Breakdown of the bounding properties of variational transition state theory and the Rayleigh quotient method

Abstract

The thermally activated escape of a Brownian particle from one metastable state to another by crossing an intervening potential barrier is studied by means of variational transition state theory (VTST) and a Rayleigh quotient method. Historically, these two methods have been shown to provide an upper bound to the ``rate constant,'' and a restricted identity between them has been recently demonstrated by Talkner and Pollak [Phys. Rev. E 50, 2646 (1994)]. Yet, we show that while VTST gives an upper bound to a specific definition of the ``reactive flux rate,'' neither VTST nor this reactive flux rate provide a rigorous upper bound to the least nonvanishing eigenvalue of the underlying Fokker--Planck operator, as is done by the Rayleigh quotient method in the Smoluchowski limit. Numerical results for the rate in a symmetric double well show that in the spatial diffusion regime, the failure of the VTST and reactive flux methods is only significant for relatively low barriers, e.g., $\ensuremath{\beta}E\ensuremath{\lesssim}5.$