Quantum Mechanical Results Of The Matrix Elements Of The Boltzmann Operator Obtained From Series Representations

TL;DR

This paper uses series representations of the Boltzmann operator to compute quantum matrix elements for different potentials, analyzing convergence and saddle point structures at various temperatures, with implications for quantum reaction rate theories.

Contribution

It introduces a method to calculate matrix elements of the Boltzmann operator using series representations, including bare potentials, and explores their convergence and structural properties at low temperatures.

Findings

Series converge rapidly even at low temperatures.

Zeroth order approximation suffices at high temperatures.

Number of saddle points increases as temperature decreases.

Abstract

Recently developed series representations of the Boltzmann operator are used to obtain Quantum Mechanical results for the matrix elements, <x| exp(-{\beta} H)|x>, of the imaginary time propagator. The calculations are done for two different potential surfaces: one of them is an Eckart Barrier and the other one is a double well potential surface. Numerical convergence of the series are investigated. Although the zeroth order term is sufficient at high temperatures, it does not lead to the correct saddle point structure at low temperatures where the tunneling is important. Nevertheless the series converges rapidly even at low temperatures. Some of the double well calculations are also done with the bare potential (without Gaussian averaging). Some equations of motion related with bare potentials are also derived. The use of the bare potential results in faster integrations of equations of motion. Although, it causes lower accuracy in the zeroth order approximation, the series show similar convergence properties both for Gaussian averaged calculations and the bare potential calculations. However, the series may not converge for bare potential calculations at low temperatures because of the low accuracy of zeroth order approximation. Interestingly, it is found that the number of saddle points of <x| exp(-{\beta} H)|x> increases as the temperature is lowered. An explanation of observed structures at low temperatures remains as a challenge. Besides, it has implications for the quantum instanton theory of reaction rates at very low temperatures.