Systematic incorporation of nuclear quantum effects into atomistic simulations by smoothed trajectory analysis

Nuclear quantum effects (NQEs) play an essential role in many atomistic systems, yet their explicit inclusion in molecular simulations remains challenging. Path-integral molecular dynamics (PIMD) provides a rigorous framework for incorporating NQEs, but its practical applicability is often limited by the slow and strongly system-dependent convergence with respect to the number of beads. Here we introduce path-integral generalized smoothed trajectory analysis (PIGSTA), a post-processing framework for the systematic incorporation of NQEs into atomistic simulations, using either classical or path-integral molecular dynamics trajectories. By applying analytically defined convolution kernels to simulation trajectories, PIGSTA corrects the frequency-dependent discretization error associated with a finite number of beads, without modifying the underlying dynamics. For harmonic systems, PIGSTA recovers the exact quantum-mechanical limit at any bead number, whereas standard PIMD becomes exact only in the infinite-bead limit. More generally, the method significantly improves the convergence of thermodynamic and structural observables at finite bead numbers and provides an internal, reference-free diagnostic of bead-number convergence based on the consistency of energy and force estimators. We assess PIGSTA for ambient liquid water and for the Zundel cation at ultralow temperature, representing a particularly demanding case for bead-number convergence. In both systems, PIGSTA reproduces the converged PIMD limit and enables physically consistent results at reduced bead numbers when convergence criteria are satisfied. Owing to its post-processing nature and negligible additional computational cost, PIGSTA offers a practical and broadly applicable approach for incorporating NQEs into atomistic simulations.