Exact tunneling splittings from path-integral hybrid Monte Carlo with enveloping bridging potentials

A path-integral hybrid Monte Carlo approach with enveloping bridging potentials (PIHMC-EBP) is proposed for calculating numerically exact tunneling splittings in molecular systems. The central idea is to construct an approximately barrierless bridging potential that smoothly connects symmetry-related regions of ring-polymer phase space, enabling direct sampling of the free-energy profile from which the relevant splittings are obtained. Two tailored nonlocal updates are designed to enhance the sampling of slow collective motions. Compared with path-integral molecular dynamics using thermodynamic integration, PIHMC-EBP obviates the need for quadrature and time step convergence checks, thereby substantially reducing the manual effort required to analyze the results. Applications to malonaldehyde (and its deuterated isotopologue) and the HCl dimer using state-of-the-art potential energy surfaces (PESs) provide the most precise tunneling splittings reported to date for both systems, while simultaneously reducing the overall computational cost by several times and three orders of magnitude, respectively. Finally, application to the water dimer yields the first numerically exact path-integral calculations of the ground-state tunneling splittings on three different PESs, all obtained simultaneously by reweighting a single set of trajectories.