Accurate and Fast Estimation of the Continuum Limit in Path Integral Simulations of Quantum Oscillators and Crystals

Convergence of imaginary-time path integral results requires a substantial number of beads (n) when quantum effects are significant. Traditional Trotter scaling approaches estimate the continuum limit (n → ∞) through extrapolation; however, their validity is restricted to the asymptotic domain of large n. We introduce an efficient extrapolation approach for quantum oscillators and crystals with harmonic character. The fitting function is inspired by the analytic solution of the harmonic oscillator (HO), or the Einstein crystal for solids. While the formulation is designed for first derivative properties (energy and pressure), its extension to second derivative properties (e.g., heat capacity) is straightforward. We apply the method to a one-dimensional HO and anharmonic oscillator (AO), as well as a three-dimensional Lennard-Jones crystal. Configurations are sampled using path integral molecular dynamics simulations in the canonical ensemble, at a low temperature of T = 0.1 (simulation units). Compared to Trotter extrapolation approaches, the new method demonstrates a substantial accuracy in estimating the continuum limit, using only a few simulations of relatively small system sizes. This efficiency significantly reduces computational cost, providing a powerful tool to facilitate computations for more complex and challenging systems, such as molecules and crystals modeled using ab initio methods.