Direct Sampling of Nuclear-Quantum Chemical Equilibria by Path-Integral Reaction Ensemble Monte Carlo

Abstract Reaction ensemble Monte Carlo samples chemical composition directly but usually treats the explicit environment with classical nuclei, whereas path-integral methods represent nuclear quantum effects explicitly but generally infer reaction equilibria from partition-function or free-energy ratios between predefined states. We formulate path-integral reaction ensemble Monte Carlo (PI-RxMC), a reaction-constrained expanded ensemble that samples chemical sectors and nuclear paths jointly, and derive the Metropolis-Hastings rule for sector changes, including sector weights, normalization factors, proposal probabilities, coordinate transformations, and Jacobians; for fixed nuclei with changing bonding topology, the free-ring-polymer contribution cancels exactly. Independent finite-P analytical, bond-topology, and rovibrational benchmarks validate the method, and direct sector-population estimates for H2 + D2 ⇌ 2HD agree with independent quantum results from 200 to 700 K. A second benchmark, AB ⇌ A + B with Δν = +1, explicitly recovers the reaction-volume and 1 M standard-state factors and exhibits a nearly 3-fold nuclear-quantum enhancement of the equilibrium constant. A bead-level slit field further demonstrates coupling to an explicit environment.