Ring-polymer instanton theory for tunneling between asymmetric wells

TL;DR

The paper extends instanton theory to asymmetric double-well systems by formulating a projected flux correlation function that remains well-defined when wells differ in depth and zero-point energy. It derives a ring-polymer instanton (RP-Instanton) formulation and provides practical expressions for the tunneling frequency $\ abla$ in multidimensional systems, validated against 1D and 2D benchmarks and applied to full-dimensional tunneling in -fenchol using a Gaussian process regression PES to manage computational cost. The results show high accuracy relative to exact quantum benchmarks and experiments, and establish a quantitative link between tunneling splittings and low-temperature reaction rates, specifically demonstrating the expected $k \propto \Omega^2$ scaling. The work thus enables reliable, first-principles tunneling studies in asymmetric molecular systems and offers a practical bridge between spectroscopic splittings and kinetic rates with potential broad applicability to complex environments and glasses.

Abstract

Instanton theory has arisen as a practical tool for calculating tunneling splittings in molecular systems. Unfortunately, the original formulation of instanton theory fundamentally breaks down when trying to calculate the level splitting in asymmetric double wells, as there is no imaginary-time periodic orbit connecting the two non-degenerate minima. We have therefore developed a new formulation of instanton theory based on a projected flux correlation function that is applicable to these asymmetric systems. Comparison with exact quantum-mechanical results in one- and two-dimensional models demonstrates that it has a reasonably high accuracy, similar to that reported for instanton theory in the symmetric case. The theory is then applied to study tunneling between non-degenerate minima in the biomolecule $α$-fenchol, for which we find good agreement with experiment. Finally, we use the connection to instanton rate theory, which is also derived from flux correlation functions, to discuss the often misunderstood relationship between tunneling splittings and reaction rate constants.

Figures (7)

• Figure 1: A double-well potential for which the minima $x_{\ell/r}$, the location of the dividing surface $x_\sigma$, and bounce point $x_b$ are indicated. The localized states have energies $E_{\ell/r}$, whereas the eigenstates have energies $E_\pm$. The level splitting, $\Delta$, is determined by the asymmetry, $2d$, in addition to the tunneling frequency.
• Figure 2: Trajectories of interest in the 1D asymmetric double well. a) Representation of the left (red) and right (blue) bouncing trajectories defined in Eq. \ref{['eq:cff-inst']}; b) Representation of the trajectory associated with $K_\mathrm{pin}$ defined in Eq. \ref{['eq:Kpin']}.
• Figure 3: Schematic illustration of the bead indexing convention of the ring-polymer, as described in the main text. The dividing surface is indicated by the vertical line. In practice, we only need to optimize half a ring polymer, indicated by the dark green dots (beads); the positions of the semi-transparent beads follows from the opaque ones.
• Figure 4: Plots of the 2D PES defined in Eq. \ref{['eq:2d-dw']}, with (a) being fully symmetric (i.e. with $a=0$), and (b) being a highly exaggerated asymmetric system, with $a=0.01$. The instanton trajectory is also shown for each case, with its beads represented as blue circles. Note that the ring polymer folds back on itself.
• Figure 5: The instanton representation of the tunneling process in -fenchol's H isotopomer.