Exact analytical evaluation of time dependent transmission coefficient from the method of reactive flux for an inverted parabolic barrier

TL;DR

This paper derives a comprehensive analytical expression for the time-dependent transmission coefficient of a particle crossing an inverted parabolic barrier, applicable beyond overdamped conditions, using a normal mode analysis.

Contribution

It introduces a general formula for the transmission coefficient using reactive flux and normal mode analysis, extending previous methods to arbitrary damping regimes.

Findings

The formula reproduces transition state theory results at short times.

It aligns with Kramers theory at long times.

The approach is applicable with various spectral densities.

Abstract

In this paper we derive a general expression for the transmission coefficient using the method of reactive flux for a particle coupled to a harmonic bath surmounting a one dimensional inverted parabolic barrier. Unlike Kohen and Tannor [J. Chem. Phys. 103, 6013 (1995)] we use a normal mode analysis where the unstable and the other modes have a complete physical meaning. Importantly our approach results a very general expression for the time dependent transmission coefficient not restricted to overdamped limit. Once the spectral density for the problem is know one can use our formula to evaluate the time dependent transmission coefficient. We have done the calculations with time dependent friction used by Xie [Phys. Rev. Lett 93, 180603 (2004)] and also the one used by Kohen and Tannor [J. Chem. Phys. 103, 6013 (1995)]. Like the formula of Kohen and Tannor our formula also reproduces the results of transition state theory as well as the Kramers theory in the limits t->0 and t->infinity respectively.