Rigorous Quantum Thermodynamics from Extended Path Integral Coarse-Graining

TL;DR

EPIGS transforms the qualitative PIGS approach into an accurate and scalable route to quantum thermodynamic simulations in complex systems, and reproduces free energies and enthalpies within 0.2 meV/atom at near-classical cost.

Abstract

Recent advances in machine-learning potentials have made molecular dynamics (MD) simulations increasingly predictive by enabling efficient descriptions of electronic quantum effects. However, nuclear quantum effects (NQEs) remain largely neglected due to their significant computational cost, limiting MD's ability to capture isotope effects and quantum tunnelling and leading to systematic thermodynamic errors. We extend path-integral coarse-graining (PIGS), a machine-learning approach for qualitatively incorporating NQEs, into a rigorous and accurate method for quantum thermodynamics. Specifically, this extended framework (EPIGS) enables the construction of temperature-transferable effective potentials that rigorously incorporate NQEs into classical MD simulations. Central to EPIGS is an instanton-based free-energy perturbation scheme (RPI-FEP) that provides direct and efficient estimation of the path-integral centroid potential. Benchmarks against full path-integral simulations on representative hydrogen-bonded systems, including liquid water, show that EPIGS reproduces free energies and enthalpies within 0.2 meV/atom at near-classical cost. EPIGS transforms the qualitative PIGS approach into an accurate and scalable route to quantum thermodynamic simulations in complex systems.

Figures (3)

• Figure 1: Schematic of temperature-dependent message-passing neural network architecture. The inverse temperature $\beta$ is encoded via a multilayer perceptron into a latent embedding vector $\boldsymbol{\tau}$, which is broadcast to all atoms and concatenated ($\oplus$) with the node features $\boldsymbol{h}_{i}$ (initialized from atomic numbers $z_i$) prior to message passing. Edge features $\boldsymbol{e}_{ji}$ encode interatomic distances and directions ($\boldsymbol{r}_{ji}$). The network outputs the quantum contributions $\Delta A_\text{c}$ and $\Delta\boldsymbol{f}_\text{c}$, and the temperature derivative $\frac{\partial A_\text{c}}{\partial \beta}$.
• Figure 2: (a) Representative snapshots of the ring-polymer instanton configurations for adenine (top) and water trimer (bottom). (b) Benchmark of RPI-FEP against fixed-centroid mass-TI for the centroid free energy. Deviation ($A_{\mathrm{c}}(\mathrm{FEP}) - A_{\mathrm{c}}(\mathrm{TI})$) is shown for small molecules ($(\text{H}_2\text{O})_{1\sim3}$, FAD, adenine) at 300 K, and for 30-molecule water clusters at 250, 300, and 400 K. Horizontal dashes indicate the ring-polymer reference values, and the shaded region represents the statistical uncertainty of the mass-TI reference. System sizes range from 3 to 90 atoms.
• Figure 3: Accuracy and temperature transferability of the EPIGS model for water. (a–f): parity plots for quantum contribution to centroid free energy $\Delta A_\text{c}$ at five temperatures, with panel (f) showing extrapolation to an unseen 45-molecule test cluster. (g–l): corresponding results for the entropic term $\beta\frac{\partial A_\text{c}}{\partial \beta}$. Marker shapes distinguish between monomer (circles) and cluster (diamonds) configurations, while fill styles indicate training (open), validation (solid), and test (solid with black border) sets. RMSE values are computed over the full dataset. Panels (f) and (l) show temperature-dependent predictions to a larger 45-molecule cluster not included in training: solid lines denote continuous MACE predictions, and pentagon markers are reference calculations at discrete temperatures obtained from RPI-FEP and PIMD samplings.