Variational transition-state theory and multidimensional tunneling for simple and complex reactions in the gas phase, solids, liquids, and enzymes

The theory of absolute reaction rates, or transition-state-theory, forms the basis of almost all of our discussions of isotope effects in chemical reactions.1 This is as true now as it was 25 years ago, except that now we have a much better appreciation of the high accuracy afforded by transition-state theory in its modern form. This chapter is concerned with modern transition-state theory, which differs from that in use 25 years ago in two key ways: • Consistent incorporation of variational effects • Inclusion of multidimensional corner-cutting tunneling contributions By variational effects we mean the use of variational transition-state theory2 - 6 to optimize the transition state dividing surface. By multidimensional tunneling we mean, at a minimum, accounting for the change in vibrational frequencies of modes perpendicular to the reaction coordinate.7 For quantative accuracy though, we know that tunneling calculations must allow for the possibility of corner cutting, that is the tendency of the dominant tunneling paths to lie on the concave side of the minimum energy path.8 - 12 The incorporation of tunneling is usually accomplished by a transmission coefficient, which can also be used for other purposes, as discussed in Section IV.A. The consistent inclusion of quantization effects, especially zero-point energy, on perpendicular vibrations is implicit in the above description, and in fact for quantitative calculations of reaction rates and even for qualitative discussions of kinetic isotope effects (KIEs), purely classical variational transition state theory2 - 4 is only of historical and heuristic importance.