors which might strongly militate against synchronicity.<ref name="dielsalder" /> Since concerted approaches involve cyclic aromatic transition states, the transition structures are expected to be highly symmetric based on the analogy to the structure of benzene (equal C-C bond lengths of approximately 1.40 Å). <ref name="dielsalder" />. Therefore, the equal length of both C-C bonds formed in the cycloaddition reaction suggests that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cis-butadiene and ethylene. Further evidence is provided by the normal mode motion at -956.25 cm-1 corresponding to the reaction path, which depicts a simultaneous formation of the two σ C-C bonds. The vibration with the lowest positive frequency occurs at 147.30 cm-1 describes a nonsynchronous approach, but the motion is simply a twisting of both fragments rather than a bond forming process equivalent to the reaction path. Even if AM1 agrees with the ab initio calculation <ref name="abinitio"/> in predicting the reaction to be synchronous, MIND0/3 calculations are in favour of the process involving a stable biradicaloid intermediate, which collapses to cyclohexene in the rate-determining step. <ref name="rzepa"/>. The nature of the mechanism is thus still open to debate. 

The AM1 computations can be further employed to confirm theoretical observations such as the fact that the Diels-Alder reaction is an allowed process. The Woodward-Hoffmann rule for thermal pericyclic reactions states that a ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a <ref name="pericyclic"/> components is odd. "S" and "a" refer to suprafacial and antarafacial components, respectively and p and r are integers. Since the investigated Diels-Alder reaction involves only suprafacial bond formation between a diene with 4 π electrons (r=1) and a dienophile with 2 π electrons (q=0), the process can be described as having only one (4q+2)s component and no (4r)a components. The total number of components is thus 1 and the reaction between ethene and cis-butadiene is expected to be allowed.
This is confirmed by the nodal properties of the HOMO and LUMO that were previously computed for ethene and butadiene. It is clear from Figure 13 that the overlap between HOMO of one reactant and the LUMO of the other reactant is bonding at both sites, if only suprafacial components are considered. Overlap between the HOMO of butadiene and the LUMO of ethylene gives the HOMO of the Cs transition structure, whereas interaction between the LUMO of butadiene and the HOMO of ethylene gives the LUMO of the transition structure. As shown in Figure 13, the HOMO and LUMO corresponding to the transition structure are anti-symmetric and symmetric, respectively, with respect to the plane of symmetry that was previously defined.
 
<gallery heights="195" widths="170" caption=" Figure 13. Frontier orbitals of the TS and bonding interactions" >
File: bondingHOMOetheneLUMObutadiene.png| Bonding interaction between HOMO of ethene and LUMO of butadiene
File: bondingLUMOetheneLUMObutadiene.png|Bonding interaction between LUMO of ethene and HOMO of butadiene
File: HOMOtransitionstate.png|HOMO of the transition structure: anti-symmetric
File: LUMOtransitionstate.png|LUMO of the transition structure: symmetric
</gallery>
[[File:HOMOhartreefock.png|436x234 px|thumb|'''Figure 14.'''HOMO of the TS obtained at the HF/3-21G level]]
 Note: The representation of the bonding interactions is only schematic and the relative size of the butadiene and ethene frontier orbitals is not accurate. The lines identify the bonding overlap that can develop as the reaction proceeds.

An attempt was made to perform all the above computations at the HF/3-21G level of theory. However, a transition state with the symmetric HOMO shown in Figure 14 was located, suggesting that the level of theory is not appropriate for the characterisation of the frontier orbitals involved in the Diels-Alder reaction. Based on the remarks that were previously made for the Cope rearrangement, calculations at the B3LYP/6-31G* level might have given more accurate results. Even if the shape and the coefficients of the molecular orbitals should not have been affected, the DFT method might have given electronic and thermally corrected energies which are in better agreement with the experimental values.
 
 

==== Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride ====
 
The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a cyclohexa-1,3-diene and a maleic anhydride fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane shown are illustrated in Figure 15. The overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride or vice versa is bonding at both sites, if only suprafacial components are considered. The cycloaddition reaction is thus allowed, which is consistent with the predictions based on the Woodward-Hoffmann rule.
 

<gallery heights="185" widths="160" caption=" Figure 15. Frontier orbitals of cyclohexa-1,3-diene and maleic anhydride " >
File: Planeofsymmetryii.png|Plane of symmetry
File: HOMOcyclohexadiene.png|HOMO of cyclohexa-1,3-diene: anti-symmetric
File: LUMOcyclohexadiene.png|LUMO of cyclohexa-1,3-diene: symmetric
File: HOMOanhydride.png|HOMO of maleic anhydride: symmetric
File: LUMOanhydride.png|LUMO of maleic anhydride: anti-symmetric
</gallery>

As mentioned in the introduction section, the Diels-Alder between cyclohexa-1,3-diene and maleic anhydride can give an ''endo'' or an ''exo'' adduct and thus it was necessary to construct a guess transition structure for both pathways. The guess transition structures were constructed by joining together a bicyclo[2.2.2]-2-octene molecule, adjusted as for the previous part by removing a -CH2-CH2- fragment, and a five-membered cyclic anhydride fragmen, at a distance of approximately 2 Å from each other. Optimization to a transition state using the Berny algorithm at the AM1 level of theory followed by vibrational analysis gave a structure with an imaginary vibrational frequency of magnitude 806.38 and 812.02 cm-1 for the <jmolFile text=" endo">endoTSipm12.mol</jmolFile> and <jmolFile text=" exo">DIELSEXO.LOG</jmolFile> adducts, respectively. The desired normal mode motion (Animation 5 and 6) corresponding to the Diels-Alder reaction path was obtained in both cases. The relevant optimized geometrical parameters of both transition structures are reported in Figure 16 and 17.

{|
![[Image:endoTSipm12r.gif|frame| center| Animation 4. Normal mode with the single imaginary frequency in the ''endo'' transition structure ]]
![[Image:exoTS.gif|frame| center| Animation 5. Normal mode with the single imaginary frequency in the ''exo'' transition structure ]]
! [[File:geometricalfeaturesendo.png| frame| center| 208x208 px|Figure 16. Geometrical features of the endo transition structure ]]
! [[File:geometricalfeaturesexo.png| frame| center| 150x150 px|Figure 17. Geometrical features of the exo transition structure ]]
|}

The length of the σ C-C bonds which are not involved in forming/breaking processes is comparable with the length of aromatic C-C σ bonds, indicating that the reaction proceeds via a cyclic aromatic transition state. Analogous to the Diels-Alder reaction between ethene and ''cis''-butadiene, the highly symmetric transition structures suggest that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cyclohexa-1,3-diene and maleic anhydride via both ''endo'' and ''exo'' adducts.<ref name='concerted'/>. Further evidence is provided by the normal mode motion at the imaginary frequency, which depicts a simultaneous formation of the two σ C-C bonds for both transition structures. In addition, IRC calculations at the AM1 level of theory, performed in both directions in 20 steps, are consistent with the proposed synchronous mechanism. The reaction profile does not show any discrete intermediates characteristic for stepwise processes, excluding thus the possibility of a mechanism proceeding via biradical or zwitterionic intermediates. It should be noted that the in the case of the IRC computation via the ''endo'' transition structure, the first step on the reaction path corresponds to the reactants and the last step corresponds to the cycloaddition product, whereas in the case of the ''exo'' transition structure the opposite situation was observed. Thus the IRC computations performed at the AM1 level confirm the exothermic nature of the Diels-Alder cycloaddition reaction.
[[File:IRCendo20steps.png| frame| center| 300x300 px|'''Figure 18.''' Reaction profile via the endo transition structure ]]
[[File:IRCexo20steps.png| frame| center| 300x300 px|'''Figure 19.''' Reaction profile via the exo transition structure ]]

The computed electronic and thermally corrected energies of the ''endo'' and ''exo'' transition structures are reported in Table 4 below. The calculations suggest that the activation energy of the reaction pathway which proceeds via the ''endo'' transition structure is lower than the activation energy of the process involving the ''exo'' transition structure by 0.87 kcal/mol (0.0013185 hartrees) at 0 K and 0.75 kcal/mol (0.001197 hartrees) at 298.15 K. Therefore, the AM1 predictions are in agreement with the experimental observation that the Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene gives the ''endo'' product faster than it gives the ''exo'' adduct, even though the latter is thermodynamically more stable. 
 

Table 4. Summary of computed energies (hartrees) for the endo and exo transition states

{| class="wikitable"
! style="font-weight: bold;" | Optimized molecule
! style="font-weight: bold;" | Electronic energy
! style="font-weight: bold;" | Sum of electronic and zero-point energies
! style="font-weight: bold;" | Sum of electronic and thermal energies
|-
| style="font-weight: bold;" | Cyclohexa-1,3-diene
| 0.027711
| 0.152502
| 0.157726
|-
| style="font-weight: bold;" | Maleic anhydride
| -0.121824
| -0.063346
| -0.058192
|-
| style="font-weight: bold;" | Endo TS
| -0.051505
| 0.133494
| 0.143683
|-
| style="font-weight: bold;" | Exo TS
| -0.050419
| 0.134879
| 0.144880
|}

[[File:endoorbitalinteractions.png| frame| right| 208x208 px|'''Figure 20.''' Secondary orbital interaction in the ''endo'' transition structure ]]
In order to account for the formation of the ''endo'' adduct under kinetically controlled conditions, the nodal properties of the frontier orbitals of the reactants and both transition structures were examined. The interaction between the frontier orbitals can be simplified in a diagram involving only atomic orbitals (Figure 20). It can be clearly observed that the ''endo'' transition structure is characterized by an in-phase secondary overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride. Since this stabilising secondary orbital interaction is not possible for an ''exo'' adduct, the ''endo'' transition state is expected to be lowered in energy relative to the ''exo'' one. The stabilisation of the ''endo'' transition state is further reinforced by the increased C-C through space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the carbon atoms of cyclohexa-1,3-diene which are not involved in the reaction (3.897 Å c.f. 2.945 Å in the ''exo'' adduct), which indicates diminished steric repulsion.
{|
! [[File:HOMOendoTS.png| frame| 300x300 px|Figure 21. HOMO of the endo transition structure: anti-symmetric]]
! [[File:LUMOendoTS.png| frame| 300x300px|Figure 22. LUMO of the endo transition structure: anti-symmetric ]]
! [[File:HOMOexoTS.png| frame| 300x300 px|Figure 23. HOMO of the exo transition structure: anti-symmetric ]]
! [[File:LUMOexoTS.png| frame| 300x300 px|Figure 24. LUMO of the exo transition structure: anti-symmetric ]]
|}

The computations performed for the cycloaddtion reaction between butadiene and ethene demonstrated that the AM1 method provides more accurate results than Hartree-Fock, since it implicitly takes into account electronic correlation through the use parameters derived from experimental data. However, as any other semi-empirical method, it is expected to give very reliable results only for systems which are related to those from which their parameters have been fitted. A further study might involve the use of MCSCF calculations, which have been reported to give a more a balanced description of closed-shell species. <ref name="MCSCF2"/>

==Conclusions==
==References==
<references>
<ref name="gauche"> K.S. Pitzerk, Chem. Rev., 1940, 27, 39-57 {{DOI|10.1021/cr60086a003 }} </ref>
<ref name="intro"> F. Bernardi, A. Bottoni, M.J. Field, M. F. Guest, I. H. Hillier, M. A. Robb, A. Venturini, J. Am. Chem. Soc., 1988, '''110''', 3050-3055
{{DOI|10.1021/ja00218a009 }} </ref>
<ref name="pi interaction"> B.G. Rocque, J.M. Gonzales, H.F. Schaffer, Mol. Phys., 2002, 100, 441-446 {{DOI|10.1080/00268970110081412 }} </ref>
<ref name="MCSCF" > K. Morokuma, W. T. Borden, D.A. Hvorat, J. Am. Chem. Soc., 1988, '''110''', 4474. {{DOI|10.1021/ja00221a092 }} </ref>
<ref name="MM2"> F. Jensen, ''J. Am. Chem. Soc''. 1992, '''114''', 1596-1603 {{DOI|10.1021/ja00031a009}} </ref>
<ref name="energy difference"> M. J. Goldstein, M. S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''',7147–7149 {{DOI| 10.1021/ja00775a046}}</ref>
<ref name="experimental"> M.J.S. Dewar, C. Jie, ''J. Am. Chem. Soc.'', 1987, '''109''', 5897 {{DOI|10.1021/ja00254a001}} </ref>
<ref name="chairTS"> K. N. Houk, S. M. Gustafson, K. A. Black, J. Am. Chem. Soc., 1992, 114, 8565-8572 {{DOI|10.1021/ja00048a032}} </ref>
<ref name="secondary"> D. Nisipuri, Stereochemistry of Organic Compounds: Principles and Applications , New Age Publications, 2nd edn., 1996 </ref>
<ref name="dielsalder"> M. J. S. Dewar, S. Olivella,, J. P. Stewartt, J. Am. Chem. Soc., 1986, 108, 5771-5779 {{DOI |10.1021/ja00279a018}} </ref>
<ref name="abinitio"> M. Ortega, A. Oliva, J. M. Lluch, J. Bertran, ''Chem. Phys. Letr.'', 1983, '''102''', 317 {{DOI|10.1016/0009-2614(83)87048-1}} </ref>
<ref name="rzepa">M.J.S. Dewar, S. Olivella, H. Rzepa, ''J. Am. Chem. Soc'', 1978, '''100''', 5650- 5659 {{DOI|10.1021/ja00486a013}}</ref>
<ref name="pericyclic"> I. Fleming, ''Pericyclic Reactions'', Oxford Science Publications, 2002, 40 </ref>
<ref name="concerted"> R. N. Buckle, P. Liu, E. W.D Roberts, D. J. Burnell , ''Tetrahedron'', 1999, '''55''', 11455–11464 {{DOI|10.1016/S0040-4020(99)00667-5}}</ref>
<ref name="MCSCF2"> K. N. Houk, Y. Lin, F.K. Brown, ''J. Am. Chem. Soc'', 1986, '''108''', 554 {{DOI|10.1021/ja00263a059}}
</ref>
</references>

Rep:Mod:iulianamaria

2014-12-05T02:05:25Z

Ipm12:

== Third Year Computational Lab: Physical module. Transition states and reactivity ==

== Introduction ==

The structures and energy of the species involved in a chemical reaction offer valuable insight into the mechanism and general features of the process. Due to the advances in optimization algorithms, information about the most stable geometry, vibrational frequencies and in principle any observable physical quantity can be computed with a high degree of accuracy. Synthetic sequences often involve two or more reaction pathways with similar activation energies and thus the location of the transition structures corresponding to each pathway on the potential energy surface is of great importance in predicting the major products.

The Cope rearrangement reaction of 1,5-hexadiene is an archetypal [3,3] sigmatropic shift which has been the subject of numerous theoretical and experimental studies. Computational evidence has been provided to show that the reaction can proceed via either an aromatic transition state consisting of two partially bonded three-carbon units or via a cyclohexane- 1,4-diyl-like structure as shown in Figure 1. <ref name="chairTS"/>. This study employs computations at the HF/3-21G and B3LYP/6-31G* levels of theory to investigate the lowest energy conformation of the reactant molecule and to locate the two possible, chair and boat, transition structures. The results are subsequently employed to rationalize the preferred mechanism and a comparison with previous research is also provided.

[[File:dielsalderscheme.png|right|200x200px|thumb|'''Figure 2.''' Diels-Alder cycloaddtion reaction]]
The [4+2] Diels-Alder reaction between ethylene and butadiene is another example of a process which has been extensively investigated from a theoretical and experimental point of view, but which is still not completely elucidated. Possible mechanisms that have been proposed for this reaction include a synchronous concerted approach via a cyclcic aromatic transition state, a two-stage asynchronous concerted mechanism and a stepwise process, which occurs via a stable diradical intermediate. <ref name="intro"/> This study uses AM1 semi-empirical molecular orbital method to locate and investigate the geometrical features of the transition state, followed by an analysis of the possible interaction between the frontier orbitals involved in the reaction.Computations at the AM1 level of theory are also employed to probe the Diels-Alder endo rule and study the possible pathways of the cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride. Both ''endo'' and ''exo'' adducts shown in Figure 2 are considered and their relative energies and nodal properties of the molecular oribitals are compared in order to determine the product that is preferentially formed.

==Results and Computational Methods==

 

=== Exercise 1. Optimizing the Reactants and Products of the Cope Rearrangement===

Due to the presence of three C-C bonds and three rotational minima around each of these bonds <ref name="pi interaction" />, 1,5-hexadiene is expected to have 27 conformations in total and ten of them to be energetically distinct. The conformations shown in Figure 3 are optimized at the B3LYP/6-31G* and HF/3-21G levels of theory and their electronic energies are compared, in order to determine the lowest energy conformation of 1,6-hexadiene. Frequency calculations performed on the optimized structures at the same level of theory yielded only real and positive frequencies, confirming that a minimum was obtained. The electronic energies and symmetries, along with the sum of electronic and zero-point energies and the sum of electronic and thermal energies of the optimized structures are summarized in Table 1.

 

'''Table 1. Summary of energies (hartrees) of the conformations under investigation'''

{| class="wikitable"
! rowspan="2" style="font-weight: bold;" | Conformer
! rowspan="2" style="font-weight: bold;" | Point Group
! colspan="3" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
|-
| style="font-style: italic;" | Anti1
| C2
| -231.692600
| -231.539601
| -231.532644
| -234.611800
| -234.469286
| -234.461965
|-
| style="font-style: italic;" | Anti2
| Ci
| -231.692535
| -231.539539
| -231.532565
| -234.611702
| -234.469212
| -234.461856
|-
| style="font-style: italic;" | Gauche3
| C1
| -231.692661
| -231.539486
| -231.532646
| -234.611326
| -234.468718
| -234.461477
|}

 
[[File:Conformers.jpg|200px|thumb|upright=2|'''Figure 3.''' Anti1 (top), Gauche3 (middle) and Anti2 (bottom) conformations of 1,5-hexadiene]]

On the basis of conformational analysis principles, one might expect the anti1 conformation to be the lowest energy conformation of 1,5-hexadiene. However, an optimization at the HF/3-21G level of theory of the gauche3 conformation, shown in Figure 3, yields an electronic energy of -231.69266 hartrees (c.f. -231.69260 hartrees for the anti1 conformation). This suggests that, in fact, the gauche3 conformation is the global minimum.

This observation is not in accordance with the general principle that a C-C-C-C anti-periplanar arrangement enables efficient overlap between σ C-C and σ* C-C orbitals, leading to a stabilization of the filled NBO. Each C-C-C-C gauche unit has been reported to contribute 0.8-1 kcal mol-1 to the total electronic energy of saturated aliphatic hydrocarbons such as n-butane, due to close contact between hydrogen atoms and increased bond-bond Pauli repulsion energy. <ref name="gauche" />. Nonetheless, the anomalous behaviour of 1,5-hexadiene can be attributed to a favourable attractive interaction between the π electrons of the C=C bonds and the nearby vinyl hydrogen atom, referred to as the CH-π interaction. <ref name="pi interaction" /> However, the energy difference between the anti1 and gauche3 conformations of 0.037 kcal mol -1 is a tenfold smaller difference <ref name="pi interaction" /> than in the case of the gauche and anti conformations of n-butane. Thus, since the Hartree-Fock Method describes only non-interacting electrons under the influence of a mean field potential and does not the include the electron correlation effects, the CH-π interaction may not be properly considered with the Hartree-Fock Method. The density functional theory optimization at the B3LYP/6-31G* level gives a higher energy difference of 0.297 kcal mol -1 between the gauche3 and anti1 conformations, but in this case, the gauche3 is the least stable one. The density functional theory includes the exchange-correlation energy term and thus might be expected to consider electronic effects with greater accuracy. However, the results suggest that the exchange-correlation functional used in these computations is not appropriate for the treatment of CH-π interactions. A post-Hartree-Fock Method such as the Møller–Plesset perturbation theory might have been more accurate in determining the lowest energy conformation.

The slight destabilisation of the ''anti2'' conformation compared with the ''anti1'' conformation can be attributed to the closer contact between one of the allylic and the terminal vinyl protons in the '' anti2'' structure, with a distance of 2.52251 Å ('' c.f.'' 2.52266 Å in the case of the ''anti1'' conformer). The ''gauche3'' conformation shows a distance of 2.61312 Å between these protons and thus reduced repulsive steric interactions, providing further evidence to support the observation that ''gauche3'' is the lowest energy conformation of 1,5-hexadiene.

The density functional theory optimization at the B3LYP/6-31G* level of the ''anti2'' conformation yielded a significantly lower electronic energy than in the case of the Hartree-Fock computations. Since DFT includes an approximate treatment of electron correlation, it is expected to provide more accurate results. However, in the absence of experimental data this assumption cannot be confirmed. It is worth noting that even if the overall symmetry of the optimized structure was Ci for both methods, the lengths of the C=C, C-H and C-C bonds were slightly higher when employing a DFT optimization method, as shown in Figure 4. The destabilizing interaction between occupied σ orbitals is thus diminished, resulting in a lower electronic energy. A lower electronic energy was also obtained for the DFT optimization of the other two conformations as shown in Table 1.

[[File:bondsHFandDFT.jpg|center|800px|thumb|upright=4|'''Figure 4.''' Bond distances in optimized ''anti2'' at the HF/3-21G (left) and B3LYP/6-31G* (right) levels of theory ]]

=== Exercise 2. Optimizing the "Chair" and "Boat" Transition Structures of the Cope Rearrangement===

[[Image:ChairTSipm.gif|frame| right| 234 px|'''Animation 1.''' Normal mode with the single imaginary frequency in the chair transition structure ]]
Two allylic fragments optimized separately at the Hartree-Fock level with a basis set of 3-21G were oriented in a chair-like conformation, with a distance between the allyl fragments of approximately 2.2 Å.The optimization to a Berny transition state of the chair transition state was performed initially at the HF/3-21G level of theory by computing the force constants at the beginning of the calculation. A vibrational analysis revealed convergence to a geometry with two imaginary vibrational frequencies and a normal mode motion which did not correspond to the desired reaction coordinate. Thus, the proposed structure was not close enough to the transition state geometry. After a basic manipulation of the position of the two allylic fragments in order to obtain a more symmetrical guess structure, the modified arrangement was re-optimized using the previous method. A single imaginary frequency of magnitude 817.94 cm-1 and the expected normal mode motion shown in Animation 1 were obtained, confirming the successful outcome of the computation. 

The same guess structure was optimized to a minimum at the HF/3-21G level of theory with the frozen coordinates method, by fixing the distance between opposite terminal allylic carbon atoms to 2.2 Å. A structure with decreased C-C bond lengths (1.381 Å c.f. 1.389 Å for the previous method) and an electronic energy of -231.619185 hartrees was obtained. The two bond/forming breaking distances were subsequently optimized using a Hessian modified to include information about the two coordinates along which the differentiation was performed. The computation gave a geometry almost identical to the one obtained by computing force constants at the beginning of the reaction, with only negligible variations of less than 0.00005 Å in both C-C and C-H bond lengths. The imaginary frequency of magnitude 817.88 was described by the same normal mode motion as for the previous method (Animation 1). 

The distance between opposite terminal allylic carbon atoms and the C-C bond length were approximately 2.021 Å and 1.389 Å, respectively, for both computational methods. The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

 
''' Table2.''' Summary of computed energies (hartrees) for the chair and boat transition states *0 K **298.15 K 
{| class="wikitable"
! rowspan="3" style="font-weight: bold;" | Transition state
! colspan="9" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| colspan="3" style="text-align: center; font-weight: bold;" | Force constant computation
| colspan="3" style="text-align: center; font-weight: bold;" | Redundant coordinate method
| colspan="3" style="text-align: center; font-weight: bold;" | QST2
| rowspan="2" | Electronic energy
| rowspan="2" | Sum of electronic and zero-point energies*
| rowspan="2" | Sum of electronic and thermal energies**
|-
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energy*
| Sym of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
|-
| Chair (C2h)
| -231.619322
| -231.466697
| -231.461338
| -231.619322
| -231.466705
| -231.461345
| -
| -
| -
| -234.556932
| -234.414879
| -234.408950
|-
| Boat (C2v)
| -
| -
| -
| -
| -
| -
| -231.602802
| -231.450926
| -231.445300
| -234.543093
| -234.402342
| -234.396008
|}
[[File:Boatinputfile.png|436x234 px|thumb|'''Figure 5.''' Input reactant and product geometries for the boat transition stucture computation]]
[[Image:BoatTSipm.gif|frame| right| 244 px|'''Animation 2.''' Normal mode with the single imaginary frequency in the boat transition structure ]]
The boat transition structure was located using a QST2 approach at the HF/3-21G level, by searching for a maximum along a parabola connecting the reactant and the product, and for a minimum in all directions perpendicular to this parabola. An initial calculation was performed using the ''anti2'' structure that was previously optimized to a minimum at the HF/3-21G level, as the reactant and the corresponding rearranged molecule. However, the calculation starting from these reactant and product structures yielded a chair transition structure with C2h symmetry, with a distance between the central allylic carbon atoms of 2.021 Å. The geometry was thus identical to the chair structure that was previously obtained by optimization to a transition state with the Berny algorithm. (Animation 1). This shows a limitation of the QST2 method, which is not capable of locating the desired first order saddle point if the reactant and product geometries are not similar to the geometry of the boat transition structure. Therefore, the geometry of the optimized ''anti2'' conformation was adjusted by changing the central dihedral C-C-C-C angle to 0 о and the angles formed by the central sp3 carbon atoms with each terminal allylic carbon atom to 100 о for both the reactant and the product as shown in Figure 5. Another QST2 calculation performed on the modified reactant and product structures gave a boat transition structure with an imaginary frequency of magnitude 839.87 (Animation 2). The distance between opposite terminal allylic carbon atoms was found to be approximately 2.140 Å and the C-C bond length was 1.381 Å.The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

Both optimized chair and boat transition structures at the HF/3-21G level were re-optimized using the B3LYP/6-31G* level of theory. Vibrational analysis at the same level of theory confirmed that each of the determined stationary points is a transition structure with a single negative eigenvalue at 569.16 and 530.35 cm-1 for the chair and boat structures, respectively. The optimized chair transition structure illustrated in <jmolFile text="Figure 6">ChairDFT.LOG</jmolFile> shows a distance of 1.967 Å between opposite terminal allylic carbon atoms and a C-C bond length of 1.408 Å. The optimized boat transition structure shown in <jmolFile text="Figure 7">BoatDFT.LOG</jmolFile> has interallylic distance of 2.207 Å and a C-C bond length of 1.393 Å. Thus, both optimized transition structures have similar interallylic distances and C-C bond length as the structures that were previously located using the Hartee-Fock method. As noted in the previous discussion, the DFT computations yield significantly lower energies than Hartree-Fock, as a result of the inclusion of electron correlation. The breaking/forming C-C bond length for the boat transition structure is 0.240 Å longer than for the chair transition structure, which is comparable with the value of 0.230 Å found by MCSCF calculations. <ref name="MCSCF" />. The energy difference at 298.15 K between the boat and chair TS was found to be 8.12 kcal/mol, which is slightly lower than the experimental value of 11.2 ± 2.5 kcal/mol <ref name="energy difference" />. However, these calculations at the B3LYP/6-31G* gave more accurate results than the MM2 method, which has been reported to yield an energy difference of 6.4 kcal/mol <ref name="MM2"/> 

The length of the C-C bond which are not involved in the Cope rearrangement varies between 1.381 and 1.408 Å, being comparable with a benzene C-C bond (1.40 Å). The present results suggest that the reaction proceeds via an aromatic transition state consisting
of two partially bonded three-carbon units. However, this observation is not in agreement with an AM1 study which found the biradicaloid transition structure to be lower in energy than an aromatic transition state.<ref name="experimental"/>. Thus, the preferred mechanism of the Cope rearrangement of 1,5-hexadiene is still not completely elucidated and further calculations are required to predict it.
[[Image:step44.gif|frame| right | 244 px|'''Animation 3.'''Reaction coordinate in the forward direction ]]
All calculations converged to transition structures with lengths of both C-C bonds that are forming or breaking during the rearrangement greater than a typical sp3 C-C bond, varying between 1.967 Å and 2.207 Å. Since all the pericyclic reactions that have been theoretically studied and predicted to be synchronous exhibit forming/breaking bond lengths higher than 2 Å <ref name="experimental"/>, it is reasonable to state that computations performed at HF/3-21G and B3LYP/6-31G* levels are consistent with a synchronous mechanism for the Cope rearrangement. 

IRC computations at the HF/3-21G level provide further evidence to support this affirmation. Calculations for the rearrangement via the chair transition state converged to the structure shown in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> after 44 steps and led to a reaction profile of the type indicated in Figure 9. Since the reaction coordinate is symmetrical, computations were performed only in the forward direction. It can be observed that there are no discrete intermediates characteristic for stepwise processes and thus the reaction is concerted. The 1,5-hexadiene conformation to which the calculations converged with an electronic energy of -231.691579 hartrees was further optimized at the HF/3-21G level of theory. The C2 symmetric gauche2 structure shown in <jmolFile text="Figure 10">OptimizedIRC.mol</jmolFile> was located as a minimum on the potential energy surface, with an electronic energy of -231.691667 hartrees. A further attempt was made to perform the IRC calculations in 100 steps, but the same configuration as the one illustrated in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> was obtained after 44 steps. Even if it would have been useful to verify that the conformation optimized at the HF/3-21G level is the correct minimum, the number of steps was not further increased due to time constraints.
[[File:IRC 50 steps.png|center|536x334 px|thumb|'''Figure 9.''' Reaction profile in the forward direction]] 
[[File:OrbitalsTS.png|right|294x294 px|thumb|'''Figure 10.''' Frontier orbitals in the Cope rearrangement ]]

Activation energies were calculated relative to the Ci symmetric configuration of 1,5-hexadiene (anti2) for both chair and boat reaction pathways at 0 K and 298.15 K. As shown in Table 3, inclusion of electron correlation terms enhances the accuracy of the results, the computations performed at the B3LYP/6-31G* level being in reasonable agreement with the experimental values <ref name="experimental"/>. The significantly lower activation energies for the chair pathway supports the experimental evidence that the transition structure adopts a chair-like conformation with C2h symmetry. <ref name="chairTS"/>. A possible explanation for the preference of the chair conformation is the presence of an antibonding secondary interaction in the boat form. <ref name="secondary"/> as shown in Figure 10. By dividing the 1,5-hexadiene into an ethylene moiety and a butadiene equivalent and considering the interaction between their HOMO and LUMO, it can be noticed that in the chair transition structure, the orbitals are too far apart to exhibit significant out-of-phase overlap.

''' Table 3. Summary of activation energies (kcal/mol) '''

{| class="wikitable"
| rowspan="2" width="125" align="center" | '''Activation energy'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Experimental'''
|-
| width="125" align="center" | '''0 K '''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
|-
| '''ΔE (chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.09
| align="center" | 33.20
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (boat)'''
| align="center" | 55.61
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

=== Exercise 3. The Diels-Alder Cycloaddition===
 
====Cycloaddition reaction between ethene and butadiene====
 

The AM1 semi-empirical molecular orbital method was employed to optimize to a minimum a butadiene and an ethylene fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 11. 

<gallery heights="185" widths="160" caption=" Figure 11. Frontier orbitals of ethene and butadiene" >
File: Planeofsymmetry.png|Plane of symmetry
File: HOMOethene.png|HOMO of ethene: symmetric
File: LUMOethene.png|LUMO of ethene: anti-symmetric
File: HOMObutadiene.png|HOMO of butadiene: anti-symmetric
File: LUMObutadiene.png|LUMO of butadiene: symmetric
</gallery>

The envelope-like guess transition structure for the Diels-Alder reaction was constructed starting from a bicyclo[2.2.2]-2-octene molecule which was adjusted by removing a -CH2-CH2- fragment. Optimization to a transition state using the Berny algorithm at the AM1 level of theory gave a structure with an imaginary frequency of magnitude 623 cm-1 and a normal mode which did not correspond to the desired reaction path. The initial bicyclo[2.2.2]-2-octene molecule was symmetrized and another optimization using the same method was performed. A critical point of Cs symmetry was located on the potential energy surface and vibrational analysis revealed a single imaginary vibrational frequency of magnitude 956.25 cm-1 with the desired normal mode motion (Animation 4). The relevant optimized geometrical parameters of the envelope type transition structure are reported in Figure 12. 

[[File:Dielsaldertransitionstategeometry2.png|300px|thumb|center|upright=2| '''Figure 12.''' Geometrical features of the optimized transition state for the Diels-Alder reaction (Å)]]

[[Image:Transitionstateanimation.gif|frame| right| 234 px|'''Animation 4.''' Normal mode with the single imaginary frequency in the transition structure ]]

The length of the σ C-C bonds which are not involved in forming/breaking processes indicates that the reaction proceeds via a cyclic aromatic transition state. The length of the σ C-C bonds that are expected to form (2.119 Å) is greater than a typical sp3 (~ 1.54 Å) and sp2 (~ 1.34 Å) bond length, but significantly lower than the sum of the corresponding van der Waals radii (~ 3.4 Å). Thus, there is a bonding interaction between the butadiene and ethylene fragments and the new σ C-C bonds are partly formed as expected for a transition structure. It has been generally reported that allowed pericyclic reactions are invariably synchronous if there are no factors which might strongly militate against synchronicity.<ref name="dielsalder" /> Since concerted approaches involve cyclic aromatic transition states, the transition structures are expected to be highly symmetric based on the analogy to the structure of benzene (equal C-C bond lengths of approximately 1.40 Å). <ref name="dielsalder" />. Therefore, the equal length of both C-C bonds formed in the cycloaddition reaction suggests that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cis-butadiene and ethylene. Further evidence is provided by the normal mode motion at -956.25 cm-1 corresponding to the reaction path, which depicts a simultaneous formation of the two σ C-C bonds. The vibration with the lowest positive frequency occurs at 147.30 cm-1 describes a nonsynchronous approach, but the motion is simply a twisting of both fragments rather than a bond forming process equivalent to the reaction path. Even if AM1 agrees with the ab initio calculation <ref name="abinitio"/> in predicting the reaction to be synchronous, MIND0/3 calculations are in favour of the process involving a stable biradicaloid intermediate, which collapses to cyclohexene in the rate-determining step. <ref name="rzepa"/>. The nature of the mechanism is thus still open to debate. 

The AM1 computations can be further employed to confirm theoretical observations such as the fact that the Diels-Alder reaction is an allowed process. The Woodward-Hoffmann rule for thermal pericyclic reactions states that a ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a <ref name="pericyclic"/> components is odd. "S" and "a" refer to suprafacial and antarafacial components, respectively and p and r are integers. Since the investigated Diels-Alder reaction involves only suprafacial bond formation between a diene with 4 π electrons (r=1) and a dienophile with 2 π electrons (q=0), the process can be described as having only one (4q+2)s component and no (4r)a components. The total number of components is thus 1 and the reaction between ethene and cis-butadiene is expected to be allowed.
This is confirmed by the nodal properties of the HOMO and LUMO that were previously computed for ethene and butadiene. It is clear from Figure 13 that the overlap between HOMO of one reactant and the LUMO of the other reactant is bonding at both sites, if only suprafacial components are considered. Overlap between the HOMO of butadiene and the LUMO of ethylene gives the HOMO of the Cs transition structure, whereas interaction between the LUMO of butadiene and the HOMO of ethylene gives the LUMO of the transition structure. As shown in Figure 13, the HOMO and LUMO corresponding to the transition structure are anti-symmetric and symmetric, respectively, with respect to the plane of symmetry that was previously defined.
 
<gallery heights="195" widths="170" caption=" Figure 13. Frontier orbitals of the TS and bonding interactions" >
File: bondingHOMOetheneLUMObutadiene.png| Bonding interaction between HOMO of ethene and LUMO of butadiene
File: bondingLUMOetheneLUMObutadiene.png|Bonding interaction between LUMO of ethene and HOMO of butadiene
File: HOMOtransitionstate.png|HOMO of the transition structure: anti-symmetric
File: LUMOtransitionstate.png|LUMO of the transition structure: symmetric
</gallery>
[[File:HOMOhartreefock.png|436x234 px|thumb|'''Figure 14.'''HOMO of the TS obtained at the HF/3-21G level]]
 Note: The representation of the bonding interactions is only schematic and the relative size of the butadiene and ethene frontier orbitals is not accurate. The lines identify the bonding overlap that can develop as the reaction proceeds.

An attempt was made to perform all the above computations at the HF/3-21G level of theory. However, a transition state with the symmetric HOMO shown in Figure 14 was located, suggesting that the level of theory is not appropriate for the characterisation of the frontier orbitals involved in the Diels-Alder reaction. Based on the remarks that were previously made for the Cope rearrangement, calculations at the B3LYP/6-31G* level might have given more accurate results. Even if the shape and the coefficients of the molecular orbitals should not have been affected, the DFT method might have given electronic and thermally corrected energies which are in better agreement with the experimental values.
 
 

==== Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride ====
 
The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a cyclohexa-1,3-diene and a maleic anhydride fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane shown are illustrated in Figure 15. The overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride or vice versa is bonding at both sites, if only suprafacial components are considered. The cycloaddition reaction is thus allowed, which is consistent with the predictions based on the Woodward-Hoffmann rule.
 

<gallery heights="185" widths="160" caption=" Figure 15. Frontier orbitals of cyclohexa-1,3-diene and maleic anhydride " >
File: Planeofsymmetryii.png|Plane of symmetry
File: HOMOcyclohexadiene.png|HOMO of cyclohexa-1,3-diene: anti-symmetric
File: LUMOcyclohexadiene.png|LUMO of cyclohexa-1,3-diene: symmetric
File: HOMOanhydride.png|HOMO of maleic anhydride: symmetric
File: LUMOanhydride.png|LUMO of maleic anhydride: anti-symmetric
</gallery>

As mentioned in the introduction section, the Diels-Alder between cyclohexa-1,3-diene and maleic anhydride can give an ''endo'' or an ''exo'' adduct and thus it was necessary to construct a guess transition structure for both pathways. The guess transition structures were constructed by joining together a bicyclo[2.2.2]-2-octene molecule, adjusted as for the previous part by removing a -CH2-CH2- fragment, and a five-membered cyclic anhydride fragmen, at a distance of approximately 2 Å from each other. Optimization to a transition state using the Berny algorithm at the AM1 level of theory followed by vibrational analysis gave a structure with an imaginary vibrational frequency of magnitude 806.38 and 812.02 cm-1 for the <jmolFile text=" endo">endoTSipm12.mol</jmolFile> and <jmolFile text=" exo">DIELSEXO.LOG</jmolFile> adducts, respectively. The desired normal mode motion (Animation 5 and 6) corresponding to the Diels-Alder reaction path was obtained in both cases. The relevant optimized geometrical parameters of both transition structures are reported in Figure 16 and 17.

{|
![[Image:endoTSipm12r.gif|frame| center| Animation 4. Normal mode with the single imaginary frequency in the ''endo'' transition structure ]]
![[Image:exoTS.gif|frame| center| Animation 5. Normal mode with the single imaginary frequency in the ''exo'' transition structure ]]
! [[File:geometricalfeaturesendo.png| frame| center| 208x208 px|Figure 16. Geometrical features of the endo transition structure ]]
! [[File:geometricalfeaturesexo.png| frame| center| 150x150 px|Figure 17. Geometrical features of the exo transition structure ]]
|}

The length of the σ C-C bonds which are not involved in forming/breaking processes is comparable with the length of aromatic C-C σ bonds, indicating that the reaction proceeds via a cyclic aromatic transition state. Analogous to the Diels-Alder reaction between ethene and ''cis''-butadiene, the highly symmetric transition structures suggest that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cyclohexa-1,3-diene and maleic anhydride via both ''endo'' and ''exo'' adducts.<ref name='concerted'/>. Further evidence is provided by the normal mode motion at the imaginary frequency, which depicts a simultaneous formation of the two σ C-C bonds for both transition structures. In addition, IRC calculations at the AM1 level of theory, performed in both directions in 20 steps, are consistent with the proposed synchronous mechanism. The reaction profile does not show any discrete intermediates characteristic for stepwise processes, excluding thus the possibility of a mechanism proceeding via biradical or zwitterionic intermediates. It should be noted that the in the case of the IRC computation via the ''endo'' transition structure, the first step on the reaction path corresponds to the reactants and the last step corresponds to the cycloaddition product, whereas in the case of the ''exo'' transition structure the opposite situation was observed. Thus the IRC computations performed at the AM1 level confirm the exothermic nature of the Diels-Alder cycloaddition reaction.
[[File:IRCendo20steps.png| frame| center| 300x300 px|'''Figure 18.''' Reaction profile via the endo transition structure ]]
[[File:IRCexo20steps.png| frame| center| 300x300 px|'''Figure 19.''' Reaction profile via the exo transition structure ]]

The computed electronic and thermally corrected energies of the ''endo'' and ''exo'' transition structures are reported in Table 4 below. The calculations suggest that the activation energy of the reaction pathway which proceeds via the ''endo'' transition structure is lower than the activation energy of the process involving the ''exo'' transition structure by 0.87 kcal/mol (0.0013185 hartrees) at 0 K and 0.75 kcal/mol (0.001197 hartrees) at 298.15 K. Therefore, the AM1 predictions are in agreement with the experimental observation that the Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene gives the ''endo'' product faster than it gives the ''exo'' adduct, even though the latter is thermodynamically more stable. 
 

Table 4. Summary of computed energies (hartrees) for the endo and exo transition states

{| class="wikitable"
! style="font-weight: bold;" | Optimized molecule
! style="font-weight: bold;" | Electronic energy
! style="font-weight: bold;" | Sum of electronic and zero-point energies
! style="font-weight: bold;" | Sum of electronic and thermal energies
|-
| style="font-weight: bold;" | Cyclohexa-1,3-diene
| 0.027711
| 0.152502
| 0.157726
|-
| style="font-weight: bold;" | Maleic anhydride
| -0.121824
| -0.063346
| -0.058192
|-
| style="font-weight: bold;" | Endo TS
| -0.051505
| 0.133494
| 0.143683
|-
| style="font-weight: bold;" | Exo TS
| -0.050419
| 0.134879
| 0.144880
|}

[[File:endoorbitalinteractions.png| frame| right| 208x208 px|'''Figure 20.''' Secondary orbital interaction in the ''endo'' transition structure ]]
In order to account for the formation of the ''endo'' adduct under kinetically controlled conditions, the nodal properties of the frontier orbitals of the reactants and both transition structures were examined. The interaction between the frontier orbitals can be simplified in a diagram involving only atomic orbitals (Figure 20). It can be clearly observed that the ''endo'' transition structure is characterized by an in-phase secondary overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride. Since this stabilising secondary orbital interaction is not possible for an ''exo'' adduct, the ''endo'' transition state is expected to be lowered in energy relative to the ''exo'' one. The stabilisation of the ''endo'' transition state is further reinforced by the increased C-C through space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the carbon atoms of cyclohexa-1,3-diene which are not involved in the reaction (3.897 Å c.f. 2.945 Å in the ''exo'' adduct), which indicates diminished steric repulsion.
{|
! [[File:HOMOendoTS.png| frame| 300x300 px|Figure 21. HOMO of the endo transition structure: anti-symmetric]]
! [[File:LUMOendoTS.png| frame| 300x300px|Figure 22. LUMO of the endo transition structure: anti-symmetric ]]
! [[File:HOMOexoTS.png| frame| 300x300 px|Figure 23. HOMO of the exo transition structure: anti-symmetric ]]
! [[File:LUMOexoTS.png| frame| 300x300 px|Figure 24. LUMO of the exo transition structure: anti-symmetric ]]
|}

The computations performed for the cycloaddtion reaction between butadiene and ethene demonstrated that the AM1 method provides more accurate results than Hartree-Fock, since it implicitly takes into account electronic correlation through the use parameters derived from experimental data. However, as any other semi-empirical method, it is expected to give very reliable results only for systems which are related to those from which their parameters have been fitted. A further study might involve the use of MCSCF calculations, which have been reported to give a more a balanced description of closed-shell species. <ref name="MCSCF2"/>

==Conclusions==
==References==
<references>
<ref name="gauche"> K.S. Pitzerk, Chem. Rev., 1940, 27, 39-57 {{DOI|10.1021/cr60086a003 }} </ref>
<ref name="intro"> F. Bernardi, A. Bottoni, M.J. Field, M. F. Guest, I. H. Hillier, M. A. Robb, A. Venturini, J. Am. Chem. Soc., 1988, '''110''', 3050-3055
{{DOI|10.1021/ja00218a009 }} </ref>
<ref name="pi interaction"> B.G. Rocque, J.M. Gonzales, H.F. Schaffer, Mol. Phys., 2002, 100, 441-446 {{DOI|10.1080/00268970110081412 }} </ref>
<ref name="MCSCF" > K. Morokuma, W. T. Borden, D.A. Hvorat, J. Am. Chem. Soc., 1988, '''110''', 4474. {{DOI|10.1021/ja00221a092 }} </ref>
<ref name="MM2"> F. Jensen, ''J. Am. Chem. Soc''. 1992, '''114''', 1596-1603 {{DOI|10.1021/ja00031a009}} </ref>
<ref name="energy difference"> M. J. Goldstein, M. S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''',7147–7149 {{DOI| 10.1021/ja00775a046}}</ref>
<ref name="experimental"> M.J.S. Dewar, C. Jie, ''J. Am. Chem. Soc.'', 1987, '''109''', 5897 {{DOI|10.1021/ja00254a001}} </ref>
<ref name="chairTS"> K. N. Houk, S. M. Gustafson, K. A. Black, J. Am. Chem. Soc., 1992, 114, 8565-8572 {{DOI|10.1021/ja00048a032}} </ref>
<ref name="secondary"> D. Nisipuri, Stereochemistry of Organic Compounds: Principles and Applications , New Age Publications, 2nd edn., 1996 </ref>
<ref name="dielsalder"> M. J. S. Dewar, S. Olivella,, J. P. Stewartt, J. Am. Chem. Soc., 1986, 108, 5771-5779 {{DOI |10.1021/ja00279a018}} </ref>
<ref name="abinitio"> M. Ortega, A. Oliva, J. M. Lluch, J. Bertran, ''Chem. Phys. Letr.'', 1983, '''102''', 317 {{DOI|10.1016/0009-2614(83)87048-1}} </ref>
<ref name="rzepa">M.J.S. Dewar, S. Olivella, H. Rzepa, ''J. Am. Chem. Soc'', 1978, '''100''', 5650- 5659 {{DOI|10.1021/ja00486a013}}</ref>
<ref name="pericyclic"> I. Fleming, ''Pericyclic Reactions'', Oxford Science Publications, 2002, 40 </ref>
<ref name="concerted"> R. N. Buckle, P. Liu, E. W.D Roberts, D. J. Burnell , ''Tetrahedron'', 1999, '''55''', 11455–11464 {{DOI|10.1016/S0040-4020(99)00667-5}}</ref>
<ref name="MCSCF2"> K. N. Houk, Y. Lin, F.K. Brown, ''J. Am. Chem. Soc'', 1986, '''108''', 554 {{DOI|10.1021/ja00263a059}}
</ref>
</references>

2014-12-04T23:52:19Z

Ipm12: /* Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride */

== Third Year Computational Lab: Physical module. Transition states and reactivity ==

== Introduction ==
* * *- discutie despre computational methods: HF, QST2, DFT
- transition state, imaginary frequency
- utilizare pt conformational analysis si locating transition structures
- cope rearrangement mechanism, current research, discuss possible routes
- ....****
Figure1,2

==Results and Computational Methods==

 

=== Exercise 1. Optimizing the Reactants and Products of the Cope Rearrangement===

Due to the presence of three C-C bonds and three rotational minima around each of these bonds <ref name="pi interaction" />, 1,5-hexadiene is expected to have 27 conformations in total and ten of them to be energetically distinct. The conformations shown in Figure 3 are optimized at the B3LYP/6-31G* and HF/3-21G levels of theory and their electronic energies are compared, in order to determine the lowest energy conformation of 1,6-hexadiene. Frequency calculations performed on the optimized structures at the same level of theory yielded only real and positive frequencies, confirming that a minimum was obtained. The electronic energies and symmetries, along with the sum of electronic and zero-point energies and the sum of electronic and thermal energies of the optimized structures are summarized in Table 1.

 

'''Table 1. Summary of energies (hartrees) of the conformations under investigation'''

{| class="wikitable"
! rowspan="2" style="font-weight: bold;" | Conformer
! rowspan="2" style="font-weight: bold;" | Point Group
! colspan="3" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
|-
| style="font-style: italic;" | Anti1
| C2
| -231.692600
| -231.539601
| -231.532644
| -234.611800
| -234.469286
| -234.461965
|-
| style="font-style: italic;" | Anti2
| Ci
| -231.692535
| -231.539539
| -231.532565
| -234.611702
| -234.469212
| -234.461856
|-
| style="font-style: italic;" | Gauche3
| C1
| -231.692661
| -231.539486
| -231.532646
| -234.611326
| -234.468718
| -234.461477
|}

 
[[File:Conformers.jpg|200px|thumb|upright=2|'''Figure 3.''' Anti1 (top), Gauche3 (middle) and Anti2 (bottom) conformations of 1,5-hexadiene]]

On the basis of conformational analysis principles, one might expect the anti1 conformation to be the lowest energy conformation of 1,5-hexadiene. However, an optimization at the HF/3-21G level of theory of the gauche3 conformation, shown in Figure 3, yields an electronic energy of -231.69266 hartrees (c.f. -231.69260 hartrees for the anti1 conformation). This suggests that, in fact, the gauche3 conformation is the global minimum.

This observation is not in accordance with the general principle that a C-C-C-C anti-periplanar arrangement enables efficient overlap between σ C-C and σ* C-C orbitals, leading to a stabilization of the filled NBO. Each C-C-C-C gauche unit has been reported to contribute 0.8-1 kcal mol-1 to the total electronic energy of saturated aliphatic hydrocarbons such as n-butane, due to close contact between hydrogen atoms and increased bond-bond Pauli repulsion energy. <ref name="gauche" />. Nonetheless, the anomalous behaviour of 1,5-hexadiene can be attributed to a favourable attractive interaction between the π electrons of the C=C bonds and the nearby vinyl hydrogen atom, referred to as the CH-π interaction. <ref name="pi interaction" /> However, the energy difference between the anti1 and gauche3 conformations of 0.037 kcal mol -1 is a tenfold smaller difference <ref name="pi interaction" /> than in the case of the gauche and anti conformations of n-butane. Thus, since the Hartree-Fock Method describes only non-interacting electrons under the influence of a mean field potential and does not the include the electron correlation effects, the CH-π interaction may not be properly considered with the Hartree-Fock Method. The density functional theory optimization at the B3LYP/6-31G* level gives a higher energy difference of 0.297 kcal mol -1 between the gauche3 and anti1 conformations, but in this case, the gauche3 is the least stable one. The density functional theory includes the exchange-correlation energy term and thus might be expected to consider electronic effects with greater accuracy. However, the results suggest that the exchange-correlation functional used in these computations is not appropriate for the treatment of CH-π interactions. A post-Hartree-Fock Method such as the Møller–Plesset perturbation theory might have been more accurate in determining the lowest energy conformation.

The slight destabilisation of the ''anti2'' conformation compared with the ''anti1'' conformation can be attributed to the closer contact between one of the allylic and the terminal vinyl protons in the '' anti2'' structure, with a distance of 2.52251 Å ('' c.f.'' 2.52266 Å in the case of the ''anti1'' conformer). The ''gauche3'' conformation shows a distance of 2.61312 between these protons and thus reduced repulsive steric interactions, providing further evidence to support the observation that ''gauche3'' is the lowest energy conformation of 1,5-hexadiene.

The density functional theory optimization at the B3LYP/6-31G* level of the ''anti2'' conformation yielded a significantly lower electronic energy than in the case of the Hartree-Fock computations. Since DFT includes an approximate treatment of electron correlation, it is expected to provide more accurate results. However, in the absence of experimental data this assumption cannot be confirmed. It is worth noting that even if the overall symmetry of the optimized structure was Ci for both methods, the lengths of the C=C, C-H and C-C bonds were slightly higher when employing a DFT optimization method, as shown in Figure 4. The destabilizing interaction between occupied σ orbitals is thus diminished, resulting in a lower electronic energy. A lower electronic energy was also obtained for the DFT optimization of the other two conformations as shown in Table 1.

[[File:bondsHFandDFT.jpg|center|800px|thumb|upright=4|'''Figure 4.''' Bond distances in optimized ''anti2'' at the HF/3-21G (left) and B3LYP/6-31G* (right) levels of theory ]]

=== Exercise 2. Optimizing the "Chair" and "Boat" Transition Structures of the Cope Rearrangement===

[[Image:ChairTSipm.gif|frame| right| 234 px|'''Animation 1.''' Normal mode with the single imaginary frequency in the chair transition structure ]]
Two allylic fragments optimized separately at the Hartree-Fock level with a basis set of 3-21G were oriented in a chair-like conformation, with a distance between the terminal allyl fragments which are involved in bond forming or breaking processes of approximately 2.2 Å.The optimization to a Berny transition state of the chair transition state was performed initially at the HF/3-21G level of theory by computing the force constants at the beginning of the calculation. A vibrational analysis revealed convergence to a geometry with two imaginary vibrational frequencies and a normal mode motion which did not correspond to the desired reaction coordinate. Thus, the proposed structure was not close enough to the transition state geometry. After a basic manipulation of the position of the two allylic fragments in order to obtain a more symmetrical guess structure, the modified arrangement was re-optimized using the previous method. A single imaginary frequency of magnitude 817.94 cm-1 and the expected normal mode motion shown in Animation 1 were obtained, confirming the successful outcome of the computation. 

The same guess structure was optimized to a minimum at the HF/3-21G level of theory with the frozen coordinates method, by fixing the distance between opposite terminal allylic carbon atoms to 2.2 Å. A structure with decreased C-C bond lengths (1.381 Å c.f. 1.389 Å for the previous method) and an electronic energy of -231.619185 hartrees was obtained. The two bond/forming breaking distances were subsequently optimized using a Hessian modified to include information about the two coordinates along which the differentiation was performed. The computation gave a geometry almost identical to the one obtained by computing force constants at the beginning of the reaction, with only negligible variations of less than 0.00005 Å in both C-C and C-H bond lengths. The imaginary frequency of magnitude 817.88 was described by the same normal mode motion as for the previous method (Animation 1). 

The distance between opposite terminal allylic carbon atoms and the C-C bond length were approximately 2.021 Å and 1.389 Å, respectively, for both computational methods. The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

 
''' Table2. Summary of computed energies (hartrees) for the chair and boat transition states''' *0 K **298.15 K 
{| class="wikitable"
! rowspan="3" style="font-weight: bold;" | Transition state
! colspan="9" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| colspan="3" style="text-align: center; font-weight: bold;" | Force constant computation
| colspan="3" style="text-align: center; font-weight: bold;" | Redundant coordinate method
| colspan="3" style="text-align: center; font-weight: bold;" | QST2
| rowspan="2" | Electronic energy
| rowspan="2" | Sum of electronic and zero-point energies*
| rowspan="2" | Sum of electronic and thermal energies**
|-
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energy*
| Sym of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
|-
| Chair (C2h)
| -231.619322
| -231.466697
| -231.461338
| -231.619322
| -231.466705
| -231.461345
| -
| -
| -
| -234.556932
| -234.414879
| -234.408950
|-
| Boat (C2v)
| -
| -
| -
| -
| -
| -
| -231.602802
| -231.450926
| -231.445300
| -234.543093
| -234.402342
| -234.396008
|}
[[File:Boatinputfile.png|436x234 px|thumb|'''Figure 5.''' Input reactant and product geometries for the boat transition stucture computation]]
[[Image:BoatTSipm.gif|frame| right| 244 px|'''Animation 2.''' Normal mode with the single imaginary frequency in the boat transition structure ]]
The boat transition structure was located using a QST2 approach at the HF/3-21G level, by searching for a maximum along a parabola connecting the reactant and the product, and for a minimum in all directions perpendicular to this parabola. An initial calculation was performed using the ''anti2'' structure that was previously optimized to a minimum at the HF/3-21G level, as the reactant and the corresponding rearranged molecule. However, the calculation starting from these reactant and product structures yielded a chair transition structure with C2h symmetry, with a distance between the central allylic carbon atoms of 2.021 Å. The geometry was thus identical to the structure that was previously obtained by optimization to a transition state with the Berny algorithm. (Animation 1). This illustrates a limitation of the QST2 method, which is not capable of locating the desired first order saddle point if the reactant and product geometries are not similar to the geometry of the boat transition structure. Therefore, the geometry of the optimized ''anti2'' conformation was adjusted by changing the central dihedral C-C-C-C angle to 0 о and the angles formed by the central sp3 carbon atoms with each terminal allylic carbon atom to 100 о for both the reactant and the product as shown in Figure 5. Another QST2 calculation performed on the modified reactant and product structures gave a boat transition structure with an imaginary frequency of magnitude 839.87, the normal mode being illustrated in Animation 2. The distance between opposite terminal allylic carbon atoms was found to be approximately 2.140 Å and the C-C bond length was 1.381 Å.The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

Both optimized chair and boat transition structures at the HF/3-21G level were re-optimized using the B3LYP/6-31G* level of theory. Vibrational analysis at the same level of theory confirmed that each of the determined stationary points is a transition structure with a single negative eigenvalue at 569.16 and 530.35 cm-1 for the chair and boat structures, respectively. The optimized chair transition structure illustrated in <jmolFile text="Figure 6">ChairDFT.LOG</jmolFile> shows a distance of 1.967 Å between opposite terminal allylic carbon atoms and a C-C bond length of 1.408 Å. The optimized boat transition structure shown in <jmolFile text="Figure 7">BoatDFT.LOG</jmolFile> has interallylic distance of 2.207 Å and a C-C bond length of 1.393 Å. Thus, both optimized transition structures have similar interallylic distances and C-C bond length as the structures that were previously located using the Hartee-Fock method. As noted in the previous discussion, the DFT computations yield significantly lower energies than Hartree-Fock, as a result of the inclusion of electron correlation. The breaking/forming C-C bond length for the boat transition structure is 0.240 Å longer than for the chair transition structure, which is comparable with the value of 0.230 Å found by MCSCF calculations. <ref name="MCSCF" />. The energy difference at 298.15 K between the boat and chair TS was found to be 8.12 kcal/mol, which is slightly lower than the experimental value of 11.2 ± 2.5 kcal/mol <ref name="energy difference" />. However, these calculations at the B3LYP/6-31G* gave more accurate results than the MM2 method, which has been reported to yield an energy difference of 6.4 kcal/mol <ref name="MM2"/> 

The length of the C-C bond which are not involved in the Cope rearrangement varies between 1.381 and 1.408 Å, being comparable with a benzene C-C bond (1.40 Å). The present results suggest that the reaction proceeds via an aromatic transition state consisting
of two partially bonded three-carbon units. However, this observation is not in agreement with a AM1 study which found the biradicaloid transition structure to be lower in energy than an aromatic transition state.<ref name="experimental"/>. Thus, the preferred mechanism of the Cope rearrangement of 1,5-hexadiene is still not completely elucidated and further calculations are required to predict it.
[[Image:step44.gif|frame| right | 244 px|'''Animation 3.'''Reaction coordinate in the forward direction ]]
All calculations converged to transition structures with lengths of both C-C bonds that are forming or breaking during the rearrangement greater than a typical sp3 C-C bond, varying between 1.967 Å and 2.207 Å. Since all the pericyclic reactions that have been theoretically studied and predicted to be synchronous exhibit forming/breaking bond lengths higher than 2 Å <ref name="experimental"/>, it is reasonable to state that computations performed at HF/3-21G and B3LYP/6-31G* levels are consistent with a synchronous mechanism for the Cope rearrangement. 

IRC computations at the HF/3-21G level provide further evidence to support this affirmation. Calculations for the rearrangement via the chair transition state converged to the structure shown in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> after 44 steps and led to a reaction profile of the type indicated in Figure 9. Since the reaction coordinate is symmetrical, computations were performed only in the forward direction. It can be observed that there are no discrete intermediates characteristic for stepwise processes and thus the reaction is concerted. The 1,5-hexadiene conformation to which the calculations converged with an electronic energy of -231.691579 hartrees was further optimized at the HF/3-21G level of theory. The C2 symmetric gauche2 structure shown in <jmolFile text="Figure 10">OptimizedIRC.mol</jmolFile> was located as a minimum on the potential energy surface, with an electronic energy of -231.691667 hartrees. A further attempt was made to perform the IRC calculations in 100 steps, but the same configuration as the one illustrated in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> was obtained after 44 steps. Even if it would have been a good method of verifying that the conformation optimized at the HF/3-21G level is the correct minimum, the number of steps was not further increased due to time constraints.
[[File:IRC 50 steps.png|center|536x334 px|thumb|'''Figure 9.''' Reaction profile in the forward direction]] 
[[File:OrbitalsTS.png|right|294x294 px|thumb|'''Figure 10.''' Frontier orbitals in the Cope rearrangement ]]

Activation energies were calculated relative to the Ci symmetric configuration of 1,5-hexadiene (anti2) for both chair and boat reaction pathways at 0 K and 298.15 K. As shown in Table 3, inclusion of electron correlation terms enhances the accuracy of the results, the computations performed at the B3LYP/6-31G* level being in reasonable agreement with the experimental values <ref name="experimental"/>. The significantly lower activation energies for the chair pathway supports the experimental evidence that the transition structure adopts a chair-like conformation with C2h symmetry. <ref name="chairTS"/>. A possible explanation for the preference of the chair conformation is the presence of an antibonding secondary interaction in the boat form. <ref name="secondary"/> as shown in Figure 10. By dividing the 1,5-hexadiene into an ethylene moiety and a butadiene equivalent and considering the interaction between their HOMO and LUMO, it can be noticed that in the chair transition structure, the orbitals are too far apart to exhibit efficient out-of-phase overlap.

''' Table 3. Summary of activation energies (kcal/mol) '''

{| class="wikitable"
| rowspan="2" width="125" align="center" | '''Activation energy'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Experimental'''
|-
| width="125" align="center" | '''0 K '''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
|-
| '''ΔE (chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.09
| align="center" | 33.20
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (boat)'''
| align="center" | 55.61
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

=== Exercise 3. The Diels-Alder Cycloaddition===
 
====Cycloaddition reaction between ethene and butadiene====
 

The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a butadiene and an ethylene fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 11. 

<gallery heights="185" widths="160" caption=" Figure 11. Frontier orbitals of ethene and butadiene" >
File: Planeofsymmetry.png|Plane of symmetry
File: HOMOethene.png|HOMO of ethene: symmetric
File: LUMOethene.png|LUMO of ethene: anti-symmetric
File: HOMObutadiene.png|HOMO of butadiene: anti-symmetric
File: LUMObutadiene.png|LUMO of butadiene: symmetric
</gallery>

The envelope-like guess transition structure for the Diels-Alder reaction was constructed starting from a bicyclo[2.2.2]-2-octene molecule which was adjusted by removing a -CH2-CH2- fragment. Optimization to a transition state using the Berny algorithm at the AM1 level of theory gave a structure with an imaginary frequency of magnitude 623 cm-1 and a normal mode which did not correspond to the desired reaction path. The initial bicyclo[2.2.2]-2-octene molecule was symmetrized and another optimization using the same method was performed. A critical point of Cs symmetry was located on the potential energy surface and vibrational analysis revealed a single imaginary vibrational frequency of magnitude 956.25 cm-1 with the desired normal mode motion (Animation 4) corresponding to the Diels-Alder cycloaddition. The relevant optimized geometrical parameters of the envelope type transition structure are reported in Figure 12. 

[[File:Dielsaldertransitionstategeometry2.png|300px|thumb|center|upright=2| '''Figure 12.''' Geometrical features of the optimized transition state for the Diels-Alder reaction (Å)]]

[[Image:Transitionstateanimation.gif|frame| right| 234 px|'''Animation 4.''' Normal mode with the single imaginary frequency in the transition structure ]]

The length of the σ C-C bonds which are not involved in forming/breaking processes indicates that the reaction proceeds via a cyclic aromatic transition state. The length of the σ C-C bonds that are expected to form (2.119 Å) is greater than a typical sp3 (~ 1.54 Å) and sp2 (~ 1.34 Å) bond length, but significantly lower than the sum of the corresponding van der Waals radii (~ 3.4 Å). Thus, there is a bonding interaction between the butadiene and ethylene fragments and the new σ C-C bonds are partly formed as expected for a transition structure. It has been generally reported that allowed pericyclic reactions are invariably synchronous in the absence of factors which might strongly militate against synchronicity. Since concerted approaches involve cyclic aromatic transition states, the transition structures are expected to be highly symmetric based on the analogy to the structure of benzene (equal C-C bond lengths of approximately 1.40 Å). <ref name="dielsalder" />. Therefore, the equal length of both C-C bonds formed in the cycloaddition reaction suggests that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cis-butadiene and ethylene. Further evidence is provided by the normal mode motion at -956.25 cm-1 corresponding to the reaction path, which depicts a simultaneous formation of the two σ C-C bonds. The vibration with the lowest positive frequency occurs at 147.30 cm-1 describes a nonsynchronous approach, but the motion is simply a twisting of both fragments rather than a bond forming process equivalent to the reaction path. Even if AM1 agrees with the ab initio calculation <ref name="abinitio"/> in predicting the reaction to be synchronous, MIND0/3 calculations are in favour of the process involving a stable biradicaloid intermediate, which collapses to cyclohexene in the rate-determining step. <ref name="rzepa"/> and thus the nature of the mechanism is still open to debate. 

The AM1 computations can be further employed to confirm theoretical observations such as the fact that the Diels-Alder reaction is an allowed process. The Woodward-Hoffmann rule for thermal pericyclic reactions states that a ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a <ref name="pericyclic"/> components is odd, where s and a refer to suprafacial and antarafacial components, respectively and p and r are integers. Since the investigated Diels-Alder reaction involves only suprafacial bond formation between a diene with 4 π electrons (r=1) and a dienophile with 2 π electrons (q=0), the process can be described as having only one (4q+2)s component and no (4r)a components. The total number of components is thus 1 and the reaction between ethene and cis-butadiene is expected to be allowed.
This is confirmed by the nodal properties of the HOMO and LUMO that were previously computed for ethene and butadiene. It is clear from Figure 13 that the overlap between HOMO of one reactant and the LUMO of the other reactant is bonding at both sites, if only suprafacial components are considered. Overlap between the HOMO of butadiene and the LUMO of ethylene give the HOMO of the Cs transition structure, whereas interaction between the LUMO of butadiene and the HOMO of ethylene give the LUMO of the transition structure. As shown in Figure 13, the HOMO and LUMO corresponding to the transition structure are anti-symmetric and symmetric, respectively, with respect to the plane of symmetry that was previously defined.
 
<gallery heights="195" widths="170" caption=" Figure 13. Frontier orbitals of the TS and bonding interactions" >
File: bondingHOMOetheneLUMObutadiene.png| Bonding interaction between HOMO of ethene and LUMO of butadiene
File: bondingLUMOetheneLUMObutadiene.png|Bonding interaction between LUMO of ethene and HOMO of butadiene
File: HOMOtransitionstate.png|HOMO of the transition structure: anti-symmetric
File: LUMOtransitionstate.png|LUMO of the transition structure: symmetric
</gallery>
[[File:HOMOhartreefock.png|436x234 px|thumb|'''Figure 14.'''HOMO of the TS obtained at the HF/3-21G level]]
 Note: The representation of the bonding interactions is only schematic and the relative size of the butadiene and ethene frontier orbitals is not accurate. The lines identify the bonding overlap that can develop as the reaction proceeds.

An attempt was made to perform all the above computations at the HF/3-21G level of theory. However, a transition state with the symmetric HOMO shown in Figure 14 was located, suggesting that the level of theory is not appropriate for the characterisation of the frontier orbitals involved in the Diels-Alder reaction. Based on the remarks that were previously made for the Cope rearrangement, calculations at the B3LYP/6-31G* level might have given more accurate results. Even if the shape and the coefficients of the molecular orbitals should not have been affected, the DFT method might have given electronic and thermally corrected energies which are in better agreement with the experimental values.
 
 

==== Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride ====
 
The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a cyclohexa-1,3-diene and a maleic anhydride fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 15. The overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride or vice versa is bonding at both sites, if only suprafacial components are considered. The cycloaddition reaction is thus allowed, which is consistent with the predictions based on the Woodward-Hoffmann rule.
 

<gallery heights="185" widths="160" caption=" Figure 15. Frontier orbitals of cyclohexa-1,3-diene and maleic anhydride " >
File: Planeofsymmetryii.png|Plane of symmetry
File: HOMOcyclohexadiene.png|HOMO of cyclohexa-1,3-diene: anti-symmetric
File: LUMOcyclohexadiene.png|LUMO of cyclohexa-1,3-diene: symmetric
File: HOMOanhydride.png|HOMO of maleic anhydride: symmetric
File: LUMOanhydride.png|LUMO of maleic anhydride: anti-symmetric
</gallery>

As mentioned in the introduction section, the Diels-Alder between cyclohexa-1,3-diene and maleic anhydride can give an ''endo'' or an ''exo'' adduct and thus a guess transition structure was constructed for both pathways. The guess transition structures were constructed starting from a bicyclo[2.2.2]-2-octene molecule, adjusted as for the previous part by removing a -CH2-CH2- fragment, and a five-membered cyclic anhydride fragement positioned at a distance of approximately 2 Å from each other. Optimization to a transition state using the Berny algorithm at the AM1 level of theory followed by vibrational analysis gave a structure with an imaginary vibrational frequency of magnitude 806.38 and 812.02 cm-1 for the <jmolFile text=" endo">endoTSipm12.mol</jmolFile> and <jmolFile text=" exo">DIELSEXO.LOG</jmolFile> adducts, respectively. The desired normal mode motion (Animation 5 and 6) corresponding to the Diels-Alder reaction path was obtained in both cases. The relevant optimized geometrical parameters of both transition structures are reported in Figure 16 and 17.

{|
![[Image:endoTSipm12r.gif|frame| center| Animation 4. Normal mode with the single imaginary frequency in the ''endo'' transition structure ]]
![[Image:exoTS.gif|frame| center| Animation 5. Normal mode with the single imaginary frequency in the ''exo'' transition structure ]]
! [[File:geometricalfeaturesendo.png| frame| center| 208x208 px|Figure 16. Geometrical features of the endo transition structure ]]
! [[File:geometricalfeaturesexo.png| frame| center| 150x150 px|Figure 17. Geometrical features of the exo transition structure ]]
|}

The length of the σ C-C bonds which are not involved in forming/breaking processes is comparable with the length of aromatic C-C σ bonds, indicating that the reaction proceeds via a cyclic aromatic transition state. Analogous to the Diels-Alder reaction between ethene and ''cis''-butadiene, the highly symmetric transition structures suggest that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cyclohexa-1,3-diene and maleic anhydride via both ''endo'' and ''exo'' adducts.<ref name='concerted'/>. Further evidence is provided by the normal mode motion at the imaginary frequency, which depicts a simultaneous formation of the two σ C-C bonds for both transition structures. In addition, IRC calculations at the AM1 level of theory, performed in both directions in 20 steps, are consistent with the proposed synchronous mechanism. The reaction profile does not show any discrete intermediates characteristic for stepwise processes, excluding thus the possibility of a mechanism proceeding via biradical or zwitterionic intermediates. It should be noted that the in the case of the IRC computation via the ''endo'' transition structure, the first step on the reaction path corresponds to the reactants and the last step corresponds to the cycloaddition product, whereas in the of the ''exo'' transition structure the opposite situation was observed. Thus the IRC computations performed at the AM1 level confirm the exothermic nature of the Diels-Alder cycloaddition reaction.
[[File:IRCendo20steps.png| frame| center| 300x300 px|'''Figure 18.''' Reaction profile via the endo transition structure ]]
[[File:IRCexo20steps.png| frame| center| 300x300 px|'''Figure 19.''' Reaction profile via the exo transition structure ]]

The computed electronic and thermally corrected energies of the ''endo'' and ''exo'' transition structures are reported in Table 5 below. The calculations suggest that the activation energy of the reaction pathway which proceeds via the ''endo'' transition structure is lower than the activation energy of the process involving the ''exo'' transition structure by 0.87 kcal/mol (0.0013185 hartrees) at 0 K and 0.75 kcal/mol (0.001197 hartrees) at 298.15 K. Therefore, the AM1 predictions are in agreement with the experimental observation that the Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene gives the ''endo'' product faster than it gives the ''exo'' adduct, even though the latter is thermodynamically more stable. 
 

Table 4. Summary of computed energies (hartrees) for the endo and exo transition states

{| class="wikitable"
! style="font-weight: bold;" | Optimized molecule
! style="font-weight: bold;" | Electronic energy
! style="font-weight: bold;" | Sum of electronic and zero-point energies
! style="font-weight: bold;" | Sum of electronic and thermal energies
|-
| style="font-weight: bold;" | Cyclohexa-1,3-diene
| 0.027711
| 0.152502
| 0.157726
|-
| style="font-weight: bold;" | Maleic anhydride
| -0.121824
| -0.063346
| -0.058192
|-
| style="font-weight: bold;" | Endo TS
| -0.051505
| 0.133494
| 0.143683
|-
| style="font-weight: bold;" | Exo TS
| -0.050419
| 0.134879
| 0.144880
|}

[[File:endoorbitalinteractions.png| frame| right| 208x208 px|'''Figure 20.''' Secondary orbital interaction in the ''endo'' transition structure ]]
In order to account for the formation of the ''endo'' adduct under kinetically controlled conditions, the nodal properties of the frontier orbitals of the reactants and both transition structures were examined. The interaction between the frontier orbitals can be simplified in a diagram involving only atomic orbitals (Figure 20). It can be clearly observed that the ''endo'' transition structure is characterized by an in-phase secondary overlap between the HOMO of cyclohexa-1,3-diene and LUMO of maleic anhydride. Since this stabilising secondary orbital interaction is not possible for an ''exo'' adduct, the ''endo'' transition state is expected to be lowered in energy relative to the ''exo''one. The stabilisation of the ''endo'' transition state is further reinforced by the increased C-C through space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the carbon atoms of cyclohexa-1,3-diene which ae not involved in the reaction (3.897 Å c.f. 2.945 Å in the ''exo'' adduct), which indicates diminished steric repulsion.
{|
! [[File:HOMOendoTS.png| frame| 300x300 px|Figure 21. HOMO of the endo transition structure: anti-symmetric]]
! [[File:LUMOendoTS.png| frame| 300x300px|Figure 22. LUMO of the endo transition structure: anti-symmetric ]]
! [[File:HOMOexoTS.png| frame| 300x300 px|Figure 23. HOMO of the exo transition structure: anti-symmetric ]]
! [[File:LUMOexoTS.png| frame| 300x300 px|Figure 24. LUMO of the exo transition structure: anti-symmetric ]]
|}

The computations performed for the cycloaddtion reaction between butadiene and ethene demonstrated that the AM1 method provides more accurate results than Hartree-Fock, due to the fact that it implicitly takes into account electronic correlation through the use parameters derived from experimental data. However, as any other semi-empirical method, it is expected to give very reliable results only for systems which are related to those from which their parameters have been fitted. A further study might involve the use of MCSCF calculations, which have been reported to give a more a balanced description of closed-shell species. <ref name="MCSCF2"/>

==Conclusions==
==References==
<references>
<ref name="gauche"> K.S. Pitzerk, Chem. Rev., 1940, 27, 39-57 {{DOI|10.1021/cr60086a003 }} </ref>
<ref name="pi interaction"> B.G. Rocque, J.M. Gonzales, H.F. Schaffer, Mol. Phys., 2002, 100, 441-446 {{DOI|10.1080/00268970110081412 }} </ref>
<ref name="MCSCF" > K. Morokuma, W. T. Borden, D.A. Hvorat, J. Am. Chem. Soc., 1988, '''110''', 4474. {{DOI|10.1021/ja00221a092 }} </ref>
<ref name="MM2"> F. Jensen, ''J. Am. Chem. Soc''. 1992, '''114''', 1596-1603 {{DOI|10.1021/ja00031a009}} </ref>
<ref name="energy difference"> M. J. Goldstein, M. S. Benzon, ''J. Am. Chem. Soc.'', 1972, '''94''',7147–7149 {{DOI| 10.1021/ja00775a046}}</ref>
<ref name="experimental"> M.J.S. Dewar, C. Jie, ''J. Am. Chem. Soc.'', 1987, '''109''', 5897 {{DOI|10.1021/ja00254a001}} </ref>
<ref name="chairTS"> K. N. Houk, S. M. Gustafson, K. A. Black, J. Am. Chem. Soc., 1992, 114, 8565-8572 {{DOI|10.1021/ja00048a032}} </ref>
<ref name="secondary"> D. Nisipuri, Stereochemistry of Organic Compounds: Principles and Applications , New Age Publications, 2nd edn., 1996 </ref>
<ref name="dielsalder"> M. J. S. Dewar, S. Olivella,, J. P. Stewartt, J. Am. Chem. Soc., 1986, 108, 5771-5779 {{DOI |10.1021/ja00279a018}} </ref>
<ref name="abinitio"> M. Ortega, A. Oliva, J. M. Lluch, J. Bertran, ''Chem. Phys. Letr.'', 1983, '''102''', 317 {{DOI|10.1016/0009-2614(83)87048-1}} </ref>
<ref name="rzepa">M.J.S. Dewar, S. Olivella, H. Rzepa, ''J. Am. Chem. Soc'', 1978, '''100''', 5650- 5659 {{DOI|10.1021/ja00486a013}}</ref>
<ref name="pericyclic"> I. Fleming, ''Pericyclic Reactions'', Oxford Science Publications, 2002, 40 </ref>
<ref name="concerted"> R. N. Buckle, P. Liu, E. W.D Roberts, D. J. Burnell , ''Tetrahedron'', 1999, '''55''', 11455–11464 {{DOI|10.1016/S0040-4020(99)00667-5}}</ref>
</references>

Rep:Mod:iulianamaria

2014-12-04T23:51:53Z

Ipm12: /* Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride */

== Third Year Computational Lab: Physical module. Transition states and reactivity ==

== Introduction ==
* * *- discutie despre computational methods: HF, QST2, DFT
- transition state, imaginary frequency
- utilizare pt conformational analysis si locating transition structures
- cope rearrangement mechanism, current research, discuss possible routes
- ....****
Figure1,2

==Results and Computational Methods==

 

=== Exercise 1. Optimizing the Reactants and Products of the Cope Rearrangement===

Due to the presence of three C-C bonds and three rotational minima around each of these bonds <ref name="pi interaction" />, 1,5-hexadiene is expected to have 27 conformations in total and ten of them to be energetically distinct. The conformations shown in Figure 3 are optimized at the B3LYP/6-31G* and HF/3-21G levels of theory and their electronic energies are compared, in order to determine the lowest energy conformation of 1,6-hexadiene. Frequency calculations performed on the optimized structures at the same level of theory yielded only real and positive frequencies, confirming that a minimum was obtained. The electronic energies and symmetries, along with the sum of electronic and zero-point energies and the sum of electronic and thermal energies of the optimized structures are summarized in Table 1.

 

'''Table 1. Summary of energies (hartrees) of the conformations under investigation'''

{| class="wikitable"
! rowspan="2" style="font-weight: bold;" | Conformer
! rowspan="2" style="font-weight: bold;" | Point Group
! colspan="3" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
| Electronic energy
| Sum of electronic and zero-point energies (0 K)
| Sum of electronic and thermal energies (298.15 K)
|-
| style="font-style: italic;" | Anti1
| C2
| -231.692600
| -231.539601
| -231.532644
| -234.611800
| -234.469286
| -234.461965
|-
| style="font-style: italic;" | Anti2
| Ci
| -231.692535
| -231.539539
| -231.532565
| -234.611702
| -234.469212
| -234.461856
|-
| style="font-style: italic;" | Gauche3
| C1
| -231.692661
| -231.539486
| -231.532646
| -234.611326
| -234.468718
| -234.461477
|}

 
[[File:Conformers.jpg|200px|thumb|upright=2|'''Figure 3.''' Anti1 (top), Gauche3 (middle) and Anti2 (bottom) conformations of 1,5-hexadiene]]

On the basis of conformational analysis principles, one might expect the anti1 conformation to be the lowest energy conformation of 1,5-hexadiene. However, an optimization at the HF/3-21G level of theory of the gauche3 conformation, shown in Figure 3, yields an electronic energy of -231.69266 hartrees (c.f. -231.69260 hartrees for the anti1 conformation). This suggests that, in fact, the gauche3 conformation is the global minimum.

This observation is not in accordance with the general principle that a C-C-C-C anti-periplanar arrangement enables efficient overlap between σ C-C and σ* C-C orbitals, leading to a stabilization of the filled NBO. Each C-C-C-C gauche unit has been reported to contribute 0.8-1 kcal mol-1 to the total electronic energy of saturated aliphatic hydrocarbons such as n-butane, due to close contact between hydrogen atoms and increased bond-bond Pauli repulsion energy. <ref name="gauche" />. Nonetheless, the anomalous behaviour of 1,5-hexadiene can be attributed to a favourable attractive interaction between the π electrons of the C=C bonds and the nearby vinyl hydrogen atom, referred to as the CH-π interaction. <ref name="pi interaction" /> However, the energy difference between the anti1 and gauche3 conformations of 0.037 kcal mol -1 is a tenfold smaller difference <ref name="pi interaction" /> than in the case of the gauche and anti conformations of n-butane. Thus, since the Hartree-Fock Method describes only non-interacting electrons under the influence of a mean field potential and does not the include the electron correlation effects, the CH-π interaction may not be properly considered with the Hartree-Fock Method. The density functional theory optimization at the B3LYP/6-31G* level gives a higher energy difference of 0.297 kcal mol -1 between the gauche3 and anti1 conformations, but in this case, the gauche3 is the least stable one. The density functional theory includes the exchange-correlation energy term and thus might be expected to consider electronic effects with greater accuracy. However, the results suggest that the exchange-correlation functional used in these computations is not appropriate for the treatment of CH-π interactions. A post-Hartree-Fock Method such as the Møller–Plesset perturbation theory might have been more accurate in determining the lowest energy conformation.

The slight destabilisation of the ''anti2'' conformation compared with the ''anti1'' conformation can be attributed to the closer contact between one of the allylic and the terminal vinyl protons in the '' anti2'' structure, with a distance of 2.52251 Å ('' c.f.'' 2.52266 Å in the case of the ''anti1'' conformer). The ''gauche3'' conformation shows a distance of 2.61312 between these protons and thus reduced repulsive steric interactions, providing further evidence to support the observation that ''gauche3'' is the lowest energy conformation of 1,5-hexadiene.

The density functional theory optimization at the B3LYP/6-31G* level of the ''anti2'' conformation yielded a significantly lower electronic energy than in the case of the Hartree-Fock computations. Since DFT includes an approximate treatment of electron correlation, it is expected to provide more accurate results. However, in the absence of experimental data this assumption cannot be confirmed. It is worth noting that even if the overall symmetry of the optimized structure was Ci for both methods, the lengths of the C=C, C-H and C-C bonds were slightly higher when employing a DFT optimization method, as shown in Figure 4. The destabilizing interaction between occupied σ orbitals is thus diminished, resulting in a lower electronic energy. A lower electronic energy was also obtained for the DFT optimization of the other two conformations as shown in Table 1.

[[File:bondsHFandDFT.jpg|center|800px|thumb|upright=4|'''Figure 4.''' Bond distances in optimized ''anti2'' at the HF/3-21G (left) and B3LYP/6-31G* (right) levels of theory ]]

=== Exercise 2. Optimizing the "Chair" and "Boat" Transition Structures of the Cope Rearrangement===

[[Image:ChairTSipm.gif|frame| right| 234 px|'''Animation 1.''' Normal mode with the single imaginary frequency in the chair transition structure ]]
Two allylic fragments optimized separately at the Hartree-Fock level with a basis set of 3-21G were oriented in a chair-like conformation, with a distance between the terminal allyl fragments which are involved in bond forming or breaking processes of approximately 2.2 Å.The optimization to a Berny transition state of the chair transition state was performed initially at the HF/3-21G level of theory by computing the force constants at the beginning of the calculation. A vibrational analysis revealed convergence to a geometry with two imaginary vibrational frequencies and a normal mode motion which did not correspond to the desired reaction coordinate. Thus, the proposed structure was not close enough to the transition state geometry. After a basic manipulation of the position of the two allylic fragments in order to obtain a more symmetrical guess structure, the modified arrangement was re-optimized using the previous method. A single imaginary frequency of magnitude 817.94 cm-1 and the expected normal mode motion shown in Animation 1 were obtained, confirming the successful outcome of the computation. 

The same guess structure was optimized to a minimum at the HF/3-21G level of theory with the frozen coordinates method, by fixing the distance between opposite terminal allylic carbon atoms to 2.2 Å. A structure with decreased C-C bond lengths (1.381 Å c.f. 1.389 Å for the previous method) and an electronic energy of -231.619185 hartrees was obtained. The two bond/forming breaking distances were subsequently optimized using a Hessian modified to include information about the two coordinates along which the differentiation was performed. The computation gave a geometry almost identical to the one obtained by computing force constants at the beginning of the reaction, with only negligible variations of less than 0.00005 Å in both C-C and C-H bond lengths. The imaginary frequency of magnitude 817.88 was described by the same normal mode motion as for the previous method (Animation 1). 

The distance between opposite terminal allylic carbon atoms and the C-C bond length were approximately 2.021 Å and 1.389 Å, respectively, for both computational methods. The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

 
''' Table2. Summary of computed energies (hartrees) for the chair and boat transition states''' *0 K **298.15 K 
{| class="wikitable"
! rowspan="3" style="font-weight: bold;" | Transition state
! colspan="9" style="text-align: center; font-weight: bold;" | HF/3-21G
! colspan="3" style="text-align: center; font-weight: bold;" | B3LYP/6-31G*
|-
| colspan="3" style="text-align: center; font-weight: bold;" | Force constant computation
| colspan="3" style="text-align: center; font-weight: bold;" | Redundant coordinate method
| colspan="3" style="text-align: center; font-weight: bold;" | QST2
| rowspan="2" | Electronic energy
| rowspan="2" | Sum of electronic and zero-point energies*
| rowspan="2" | Sum of electronic and thermal energies**
|-
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energy*
| Sym of electronic and thermal energies**
| Electronic energy
| Sum of electronic and zero-point energies*
| Sum of electronic and thermal energies**
|-
| Chair (C2h)
| -231.619322
| -231.466697
| -231.461338
| -231.619322
| -231.466705
| -231.461345
| -
| -
| -
| -234.556932
| -234.414879
| -234.408950
|-
| Boat (C2v)
| -
| -
| -
| -
| -
| -
| -231.602802
| -231.450926
| -231.445300
| -234.543093
| -234.402342
| -234.396008
|}
[[File:Boatinputfile.png|436x234 px|thumb|'''Figure 5.''' Input reactant and product geometries for the boat transition stucture computation]]
[[Image:BoatTSipm.gif|frame| right| 244 px|'''Animation 2.''' Normal mode with the single imaginary frequency in the boat transition structure ]]
The boat transition structure was located using a QST2 approach at the HF/3-21G level, by searching for a maximum along a parabola connecting the reactant and the product, and for a minimum in all directions perpendicular to this parabola. An initial calculation was performed using the ''anti2'' structure that was previously optimized to a minimum at the HF/3-21G level, as the reactant and the corresponding rearranged molecule. However, the calculation starting from these reactant and product structures yielded a chair transition structure with C2h symmetry, with a distance between the central allylic carbon atoms of 2.021 Å. The geometry was thus identical to the structure that was previously obtained by optimization to a transition state with the Berny algorithm. (Animation 1). This illustrates a limitation of the QST2 method, which is not capable of locating the desired first order saddle point if the reactant and product geometries are not similar to the geometry of the boat transition structure. Therefore, the geometry of the optimized ''anti2'' conformation was adjusted by changing the central dihedral C-C-C-C angle to 0 о and the angles formed by the central sp3 carbon atoms with each terminal allylic carbon atom to 100 о for both the reactant and the product as shown in Figure 5. Another QST2 calculation performed on the modified reactant and product structures gave a boat transition structure with an imaginary frequency of magnitude 839.87, the normal mode being illustrated in Animation 2. The distance between opposite terminal allylic carbon atoms was found to be approximately 2.140 Å and the C-C bond length was 1.381 Å.The resulting electronic energy and corrected energies, along with the point group of the optimized structure are summarized in Table 2.

Both optimized chair and boat transition structures at the HF/3-21G level were re-optimized using the B3LYP/6-31G* level of theory. Vibrational analysis at the same level of theory confirmed that each of the determined stationary points is a transition structure with a single negative eigenvalue at 569.16 and 530.35 cm-1 for the chair and boat structures, respectively. The optimized chair transition structure illustrated in <jmolFile text="Figure 6">ChairDFT.LOG</jmolFile> shows a distance of 1.967 Å between opposite terminal allylic carbon atoms and a C-C bond length of 1.408 Å. The optimized boat transition structure shown in <jmolFile text="Figure 7">BoatDFT.LOG</jmolFile> has interallylic distance of 2.207 Å and a C-C bond length of 1.393 Å. Thus, both optimized transition structures have similar interallylic distances and C-C bond length as the structures that were previously located using the Hartee-Fock method. As noted in the previous discussion, the DFT computations yield significantly lower energies than Hartree-Fock, as a result of the inclusion of electron correlation. The breaking/forming C-C bond length for the boat transition structure is 0.240 Å longer than for the chair transition structure, which is comparable with the value of 0.230 Å found by MCSCF calculations. <ref name="MCSCF" />. The energy difference at 298.15 K between the boat and chair TS was found to be 8.12 kcal/mol, which is slightly lower than the experimental value of 11.2 ± 2.5 kcal/mol <ref name="energy difference" />. However, these calculations at the B3LYP/6-31G* gave more accurate results than the MM2 method, which has been reported to yield an energy difference of 6.4 kcal/mol <ref name="MM2"/> 

The length of the C-C bond which are not involved in the Cope rearrangement varies between 1.381 and 1.408 Å, being comparable with a benzene C-C bond (1.40 Å). The present results suggest that the reaction proceeds via an aromatic transition state consisting
of two partially bonded three-carbon units. However, this observation is not in agreement with a AM1 study which found the biradicaloid transition structure to be lower in energy than an aromatic transition state.<ref name="experimental"/>. Thus, the preferred mechanism of the Cope rearrangement of 1,5-hexadiene is still not completely elucidated and further calculations are required to predict it.
[[Image:step44.gif|frame| right | 244 px|'''Animation 3.'''Reaction coordinate in the forward direction ]]
All calculations converged to transition structures with lengths of both C-C bonds that are forming or breaking during the rearrangement greater than a typical sp3 C-C bond, varying between 1.967 Å and 2.207 Å. Since all the pericyclic reactions that have been theoretically studied and predicted to be synchronous exhibit forming/breaking bond lengths higher than 2 Å <ref name="experimental"/>, it is reasonable to state that computations performed at HF/3-21G and B3LYP/6-31G* levels are consistent with a synchronous mechanism for the Cope rearrangement. 

IRC computations at the HF/3-21G level provide further evidence to support this affirmation. Calculations for the rearrangement via the chair transition state converged to the structure shown in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> after 44 steps and led to a reaction profile of the type indicated in Figure 9. Since the reaction coordinate is symmetrical, computations were performed only in the forward direction. It can be observed that there are no discrete intermediates characteristic for stepwise processes and thus the reaction is concerted. The 1,5-hexadiene conformation to which the calculations converged with an electronic energy of -231.691579 hartrees was further optimized at the HF/3-21G level of theory. The C2 symmetric gauche2 structure shown in <jmolFile text="Figure 10">OptimizedIRC.mol</jmolFile> was located as a minimum on the potential energy surface, with an electronic energy of -231.691667 hartrees. A further attempt was made to perform the IRC calculations in 100 steps, but the same configuration as the one illustrated in <jmolFile text="Figure 8">IRC step 44.mol</jmolFile> was obtained after 44 steps. Even if it would have been a good method of verifying that the conformation optimized at the HF/3-21G level is the correct minimum, the number of steps was not further increased due to time constraints.
[[File:IRC 50 steps.png|center|536x334 px|thumb|'''Figure 9.''' Reaction profile in the forward direction]] 
[[File:OrbitalsTS.png|right|294x294 px|thumb|'''Figure 10.''' Frontier orbitals in the Cope rearrangement ]]

Activation energies were calculated relative to the Ci symmetric configuration of 1,5-hexadiene (anti2) for both chair and boat reaction pathways at 0 K and 298.15 K. As shown in Table 3, inclusion of electron correlation terms enhances the accuracy of the results, the computations performed at the B3LYP/6-31G* level being in reasonable agreement with the experimental values <ref name="experimental"/>. The significantly lower activation energies for the chair pathway supports the experimental evidence that the transition structure adopts a chair-like conformation with C2h symmetry. <ref name="chairTS"/>. A possible explanation for the preference of the chair conformation is the presence of an antibonding secondary interaction in the boat form. <ref name="secondary"/> as shown in Figure 10. By dividing the 1,5-hexadiene into an ethylene moiety and a butadiene equivalent and considering the interaction between their HOMO and LUMO, it can be noticed that in the chair transition structure, the orbitals are too far apart to exhibit efficient out-of-phase overlap.

''' Table 3. Summary of activation energies (kcal/mol) '''

{| class="wikitable"
| rowspan="2" width="125" align="center" | '''Activation energy'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Experimental'''
|-
| width="125" align="center" | '''0 K '''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
| width="125" align="center" | '''298.15 K'''
| width="125" align="center" | '''0 K'''
|-
| '''ΔE (chair)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.09
| align="center" | 33.20
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (boat)'''
| align="center" | 55.61
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}

 

=== Exercise 3. The Diels-Alder Cycloaddition===
 
====Cycloaddition reaction between ethene and butadiene====
 

The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a butadiene and an ethylene fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 11. 

<gallery heights="185" widths="160" caption=" Figure 11. Frontier orbitals of ethene and butadiene" >
File: Planeofsymmetry.png|Plane of symmetry
File: HOMOethene.png|HOMO of ethene: symmetric
File: LUMOethene.png|LUMO of ethene: anti-symmetric
File: HOMObutadiene.png|HOMO of butadiene: anti-symmetric
File: LUMObutadiene.png|LUMO of butadiene: symmetric
</gallery>

The envelope-like guess transition structure for the Diels-Alder reaction was constructed starting from a bicyclo[2.2.2]-2-octene molecule which was adjusted by removing a -CH2-CH2- fragment. Optimization to a transition state using the Berny algorithm at the AM1 level of theory gave a structure with an imaginary frequency of magnitude 623 cm-1 and a normal mode which did not correspond to the desired reaction path. The initial bicyclo[2.2.2]-2-octene molecule was symmetrized and another optimization using the same method was performed. A critical point of Cs symmetry was located on the potential energy surface and vibrational analysis revealed a single imaginary vibrational frequency of magnitude 956.25 cm-1 with the desired normal mode motion (Animation 4) corresponding to the Diels-Alder cycloaddition. The relevant optimized geometrical parameters of the envelope type transition structure are reported in Figure 12. 

[[File:Dielsaldertransitionstategeometry2.png|300px|thumb|center|upright=2| '''Figure 12.''' Geometrical features of the optimized transition state for the Diels-Alder reaction (Å)]]

[[Image:Transitionstateanimation.gif|frame| right| 234 px|'''Animation 4.''' Normal mode with the single imaginary frequency in the transition structure ]]

The length of the σ C-C bonds which are not involved in forming/breaking processes indicates that the reaction proceeds via a cyclic aromatic transition state. The length of the σ C-C bonds that are expected to form (2.119 Å) is greater than a typical sp3 (~ 1.54 Å) and sp2 (~ 1.34 Å) bond length, but significantly lower than the sum of the corresponding van der Waals radii (~ 3.4 Å). Thus, there is a bonding interaction between the butadiene and ethylene fragments and the new σ C-C bonds are partly formed as expected for a transition structure. It has been generally reported that allowed pericyclic reactions are invariably synchronous in the absence of factors which might strongly militate against synchronicity. Since concerted approaches involve cyclic aromatic transition states, the transition structures are expected to be highly symmetric based on the analogy to the structure of benzene (equal C-C bond lengths of approximately 1.40 Å). <ref name="dielsalder" />. Therefore, the equal length of both C-C bonds formed in the cycloaddition reaction suggests that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cis-butadiene and ethylene. Further evidence is provided by the normal mode motion at -956.25 cm-1 corresponding to the reaction path, which depicts a simultaneous formation of the two σ C-C bonds. The vibration with the lowest positive frequency occurs at 147.30 cm-1 describes a nonsynchronous approach, but the motion is simply a twisting of both fragments rather than a bond forming process equivalent to the reaction path. Even if AM1 agrees with the ab initio calculation <ref name="abinitio"/> in predicting the reaction to be synchronous, MIND0/3 calculations are in favour of the process involving a stable biradicaloid intermediate, which collapses to cyclohexene in the rate-determining step. <ref name="rzepa"/> and thus the nature of the mechanism is still open to debate. 

The AM1 computations can be further employed to confirm theoretical observations such as the fact that the Diels-Alder reaction is an allowed process. The Woodward-Hoffmann rule for thermal pericyclic reactions states that a ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a <ref name="pericyclic"/> components is odd, where s and a refer to suprafacial and antarafacial components, respectively and p and r are integers. Since the investigated Diels-Alder reaction involves only suprafacial bond formation between a diene with 4 π electrons (r=1) and a dienophile with 2 π electrons (q=0), the process can be described as having only one (4q+2)s component and no (4r)a components. The total number of components is thus 1 and the reaction between ethene and cis-butadiene is expected to be allowed.
This is confirmed by the nodal properties of the HOMO and LUMO that were previously computed for ethene and butadiene. It is clear from Figure 13 that the overlap between HOMO of one reactant and the LUMO of the other reactant is bonding at both sites, if only suprafacial components are considered. Overlap between the HOMO of butadiene and the LUMO of ethylene give the HOMO of the Cs transition structure, whereas interaction between the LUMO of butadiene and the HOMO of ethylene give the LUMO of the transition structure. As shown in Figure 13, the HOMO and LUMO corresponding to the transition structure are anti-symmetric and symmetric, respectively, with respect to the plane of symmetry that was previously defined.
 
<gallery heights="195" widths="170" caption=" Figure 13. Frontier orbitals of the TS and bonding interactions" >
File: bondingHOMOetheneLUMObutadiene.png| Bonding interaction between HOMO of ethene and LUMO of butadiene
File: bondingLUMOetheneLUMObutadiene.png|Bonding interaction between LUMO of ethene and HOMO of butadiene
File: HOMOtransitionstate.png|HOMO of the transition structure: anti-symmetric
File: LUMOtransitionstate.png|LUMO of the transition structure: symmetric
</gallery>
[[File:HOMOhartreefock.png|436x234 px|thumb|'''Figure 14.'''HOMO of the TS obtained at the HF/3-21G level]]
 Note: The representation of the bonding interactions is only schematic and the relative size of the butadiene and ethene frontier orbitals is not accurate. The lines identify the bonding overlap that can develop as the reaction proceeds.

An attempt was made to perform all the above computations at the HF/3-21G level of theory. However, a transition state with the symmetric HOMO shown in Figure 14 was located, suggesting that the level of theory is not appropriate for the characterisation of the frontier orbitals involved in the Diels-Alder reaction. Based on the remarks that were previously made for the Cope rearrangement, calculations at the B3LYP/6-31G* level might have given more accurate results. Even if the shape and the coefficients of the molecular orbitals should not have been affected, the DFT method might have given electronic and thermally corrected energies which are in better agreement with the experimental values.
 
 

==== Cycloaddition reaction between cyclohexa-1,3-diene and maleic anhydride ====
 
The AM1 semi-empirical molecular orbital method was employed to optimize to minimum a cyclohexa-1,3-diene and a maleic anhydride fragment. The determined HOMO and LUMO for both molecules along with their symmetry with respect to the plane of symmetry shown are illustrated in Figure 15. The overlap between the HOMO of cyclohexa-1,3-diene and the LUMO of maleic anhydride or vice versa is bonding at both sites, if only suprafacial components are considered. The cycloaddition reaction is thus allowed, which is consistent with the predictions based on the Woodward-Hoffmann rule.
 

<gallery heights="185" widths="160" caption=" Figure 15. Frontier orbitals of cyclohexa-1,3-diene and maleic anhydride " >
File: Planeofsymmetryii.png|Plane of symmetry
File: HOMOcyclohexadiene.png|HOMO of cyclohexa-1,3-diene: anti-symmetric
File: LUMOcyclohexadiene.png|LUMO of cyclohexa-1,3-diene: symmetric
File: HOMOanhydride.png|HOMO of maleic anhydride: symmetric
File: LUMOanhydride.png|LUMO of maleic anhydride: anti-symmetric
</gallery>

As mentioned in the introduction section, the Diels-Alder between cyclohexa-1,3-diene and maleic anhydride can give an ''endo'' or an ''exo'' adduct and thus a guess transition structure was constructed for both pathways. The guess transition structures were constructed starting from a bicyclo[2.2.2]-2-octene molecule, adjusted as for the previous part by removing a -CH2-CH2- fragment, and a five-membered cyclic anhydride fragement positioned at a distance of approximately 2 Å from each other. Optimization to a transition state using the Berny algorithm at the AM1 level of theory followed by vibrational analysis gave a structure with an imaginary vibrational frequency of magnitude 806.38 and 812.02 cm-1 for the <jmolFile text=" endo">endoTSipm12.mol</jmolFile> and <jmolFile text=" exo">DIELSEXO.LOG</jmolFile> adducts, respectively. The desired normal mode motion (Animation 5 and 6) corresponding to the Diels-Alder reaction path was obtained in both cases. The relevant optimized geometrical parameters of both transition structures are reported in Figure 16 and 17.

{|
![[Image:endoTSipm12r.gif|frame| center| Animation 4. Normal mode with the single imaginary frequency in the ''endo'' transition structure ]]
![[Image:exoTS.gif|frame| center| Animation 5. Normal mode with the single imaginary frequency in the ''exo'' transition structure ]]
! [[File:geometricalfeaturesendo.png| frame| center| 208x208 px|Figure 16. Geometrical features of the endo transition structure ]]
! [[File:geometricalfeaturesexo.png| frame| center| 150x150 px|Figure 17. Geometrical features of the exo transition structure ]]
|}

The length of the σ C-C bonds which are not involved in forming/breaking processes is comparable with the length of aromatic C-C σ bonds, indicating that the reaction proceeds via a cyclic aromatic transition state. Analogous to the Diels-Alder reaction between ethene and ''cis''-butadiene, the highly symmetric transition structures suggest that the AM1 semi-empirical molecular orbital method predicts a synchronous concerted approach between cyclohexa-1,3-diene and maleic anhydride via both ''endo'' and ''exo'' adducts.<ref name='concerted'/>. Further evidence is provided by the normal mode motion at the imaginary frequency, which depicts a simultaneous formation of the two σ C-C bonds for both transition structures. In addition, IRC calculations at the AM1 level of theory, performed in both directions in 20 steps, are consistent with the proposed synchronous mechanism. The reaction profile does not show any discrete intermediates characteristic for stepwise processes, excluding thus the possibility of a mechanism proceeding via biradical or zwitterionic intermediates. It should be noted that the in the case of the IRC computation via the ''endo'' transition structure, the first step on the reaction path corresponds to the reactants and the last step corresponds to the cycloaddition product, whereas in the of the ''exo'' transition structure the opposite situation was observed. Thus the IRC computations performed at the AM1 level confirm the exothermic nature of the Diels-Alder cycloaddition reaction.
[[File:IRCendo20steps.png| frame| center| 300x300 px|'''Figure 18.''' Reaction profile via the endo transition structure ]]
[[File:IRCexo20steps.png| frame| center| 300x300 px|'''Figure 19.''' Reaction profile via the exo transition structure ]]

The computed electronic and thermally corrected energies of the ''endo'' and ''exo'' transition structures are reported in Table 5 below. The calculations suggest that the activation energy of the reaction pathway which proceeds via the ''endo'' transition structure is lower than the activation energy of the process involving the ''exo'' transition structure by 0.87 kcal/mol (0.0013185 hartrees) at 0 K and 0.75 kcal/mol (0.001197 hartrees) at 298.15 K. Therefore, the AM1 predictions are in agreement with the experimental observation that the Diels-Alder reaction between maleic anhydride and cyclohexa-1,3-diene gives the ''endo'' product faster than it gives the ''exo'' adduct, even though the latter is thermodynamically more stable. 
 

Table 4. Summary of computed energies (hartrees) for the endo and exo transition states

{| class="wikitable"
! style="font-weight: bold;" | Optimized molecule
! style="font-weight: bold;" | Electronic energy
! style="font-weight: bold;" | Sum of electronic and zero-point energies
! style="font-weight: bold;" | Sum of electronic and thermal energies
|-
| style="font-weight: bold;" | Cyclohexa-1,3-diene
| 0.027711
| 0.152502
| 0.157726
|-
| style="font-weight: bold;" | Maleic anhydride
| -0.121824
| -0.063346
| -0.058192
|-
| style="font-weight: bold;" | Endo TS
| -0.051505
| 0.133494
| 0.143683
|-
| style="font-weight: bold;" | Exo TS
| -0.050419
| 0.134879
| 0.144880
|}

[[File:endoorbitalinteractions.png| frame| right| 208x208 px|'''Figure 20.''' Secondary orbital interaction in the ''endo'' transition structure ]]
In order to account for the formation of the ''endo'' adduct under kinetically controlled conditions, the nodal properties of the frontier orbitals of the reactants and both transition structures were examined. The interaction between the frontier orbitals can be simplified in a diagram involving only atomic orbitals (Figure 20). It can be clearly observed that the ''endo'' transition structure is characterized by an in-phase secondary overlap between the HOMO of cyclohexa-1,3-diene and LUMO of maleic anhydride. Since this stabilising secondary orbital interaction is not possible for an ''exo'' adduct, the ''endo'' transition state is expected to be lowered in energy relative to the ''exo''one. The stabilisation of the ''endo'' transition state is further reinforced by the increased C-C through space distance between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the carbon atoms of cyclohexa-1,3-diene which ae not involved in the reaction (3.897 Å c.f. 2.945 Å in the ''exo'' adduct), which indicates diminished steric repulsion.
{|
! [[File:HOMOendoTS.png| frame| 300x300 px|Figure 21. HOMO of the endo transition structure: anti-symmetric]]
! [[File:LUMOendoTS.png| frame| 300x300px|Figure 22. LUMO of the endo transition structure: anti-symmetric ]]
! [[File:HOMOexoTS.png| frame| 300x300 px|Figure 23. HOMO of the exo transition structure: anti-symmetric ]]
! [[File:LUMOexoTS.png| frame| 300x300 px|Figure 24. LUMO of the exo transition structure: anti-symmetric ]]
|}

The computations performed for the cycloaddtion reaction between butadiene and ethene demonstrated that the AM1 method provides more accurate results than Hartree-Fock, due to the fact that it implicitly takes into account electronic correlation through the use parameters derived from experimental data. However, as any other semi-empirical method, it is expected to give very reliable results only for systems which are related to those from which their parameters have been fitted. A further study might involve the use of MCSCF calculations, which have been reported to give a more a balanced description of closed-shell species. <ref name="MCSCF"/>

==Conclusions==
==References==
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