Machine Learning-Accelerated Path Integral Molecular Dynamics and ¹³ C NMR Simulations Unlock New Insights into Quantum Effects in C ₆₀ Fullerene

A definitive answer to the existence and magnitude of the negative thermal expansion (NTE) and the ¹³C nuclear magnetic resonance (NMR) signature in C₆₀ fullerene has been previously demonstrated using quantum-mechanical treatments of thermal rovibrational motion. This approach, while accurate, is computationally expensive, lacks the implementation of dispersion corrections, and is fundamentally limited to systems with well-defined equilibrium geometries and sufficiently strong restoring forces, making it inapplicable to weakly bound van der Waals complexes. Alternative methods, such as ab initio path integral molecular dynamics (PIMD), are more flexible but remain computationally expensive, especially when combined with calculations of the ¹³C NMR parameters. To overcome these limitations, we introduce an accurate and efficient neural network-based approach that combines machine learning interatomic potentials (MLIPs) with an NMR machine learning (NMR-ML) model. MLIPs enable machine learning PIMD (MLPIMD) simulations, while the NMR-ML model computes ¹³C isotropic magnetic shielding, σiso, directly from MLPIMD snapshots. We perform temperature-dependent MLPIMD simulations with MLIPs trained at different levels of theory. In all cases, NTE is observed, and the results reveal how both dispersion effects and atomic basis set choices influence its magnitude. Furthermore, we confirm that NTE is a quantum-mechanical phenomenon, and hence, classical MD simulations cannot reproduce it. To further test our approach, we investigate fully quantum-mechanical secondary isotope shifts of ¹³C NMR magnetic shielding due to the isotope change from ¹²C to ¹³C of the immediate neighbor with hexagon-hexagon or hexagon-pentagon bonds with the observed nucleus. The results show good agreement with the experimental data, highlighting the accuracy of our approach. This work demonstrates that ML-accelerated simulations enable accurate and efficient modeling of thermally activated quantum mechanical phenomena.