Analytic continuation of Wolynes theory into the Marcus inverted regime

Abstract

The Wolynes theory of electronically nonadiabatic reaction rates [P. G. Wolynes, J. Chem. Phys. 87, 6559 (1987)] is based on a saddle point approximation to the time integral of a reactive flux autocorrelation function in the nonadiabatic (golden rule) limit. The dominant saddle point is on the imaginary time axis at t=iλℏ, and provided λ lies in the range -β/2≤λ≤β/2, it is straightforward to evaluate the rate constant using information obtained from an imaginary time path integral calculation. However, if λ lies outside this range, as it does in the Marcus inverted regime, the path integral diverges. This has led to claims in the literature that Wolynes theory cannot describe the correct behaviour in the inverted regime. Here we show how the imaginary time correlation function obtained from a path integral calculation can be analytically continued to λ<-β/2, and the continuation used to evaluate the rate in the inverted regime. Comparison with exact golden rule results for a spin-boson model and a more demanding (asymmetric and anharmonic) model of electronic predissociation shows that the theory is just as accurate in the inverted regime as it is in the normal regime.

Instanton formulation of Fermi’s golden rule in the Marcus inverted regime

“…In this respect, our method appealingly stands out from alternative approaches which effectively extrapolate the data collected in the normal regime. 54 The instanton orbit can be interpreted as the 'dominant tunnelling pathway' and identifies which nuclear modes are involved in the reaction. In cases where there are competing reactions, the instanton approach will identify the dominant mechanism.…”

“…In effect, the method of Ref. 54 analytically continued the function c(τ) into the region with τ < 0 by fitting it numerically to a suitable functional form 2 based on calculations in the normal regime and extrapolating to describe systems in the inverted regime. We will go one step further to find a semiclassical instanton approach which is directly applicable in the inverted regime and requires no extrapolation.…”

“…We chose to apply our method to the same two model systems as Lawrence and Manolopoulos in their work on the extrapolation of Wolynes theory into the inverted regime. 54 A. Spin-boson model The first model system under consideration is the spinboson model at T = 300 K defined by the potentials…”

“…The order-of-magnitude deviation of the classical rate for large ε emphasizes the remarkable relevance of nuclear tunnelling effects in these systems especially in the inverted regime, which can be almost perfectly captured with our semiclassical instanton approach. Note that although Lawrence and Manolopoulos were able to achieve similarly accurate results with their approach, 54 it was necessary for them to design a special functional form to treat this dissociative system, whereas we could apply an identical algorithm to the case of the spin-boson model.…”

An analysis of isomorphic RPMD in the golden rule limit

“…Provided such systems are in the golden rule limit, however, Wolynes theory already provides a quantitatively accurate way of calculating reaction rates in both the normal and inverted Marcus regimes. 14,33 Moreover one can connect Wolynes theory in the golden rule limit to conventional RPMD rate theory in the Born-Oppenheimer limit using a simple interpolation formula, which enables the calculation of accurate reaction rates for arbitrary electronic coupling strengths. 34,35 In order to apply iso-RPMD to such systems one would need to modify the isomorphic Hamiltonian or the dynamics to make it more closely related to an accurate approximation to the quantum rate in Eq.…”