On the derivation of semiclassical expressions for quantum reaction rate constants in multidimensional systems

Expressions for reaction rate constants in multidimensional chemical systems are derived by applying semiclassical approximation to the quantum path integrals of the ImF formulation of reaction rate theory. First, the transverse degrees of freedom orthogonal to the reaction coordinate are treated within the steepest descent approximation, after which the semiclassical approximation is applied to the remaining reaction coordinate. Thus derived, the semiclassical expressions account for the multidimensional nature of quantum effects and accurately incorporate nuclear quantum effects such as multidimensional tunneling and zero point energies. The obtained expressions are applicable in the broad temperature range from the deep tunneling to high-temperature regimes. The present paper provides derivation of the semiclassical instanton expressions proposed by Kryvohuz [J. Chem. Phys. 134, 114103 (2011)].