Accurate and efficient calculations of Hellmann–Feynman forces for quantum computation.

This article focuses on the development and evaluation of an analytical quantum computing method, termed Ana–HF–VQE, for calculating atomic forces in quantum chemistry simulations based on the Hellmann–Feynman theorem within the variational quantum eigensolver (VQE) framework. The method is benchmarked against the finite-difference-based FD–HF–VQE approach and classical full configuration interaction (FCI) calculations for molecules including H₂, LiH, H₂O, NH₃, and CH₄. Results show that Ana–HF–VQE achieves high accuracy comparable to FCI and outperforms FD–HF–VQE notably in systems with degenerate molecular orbitals, such as NH₃, where FD–HF–VQE suffers from numerical instabilities. The study further demonstrates the practical applicability of Ana–HF–VQE by successfully performing molecular geometry optimizations and ab initio molecular dynamics simulations, highlighting its potential to extend quantum computing applications in chemical modeling.