Perspective: Ring-polymer instanton theory

Abstract

Since the earliest explorations of quantum mechanics, it has been a topic of great interest that quantum tunneling allows particles to penetrate classically insurmountable barriers. Instanton theory provides a simple description of these processes in terms of dominant tunneling pathways. Using a ring-polymer discretization, an efficient computational method is obtained for applying this theory to compute reaction rates and tunneling splittings in molecular systems. Unlike other quantum-dynamics approaches, the method scales well with the number of degrees of freedom, and for many polyatomic systems, the method may provide the most accurate predictions which can be practically computed. Instanton theory thus has the capability to produce useful data for many fields of low-temperature chemistry including spectroscopy, atmospheric and astrochemistry, as well as surface science. There is however still room for improvement in the efficiency of the numerical algorithms, and new theories are under development for describing tunneling in nonadiabatic transitions.

Semiclassical analysis of the quantum instanton approximation

“…13,14 Semiclassical instanton (SCI) theory provides one of a) Electronic mail: christophe.vaillant@epfl.ch b) Electronic mail: jiri.vanicek@epfl.ch c) Electronic mail: jeremy.richardson@phys.chem.ethz.ch the simplest formulations of a QTST for multidimensional tunneling. 15,16 It can be derived rigorously by taking asymptotic ( → 0) approximations to the fluxflux correlation function 17 and is known in a number of formulations, 18 including one obtained from the "ImF" approximation, 19 which can all be shown to be equivalent. 20 SCI theory is defined in terms of the dominant tunneling pathway located on the full-dimensional potential-energy surface, and employs a harmonic approximation for fluctuations around this pathway.…”

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

“…The instanton for this system corresponds to two very short imaginary-time trajectories describing a periodic orbit which is collapsed onto an infinitesimally short line. 45 Note that unlike for instanton theory on a single Born-Oppenheimer potential, 76,77 there is no cross-over temperature for golden-rule instanton theory.…”

“…For the standard instanton approach in the Born-Oppenheimer limit, it is known that the e −W /h factor can be derived from one-dimensional WKB theory as the probability of reaction at a given energy, E. 76,77 WKB theory has also been applied to the nonadiabatic transfer problem and the result is known as the Zhu-Nakamura formula. 78,85 In the diabatic limit their result (written in our notation) for the inverted regime is e −(W 0 −W 1 )/h , where the W n values are calculated along paths from the crossing point to the turning points and back again.…”

“…In practical implementations of the instanton approach, we typically adopt a ring-polymer formulation. 47,77,[86][87][88] For golden-rule instanton calculations, we follow the approach suggested for the normal regime in Ref. 49 in which both paths are discretized into N n equally spaced imaginary-time intervals of length η n = τ n /N n .…”

On the calculation of quantum mechanical electron transfer rates

“…1 Although this approximation can be very accurate, it is only applicable to problems in which the rate is dominated by a single tunnelling path. [2][3][4] It cannot be used for reactions in solution, in which the configuration space of the solvent leads to multiple contributing tunnelling paths. In order to overcome this problem, methods which use statistical sampling of imaginary time paths are required.…”