Mixed quantum classical rate theory for dissipative systems

Numerically exact solutions for the quantum rate of potential barrier crossing in dissipative systems are only possible for highly idealized systems. It is, therefore, of interest to develop approximate theories of more general applicability. In this paper we formulate a mixed quantum classical thermodynamical rate theory for dissipative systems. The theory consists of two parts. The evaluation of a thermal flux and the computation of the classically evolved product projection operator. Since the dividing surface is perpendicular to the unstable normal mode of the dissipative system, we reformulate the theory in terms of the unstable normal mode and a collective bath mode. The influence functional for the thermal flux matrix elements in this representation is derived. The classical mechanics are reformulated in terms of the same two degrees of freedom. The one-dimensional Langevin equation for the system coordinate is replaced by a coupled set of Langevin equations for the unstable normal mode and the collective bath mode. The resulting rate expression is given in the continuum limit, so that computation of the rate does not necessitate a discretization of the bath modes. To overcome the necessity of computing a multidimensional Fourier transform of the matrix elements of the thermal flux operator, we adapt, as in previous studies, a method of Creswick [Mod. Phys. Lett. B 9, 693 (1995)], by which only a one-dimensional Fourier transform is needed. This transform is computed by quadrature. The resulting theory is tested against the landmark numerical results of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] obtained for barrier crossing in a symmetric double well potential. We find that mixed quantum classical rate theory (MQCLT) provides a substantial improvement over our previous quantum transition state theory as well as centroid transition state theory computations and is in overall good agreement with the exact results. (C) 2002 American Institute of Physics.