Cramer C.J. Essentials of Computational Chemistry Theories and Models

point. They then carried out IRC calculations to verify that the TS structure connected in one direction to the desired product and in the other direction to propylene oxide.

For the isomerizations to allyl alcohol, propanal, and acetone, they found concerted TS structures that represented the only barrier between reactant and product, and these structures were predicted to have stable, closed-shell singlet wave functions. However, for the isomerization to methyl vinyl ether, a pathway involving three TS structures and two intermediates was identified, with several stationary points having a high degree of biradical character. To deal with this problem, they used a broken-symmetry SCF procedure (see Section 8.5.3). Another multistep pathway involving a carbene intermediate was also found for the isomerization of propylene oxide to methyl vinyl ether, but Dubnikova and Lifshitz assigned it as being kinetically unimportant based on significantly higher TS energies than those found for the first pathway.

While the B3LYP/cc-pVDZ level was judged to be a good choice for locating stationary points, it was not expected to be quantitatively useful in computing activation enthalpies. For this purpose, single-point CCSD(T)/cc-pVDZ calculations were carried out. Dubnikova and Lifshitz are not clear on what, if any, special precautions were taken with the biradical species (i.e., were single-reference HF wave functions somehow generated, or were mixedstate UHF reference wave functions used?) The potential energies were combined with the ZPVEs and thermal enthalpic contributions calculated from scaled B3LYP frequency calculations to determine absolute H values for all species. Absolute entropies S were computed from the B3LYP geometries and scaled vibrational frequencies. The energies for several of the stationary points relative to propylene oxide varied by as much as 4 kcal mol−1 comparing CCSD(T) to B3LYP. Although it is not a priori obvious which might be expected to do better, the general rule that B3LYP somewhat underestimates barrier heights compared to CCSD(T) suggests the latter will be of greatest utility.

With all activation parameters in hand, Dubnikova and Lifshitz convert them to A and Ea of the Arrhenius equation (Eqs. (15.30) – (15.32)) to compare to measured values; the data are provided in Table 15.2. In the case of the rearrangement to methyl vinyl ether, the data for the highest energy TS structure along the path were used. It is interesting to note that the comparison of the rate constants derived from the activation parameters at some particular temperature – 1000 K is shown in Table 15.2 – appears more favorable than a direct comparison of the activation parameters themselves. This occurs because in every case the error in activation energy is compensated for by an error in the pre-exponential factor. That is, if the activation energy is predicted to be too high, which would predict too

Table 15.2 Predicted and measured activation parameters for unimolecular rearrangements of propylene oxide

546 15 ADIABATIC REACTION DYNAMICS

small a rate constant, the pre-exponential is predicted to be too large, which returns the rate constant to a reasonable value, and vice versa. In spite of such compensating errors, in the case of acetone the final error in the rate is almost a factor of 20. At lower temperatures, this error would increase dramatically.

Nevertheless, the agreement that is obtained – which is probably the best one should

ether pathway – suggests that the theoretically predicted TS structures are accurate representations of the actual transition states. This establishes the concerted nature of three of the rearrangements and the stepwise nature of the fourth.

Bibliography and Suggested Additional Reading

Chuang, Y.-Y., Cramer, C. J., and Truhlar, D. G. 1998. ‘The Interface of Electronic Structure and Dynamics for Reactions in Solution’, Int. J. Quantum Chem., 70, 887.

Chuang, Y.-Y., Radhakrishnan, M. L., Fast, P. L., Cramer, C. J., and Truhlar, D. G. 1999. ‘Direct Dynamics for Free Radical Kinetics in Solution: Solvent Effect on the Rate Constant for the Reaction of Methanol with Atomic Hydrogen’, J. Phys. Chem. A, 103, 4893.

Espenson, J. H. 1995. Chemical Kinetics and Reaction Mechanisms, 2nd Edn., McGraw-Hill: New York.

Garrett, B. C. and Truhlar, D. G. 1979. ‘Semiclassical Tunneling Calculations’, J. Phys. Chem., 83, 2921.

Hynes, J. T. 1996. ‘Crossing the Transition State in Solution’, in Solvent Effects and Chemical Reactivity, Tapia, O. and Bertran,´ J. Eds., Kluwer: Dordrecht, 231.

Jensen, F. 1999. Introduction to Computational Chemistry, Wiley: Chichester.

Jensen, F. and Norrby, P.-O. 2003. ‘Transition States from Empirical Force Fields’, Theor. Chem. Acc.,

109, 1.

Johnston, H. S. 1966. Gas Phase Reaction Rate Theory, Ronald Press: New York.

Lowry, T. H. and Richardson, K. S. 1981. Mechanism and Theory in Organic Chemistry, 2nd Edn., Harper & Row: New York.

Steinfeld, J. I., Francisco, J. S., and Hase, W. L. 1999. Chemical Kinetics and Dynamics, 2nd Edn., Prentice Hall: Upper Saddle River, NJ.

Truhlar, D. G., Garrett, B. C., and Klippenstein, S. J. 1996. ‘Current Status of Transition-state Theory’,

J. Phys. Chem., 100, 12771.

Tucker, S. C. and Truhlar, D. G. 1989. ‘Dynamical Formulation of Transition State Theory: Variational Transition States and Semiclassical Tunneling’, in New Theoretical Concepts for Understanding Organic Reactions, Bertran,´ J. and Czismadia, I. G., Eds., Kluwer: Berlin, 291.

Worth, G. A. and Robb, M. A. 2002. ‘Applying Direct Molecular Dynamics to Non-adiabatic Systems’,

Adv. Chem. Phys., 124, 355.

References

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Appendix A

Acronym Glossary

Note: Basis set abbreviations are detailed in Chapter 6 and are, for the most part, not included here. Only the most common combinations of exchange and correlation functionals are included as separate acronyms. Unit abbreviations are not listed.

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)