Ring-polymer instanton method for calculating tunneling splittings

Abstract

The semiclassical instanton expression for the tunneling splitting between two symmetric wells is rederived, starting from the ring-polymer representation of the quantum partition function. This leads to simpler mathematics by replacing functional determinants with matrix determinants. By exploiting the simple Hückel-like structure of the matrices, we derive an expression for the instanton tunneling splitting in terms of a minimum on the potential surface of a linear polymer. The latter is a section cut out of a ring polymer, consisting of an infinite number of beads, which describes a periodic orbit on the inverted potential surface. The approach is straightforward to generalize to multiple dimensions, and we demonstrate that it is computationally practical by carrying out instanton calculations of tunneling splittings in HO(2) and malonaldehyde in full dimensionality.

Investigation of Terahertz Vibration–Rotation Tunneling Spectra for the Water Octamer

“…Here we summarize the ring-polymer instanton method, 39,40 which was used to compute approximate tunneling splittings and tunneling pathways. The basis of the approach is that the tunneling splittings can be obtained from the limit 41−43 …”

“…The values of the tunneling matrix elements obtained are too small to give rise to experimentally measurable splitting patterns (see next section), and thus we did not try to obtain more accurate convergence of h λμ , which could have been achieved by converging with respect to N and β, as in ref 39, and also by using a better potential energy surface. 57 The reason that the values of h λμ are relatively small is clear: at least two hydrogen-bonds have to be broken along each of the four tunneling paths.…”

Kinetically constrained ring-polymer molecular dynamics for non-adiabatic chemical reactions

“…23,[59][60][61] In a particularly striking failure of instanton-based methods, the rate of deep-tunneling across strongly asymmetric barriers is significantly overestimated in RPMD and steepest-descent SCI calculations, which manifests in incorrect rate coefficients for ET in the Marcus inverted regime. 23,62 A simple and methodologically suggestive way to understand this overestimation is to recognize that ring-polymer configurations associated with transitions between asymmetric potential wells (i.e., kinked ring-polymer configurations across non-degenerate diabatic surfaces, such that |V 0 (R) − V 1 (R)| |K(R)|) appear with greater probability in the equilibrium distribution than is appropriate for an accurate transition-state theory (TST) description of the deeptunneling process.…”

“…[50][51][52] This connection between imaginary-time path-integral statistics and the diabatic coupling K lies at the heart of semiclassical instanton (SCI) theory, [53][54][55][56][57][58] and it underpins the accuracy of the RPMD method for the description of thermal reaction rates in the deep-tunneling regime. [59][60][61] For these reasons, the formation of kink-pairs during nonadiabatic transitions is an important feature to preserve in any extension of the RPMD method to multi-level systems. We thus introduce a discrete collective variable that reports on the existence of kink-pairs in the ring-polymer configuration,…”

Efficient algorithms for semiclassical instanton calculations based on discretized path integrals

“…To obtain the reliable single kink path, several variants on the least action approach have been proposed. 20,21,[35][36][37][38][39][40][41][42][43][44][45] To describe the sina) Authors to whom correspondence should be addressed. Electronic addresses: kawatsu@fukui.kyoto-u.ac.jp and smiura@mail.kanazawau.ac.jp.…”