Transition State Structure Calculations

Another use of frequency calculations is to determine the nature of a stationary point found by a geometry optimization. As we ve noted, geometry optimizations converge to a structure on the potential energy surface where the forces on the system are essentially zero. The final structure may correspond to a minimum on the potential energy surface, or it may represent a saddle point, which is a minimum with respect to some directions on the surface and a maximum in one or more others. First order saddle points—which are a maximum in exactly one direction and a minimum in all other orthogonal directions—correspond to transition state structures linking two minima. 

One cannot discuss Lewis acid-catalyzed cycloaddition reactions in the present context without trying to understand the reaction course mechanistically, e.g. using a frontier molecular orbital (FMO) point of reasoning, or theoretical calculations of transition state structures. 

The carbo-Diels-Alder reaction of acrolein with butadiene (Scheme 8.1) has been the standard reaction studied by theoretical calculations in order to investigate the influence of Lewis acids on the reaction course and several papers deal with this reaction. As an extension of an ab-initio study of the carbo-Diels-Alder reaction of butadiene with acrolein [5], Houk et al. investigated the transition-state structures and the origins of selectivity of Lewis acid-catalyzed carbo-Diels-Alder reactions [6]. Four different transition-state structures were considered (Fig. 8.4). Acrolein can add either endo (N) or exo (X), in either s-cis (C) or s-trans (T), and the Lewis acid coordinates to the carbonyl in the molecular plane, either syn or anti to the alkene. 

Both experimental [7] and theoretical [8] investigations have shown that the anti complexes of acrolein and boranes are the most stable and the transition states were located only for these four anti complexes. The most stable transition-state structure was calculated (RHF/3-21G) to be NC, while XT is the least stable of the four located. The activation energy has been calculated to be 21.6 kcal mol for the catalyzed reaction, which is substantially above the experimental value of 10.4 1.9 kcal mol for the AlCl3-catalyzed addition of methyl acrylate to butadiene [4a]. The transition-state structure NC is shown in Fig. 8.5. 

The mechanism of the carbo-Diels-Alder reaction has been a subject of controversy with respect to synchronicity or asynchronicity. With acrolein as the dieno-phile complexed to a Lewis acid, one would not expect a synchronous reaction. The C1-C6 and C4—C5 bond lengths in the NC-transition-state structure for the BF3-catalyzed reaction of acrolein with butadiene are calculated to be 2.96 A and 1.932 A, respectively [6]. The asynchronicity of the BF3-catalyzed carbo-Diels-Alder reaction is also apparent from the pyramidalization of the reacting centers C4 and C5 of NC (the short C-C bond) is pyramidalized by 11°, while Cl and C6 (the long C-C bond) are nearly planar. The lowest energy transition-state structure (NC) has the most pronounced asynchronicity, while the highest energy transition-state structure (XT) is more synchronous. 

Fig. 8.5 The calculated transition-state structure for the reaction of acrolein with butadiene leading to carbo-Diels-Alder adduct catalyzed by BH3 using a RHF/3-21G basis set [6]...

In an investigation by Yamabe et al. [9] of the fine tuning of the [4-1-2] and [2-1-4] cycloaddition reaction of acrolein with butadiene catalyzed by BF3 and AICI3 using a larger basis set and more sophisticated calculations, the different reaction paths were also studied. The activation energy for the uncatalyzed reaction were calculated to be 17.52 and 16.80 kcal mol for the exo and endo transition states, respectively, and is close to the experimental values for s-trans-acrolein. For the BF3-catalyzed reaction the transition-state energies were calculated to be 10.87 and 6.09 kcal mol , for the exo- and endo-reaction paths, respectively [9]. The calculated transition-state structures for this reaction are very asynchronous and similar to those obtained by Houk et al. The endo-reaction path for the BF3-catalyzed reaction indicates that an inverse electron-demand C3-0 bond formation (2.635 A... 

In a combined experimental and theoretical investigation it was found that the / -alkyl group in the dienophile gave a steric interaction in the transition-state structure which supported the asynchronous transition-state structure for the Lewis acid-catalyzed carbo- and hetero-Diels-Alder reactions. The calculated transition-state energies were of similar magnitude as obtained in other studies of these BF3-catalyzed carbo-Diels-Alder reactions. 

The four different transition states in Fig. 8.10 were considered with BF3 as a model for the BLA catalyst in the theoretical calculations. It was found that the lowest transition-state energy for the BF3-catalyzed reactions was calculated to be 21.3 kcal mol for anti-exo transition state, while only 1.5 kcal mol higher in energy the syn-exo transition state, was found. The uncatalyzed reaction was calculation to proceed via an exo transition state having an energy of 37.0 kcal mol . The calculations indicated that the reaction proceeds predominantly by an exo transition-state structure and that it is enhanced by the coordination of the Lewis acid. 

The transition-state structure of the hetero-Diels-Alder reaction is generally found to be unsymmetrical. Houk et al. have for the reaction of formaldehyde with 1,3-butadiene calculated the C-C and C-0 bond lengths to be 2.133 A and 1.998 A, respectively, in the transition state using ab-initio calculations shown in Fig. 8.12 [25 bj. The reaction of formaldimine follows the same trend for the transition-state structure. 

The hetero-Diels-Alder reaction of formaldehyde with 1,3-butadiene has been investigated with the formaldehyde oxygen atom coordinated to BH3 as a model for a Lewis acid [25 bj. Two transition states were located, one with BH3 exo, and one endo, relative to the diene. The former has the lowest energy and the calculated transition-state structure is much less symmetrical than for the uncatalyzed reaction shown in Fig. 8.12. The C-C bond length is calculated to be 0.42 A longer, while the C-0 bond length is 0.23 A shorter, compared to the uncatalyzed reac-... 

Fig. 8.12 Calculated transition-state structure for the hetero-Diels-Alder reaction of formaldehyde with butadiene [25 bj...

Fig. 8.14 The calculated transition-state structures along the reaction path for the step-wise formation of the hetero-Diels-Alder...

The final class of reactions to be considered will be the [4 + 2]-cycloaddition reaction of nitroalkenes with alkenes which in principle can be considered as an inverse electron-demand hetero-Diels-Alder reaction. Domingo et al. have studied the influence of reactant polarity on the reaction course of this type of reactions using DFT calculation in order to understand the regio- and stereoselectivity for the reaction, and the role of Lewis acid catalysis [29]. The reaction of e.g. ni-troethene 15 with an electron-rich alkene 16 can take place in four different ways and the four different transition-state structures are depicted in Fig. 8.16. 

Fig. 8.19 The calculated transition-state structure for the BH3-exo-selective 1,3-dipolar cycloaddition reaction of nitrone 21 with ethyl vinyl ether 22 [32 ...

By ab initio MO and density functional theoretical (DPT) calculations it has been shown that the branched isomers of the sulfanes are local minima on the particular potential energy hypersurface. In the case of disulfane the thiosulfoxide isomer H2S=S of Cg symmetry is by 138 kj mol less stable than the chain-like molecule of C2 symmetry at the QCISD(T)/6-31+G // MP2/6-31G level of theory at 0 K [49]. At the MP2/6-311G //MP2/6-3110 level the energy difference is 143 kJ mol" and the activation energy for the isomerization is 210 kJ mol at 0 K [50]. Somewhat smaller values (117/195 kJ mor ) have been calculated with the more elaborate CCSD(T)/ ANO-L method [50]. The high barrier of ca. 80 kJ mol" for the isomerization of the pyramidal H2S=S back to the screw-like disulfane structure means that the thiosulfoxide, once it has been formed, will not decompose in an unimolecular reaction at low temperature, e.g., in a matrix-isolation experiment. The transition state structure is characterized by a hydrogen atom bridging the two sulfur atoms. 

Miller and Wolfenden, 2002). This latter ratio is the inverse of the rate enhancement achieved by the enzyme. In other words, the enzyme active site will have greater affinity for the transition state structure than for the ground state substrate structure, by an amount equivalent to the fold rate enhancement of the enzyme (rearranging, we can calculate KJX = Ksik Jk, )). Table 2.2 provides some examples of enzymatic rate enhancements and the calculated values of the dissociation constant for the /A binary complex (Wolfenden, 1999). 

Stanton and Merz studied the reaction of carbon dioxide addition to zinc hydroxide, as a model for zinc metallo-enzyme human carbonic anhydrase IIJ 36. It was shown that the LDA calculations (DFT(SVWN)) were not reliable for locating transition state structures whereas the post-LDA ones (DFT(B88/P86)) led to the transition state structures and ener-... 

Theoretical calculations have also permitted one to understand the simultaneous increase of reactivity and selectivity in Lewis acid catalyzed Diels-Alder reactions101-130. This has been traditionally interpreted by frontier orbital considerations through the destabilization of the dienophile s LUMO and the increase in the asymmetry of molecular orbital coefficients produced by the catalyst. Birney and Houk101 have correctly reproduced, at the RHF/3-21G level, the lowering of the energy barrier and the increase in the endo selectivity for the reaction between acrolein and butadiene catalyzed by BH3. They have shown that the catalytic effect leads to a more asynchronous mechanism, in which the transition state structure presents a large zwitterionic character. Similar results have been recently obtained, at several ab initio levels, for the reaction between sulfur dioxide and isoprene1. ... 

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