Nonadiabatic quantum transition-state theory in the golden-rule limit. I. Theory and application to model systems

Abstract

We propose a new quantum transition-state theory for calculating Fermi's golden-rule rates in complex multidimensional systems. This method is able to account for the nuclear quantum effects of delocalization, zero-point energy and tunnelling in an electron-transfer reaction. It is related to instanton theory but can be computed by path-integral sampling and is thus applicable to treat molecular reactions in solution. A constraint functional based on energy conservation is introduced which ensures that the dominant paths contributing to the reaction rate are sampled. We prove that the theory gives exact results for a system of crossed linear potentials and also the correct classical limit for any system. In numerical tests, the new method is also seen to be accurate for anharmonic systems, and even gives good predictions for rates in the Marcus inverted regime.

Revisiting nuclear tunnelling in the aqueous ferrous–ferric electron transfer

“…In contrast, as we have shown in Ref. 40, GR-QTST always reduces to the correct classical expression (Eqn. (9)) in the high-temperature limit where the ring polymer collapses.…”

“…However, we could show the unsuitability of this assumption by comparing Wolynes theory 32 and our newly developed GR-QTST. 40,41 In an earlier study it was observed that GR-QTST and Wolynes theory predict the same rate for a spin-boson model in the quantum limit. 40 In contrast to this, the two theories predict quantum correction factors that differ by a factor of 6 for the atomistic system, therefore making it impossible to argue that a spin-boson model is a good approximation in this case.…”

“…Our aim is to quantify the quantum effects present in the aqueous ferrous-ferric system and thereby to address the controversy of the magnitude of the effect of quantum tunnelling on the reaction rate. In this section we present results from both Wolynes theory and our newly developed GR-QTST 40,41 and discuss the predictions for rate constants and isotope effects from these two different approaches. We investigate possible pitfalls of each theory and discuss how they affect the rate of this system.…”

“…59 This constraint is automatically obeyed by all instantons, which ensures a strong connection to instanton theory, and it also retains the correct classical limit. GR-QTST has been shown to perform very well for model systems in both the normal and inverted regimes, 40 including the multidimensional spin-boson model. GR-QTST was also investigated for systems with multiple transition states, where Wolynes theory breaks, and provides accurate rate predictions.…”

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

“…Nonetheless, a good understanding of the instanton paths has helped in the development of a number of methods based on path-integral sampling which are applicable to reactions in solution. 61,62,104 We hope that the information obtained on the instanton in this work will help derive novel path-integral-based rate theories which can describe the inverted regime more rigorously.…”