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

Abstract

Fermi's golden rule defines the transition rate between weakly coupled states and can thus be used to describe a multitude of molecular processes including electron-transfer reactions and light-matter interaction. However, it can only be calculated if the wave functions of all internal states are known, which is typically not the case in molecular systems. Marcus theory provides a closed-form expression for the rate constant, which is a classical limit of the golden rule, and indicates the existence of a normal regime and an inverted regime. Semiclassical instanton theory presents a more accurate approximation to the golden-rule rate including nuclear quantum effects such as tunnelling, which has so far been applicable to complex anharmonic systems in the normal regime only. In this paper we extend the instanton method to the inverted regime and study the properties of the periodic orbit, which describes the tunnelling mechanism via two imaginary-time trajectories, one of which now travels in negative imaginary time. It is known that tunnelling is particularly prevalent in the inverted regime, even at room temperature, and thus this method is expected to be useful in studying a wide range of molecular transitions occurring in this regime.

Advances and challenges for experiment and theory for multi-electron multi-proton transfer at electrified solid–liquid interfaces

“…The discussion has focused on electronically adiabatic reactions, but non-adiabatic reactions can also be treated within the general GCE rate theory. 66 Novel rate theories for including non-adiabatic effects in instanton theory 334,335 and RPMD 284,293,336 exist but thus far they have been applied only to model systems. Therefore, the non-adiabatic reactions are presented within the more established curvecrossing picture in Sections 5 and 7.…”

Spin crossover of thiophosgene via multidimensional heavy-atom quantum tunneling

“…60,61 As we cannot solve the vibrational Schrödinger equation for the full problem, we would like to approximate Fermi's golden rule as accurately as possible without making drastic simplifications for the shape of the diabatic surfaces. This can be accomplished using golden-rule instanton theory, 62,63 which is an extension of the original instanton method [64][65][66][67][68][69][70][71] for reactions within the Born-Oppenheimer approximation. The theory constitutes a rigorous semiclassical approximation to a path-integral formulation of Fermi's golden rule and has been shown to be in almost perfect agreement with the quantum rate for a number of multidimensional and anharmonic model systems.…”

“…The theory constitutes a rigorous semiclassical approximation to a path-integral formulation of Fermi's golden rule and has been shown to be in almost perfect agreement with the quantum rate for a number of multidimensional and anharmonic model systems. 63,[72][73][74][75] The method locates the ideal tunneling pathway for each temperature in a self-consistent fashion, which enables one not only to compute accurate reaction rates but also to predict temperature-dependent changes of the mechanism.…”

“…In this work we apply the newly developed golden-rule instanton theory for the inverted regime 63 to the T 1 → S 0 spin crossover of thiophosgene based on on-the-fly multireference perturbation theory electronic-structure calculations and thereby show how the method can simulate spin-crossing reactions with unprecedented accuracy. We find that our treatment of multidimensional tunneling is necessary for a correct description of the reaction mechanism, which explains the significant improvement of our results compared with those of previous studies.…”

A general non-adiabatic quantum instanton approximation

“…Early work extending the semiclassical instanton approach to treat non-adiabatic reactions was based on assuming that the "Im-F" premise 38,39 could be applied to non-adiabatic systems, and succeeded in providing a theory that bridged between the golden rule and Born-Oppenheimer limits. [41][42][43][44] More recently, Richardson et al [45][46][47] have provided a rigorous derivation of the semiclassical instanton rate in the golden rule limit and found some important differences between the resulting expression and that given by the Im-F formulation. However, their derivation has yet to be extended beyond the golden rule limit so that it can be applied to reactions with arbitrary electronic coupling strengths.…”