Nuclear Gradients from Auxiliary-Field Quantum Monte Carlo and Their Applications in ML-Driven Geometry Optimization and Transition State Search

In this article, we present a method for computing accurate and scalable nuclear forces within the phaseless auxiliary-field quantum Monte Carlo (AFQMC) framework. Our approach leverages automatic differentiation of the energy functional to obtain nuclear gradients at a computational cost comparable to that of energy evaluation. The accuracy of the method is validated against finite difference calculations, showing excellent agreement. We then explore several machine learning (ML) strategies for learning noisy AFQMC data. These ML potentials are subsequently used to perform geometry optimizations and nudged elastic band (NEB) calculations, successfully identifying the transition state of the formamide-formimidic acid tautomerization. The resulting transition-state geometry and barrier heights are in close agreement with coupled-cluster reference values. This work paves the way for highly accurate geometry optimization, molecular dynamics, or reaction path calculations.