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Physical Review Letters

Quantum Theory of Activated Events in Condensed Phases

Peter Guy Wolynes

Published Sep 28, 1981• 305 citations
Publisher

Abstract

The quantum analog of Kramers's classical theory of the rate of activated events is presented. By means of a path-integral approach, the correlation function expression for the rate of barrier crossing by a quantum mechanical particle coupled to a heat bath is evaluated for finite temperatures. Tunneling effects are dramatically diminished by the frictional influence of the heat bath.

DOI
10.1103/physrevlett.47.968

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The quantum analog of Kramers's classical theory of the rate of activated events is presented. By means of a path-integral approach, the correlation function expression for the rate of barrier crossing by a quantum mechanical particle coupled to a heat bath is evaluated for finite temperatures. Tunneling effects are dramatically diminished by the frictional influence of the heat bath.
We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.
A quantum-activated rate process in the presence of a nonlinear dissipative environment is studied with a Feynman path-integral technique. The nonlinear dissipation arises from specific nonlinear couplings between the reaction coordinate and a many-dimensional bath of effective harmonic oscillators. Such coupling results in a nonlinear influence functional for the effective Feynman path integral along the reaction coordinate. Large nonlinear effects on the rate constant are found at low temperatures.
The quantum-mechanical version of the Kramers turnover problem is considered. The multidimensional character of the problem is taken into account via transformation to normal modes. This eliminates the coupling to the bath near the barrier top allowing the use of a simple harmonic transmission coefficient for the barrier dynamics. The well dynamics is described by a continuum form of a master equation for the energy in the unstable normal mode. Within first-order perturbation theory, the equations of motion for the stable normal modes have the form of a forced oscillator. The transition probability kernel is found using the known solution for the quantum forced oscillator problem. An expression for the quantum escape rate is derived. It encompasses all previously known limiting results in the thermally activated tunneling regime. The depopulation factor, which accounts for the nonequilibrium energy distribution is evaluated. The quantum transition probability kernel is broader than the classical and is skewed towards lower energies. Interplay between these two effects, together with a positive tunneling contribution, determines the relative magnitude of the quantum rate compared to the classical one. The theory is valid for arbitrary dissipation. Its use is illustrated for the case of a cubic potential with Ohmic (Markovian) dissipation.
Abstract The quantum dissipative dynamics of a particle which tunnels through the barrier of a double well potential and is coupled to an Ohmic heat bath is studied. Exact formal expressions for correlation functions are presented and examined. Attention is focused on the region in which coherence effects are important, and on the algebraic long‐time tails at zero temperature. Coherence effects in the dissipative quantum transport of a particle in a tight‐binding lattice are also investigated.
The author studies the barrier-crossing rate Gamma for a particle in a symmetric double well coupled to a sluggish bath; the aim is to justify the quantum simulation procedure used in related work on the diffusion of hydrogen in metals. He uses path-integral methods to discuss the transition between the low-temperature tunnelling and the high-temperature barrier-jumping regimes, and suggests a unifying formula valid for both, which relates Gamma to the probability of finding the centroid of the quantum path at the barrier top. Numerical results for Gamma are presented for the case of the quartic double well.