An improved path-integral method for golden-rule rates

Abstract

We present a simple method for the calculation of reaction rates in the Fermi golden-rule limit, which accurately captures the effects of tunneling and zero-point energy. The method is based on a modification of the recently proposed golden-rule quantum transition state theory (GR-QTST) of Thapa, Fang, and Richardson [J. Chem. Phys. 150, 104107 (2019)]. While GR-QTST is not size consistent, leading to the possibility of unbounded errors in the rate, our modified method has no such issue and so can be reliably applied to condensed phase systems. Both methods involve path-integral sampling in a constrained ensemble; the two methods differ, however, in the choice of constraint functional. We demonstrate numerically that our modified method is as accurate as GR-QTST for the one-dimensional model considered by Thapa and co-workers. We then study a multidimensional spin-boson model, for which our method accurately predicts the true quantum rate, while GR-QTST breaks down with an increasing number of boson modes in the discretization of the spectral density. Our method is able to accurately predict reaction rates in the Marcus inverted regime without the need for the analytic continuation required by Wolynes theory.

Path-integral approximations to quantum dynamics

“…Recently, Chowdhury and Huo [70] have developed 'non-adiabatic Matsubara dynamics', which treats the mapping variables quantum mechanically and the nuclear coordinates by Matsubara dynamics. Progress has also been made in the treatment of charge-transfer reactions, with the development of Golden-rule ring-polymer transition-state approaches [157][158][159][160] and analytic continuations of instanton [161] and Wolynes theory [162,163,172], capable of treating the Marcus-inverted regime. However, it would be premature to say more at present about these rapidly developing areas: we leave this task to future review articles.…”

Confirming the role of nuclear tunneling in aqueous ferrous–ferric electron transfer

“…80 As we have already discussed, this was used to argue that the Wolynes theory approximation was not valid for this system and that the true reaction was not well described by a spin-boson model. 80 However, as we have recently stressed, 85 GR-QTST has a size inconsistency issue which can lead to large errors in its predictions of rates for multidimensional reactions. In the following, we will demonstrate that this size inconsistency does in fact cause a large error in the GR-QTST rate for ferrous-ferric electron transfer, and hence, this method cannot be used to decide whether Wolynes theory is accurate for this problem.…”

“…We have recently suggested an alternative method to Wolynes theory and GR-QTST, which is based on the exact result in the limit of two linear crossing diabats: the linear golden-rule (LGR) approximation. 85 Importantly, this does not suffer from the size inconsistency issue of GR-QTST, and it does not make the Gaussian assumption of Wolynes theory. It therefore provides an independent way of assessing whether the assumptions of Wolynes theory break down in ferrous-ferric electron transfer.…”

“…In a recent paper, 85 we have demonstrated that GR-QTST can suffer from a serious size inconsistency issue for large systems. Here, we argue that it is this size inconsistency that is responsible for the large difference between the Wolynes and GR-QTST predictions of the rate in ferrous-ferric electron transfer, rather than a break down of Wolynes theory.…”

“…To support this argument, and help assess both the accuracy of Wolynes theory and the spin-boson mapping for this system, we also apply an alternative path-integral method, the linear golden-rule (LGR) approximation. 85 Since it is both size consistent and does not assume the correlation function is Gaussian, LGR provides a viable alternative to Wolynes theory and the spinboson mapping method for including nuclear quantum effects in condensed phase electron transfer reactions.…”