Centroid-density quantum rate theory: Variational optimization of the dividing surface

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Abstract
A generalization of Feynman path integral quantum activated rate theory is presented that has classical variational transition state theory as its foundation. This approach is achieved by recasting the expression for the rate constant in a form that mimics the phase‐space integration over a dividing surface that is found in the classical theory. Centroid constrained partition functions are evaluated in terms of phase‐space imaginary time path integrals that have the coordinate and momenta centroids tied to the dividing surface. The present treatment extends the formalism developed by Voth, Chandler, and Miller [J. Chem. Phys. 91, 7749 (1989)] to arbitrary nonplanar and/or momentum dependent dividing surfaces. The resulting expression for the rate constant reduces to a strict variational upper bound to the rate constant in both the harmonic and classical limits. In the case of an activated system linearly coupled to a harmonic bath, the dividing surface may contain explicit solvent coordinate dependence so that one can take advantage of previously developed influence functionals associated with the harmonic bath even with nonplanar or momentum dependent dividing surfaces. The theory is tested on the model two‐dimensional system consisting of an Eckart barrier linearly coupled to a single harmonic oscillator bath. The resulting rate constants calculated from our approximate theory are in excellent agreement with previous accurate results obtained from accurate quantum mechanical calculations [McRae et al., J. Chem. Phys. 97, 7392 (1992)].