Quantum Rate Theory for Solids. III.N-Dimensional Tunneling Effects

A theory for rate processes in solids is presented which includes the effects of quantum statistics, tunneling, and the $N$-dimensional character of the problem. It utilizes the concept of an ensemble of appropriate minimum-uncertainty wave packets to describe the conditions of thermal equilibrium in an $N$-dimensional potential well. An exact solution is found for the motion of Gaussian wave packets on a second-degree $N$-dimensional potential surface. This solution is valid also when the initial principal directions of the wave packet differ from those of the potential surface so that the problem is not separable. It is used to compute the tunneling probability for a Gaussian wave packet on a second-degree $N$-dimensional potential surface containing a saddle point. This probability depends only on the wave-packet characteristics in the saddle-point direction, and this facilitates its incorporation in the rate expression. Since the initial characteristics of the wave packets of the ensemble are determined by the potential-well character, and their subsequent tunneling probability is determined by the saddle-point character, the theory includes the effect of a difference in orientation between the two sets of principal directions. For an artificial two-dimensional problem formulated specifically to study this aspect of the process, an orientation difference is found to produce large effects only at low temperatures where tunneling is important.