Noise-Tolerant Force Calculations in Density Functional Theory: A Surface Integral Approach for Wavelet-Based Methods

We introduce a method for computing quantum mechanical forces through surface integrals over the stress tensor within the framework of the density functional theory. This approach avoids the inaccuracies of traditional force calculations using the Hellmann–Feynman theorem when applied to multiresolution wavelet representations of orbitals. By integrating the quantum mechanical stress tensor over surfaces that enclose individual nuclei, we achieve highly accurate forces that exhibit superior consistency with the potential energy surface. Extensive benchmarks show that surface integrals over the stress tensor offer a robust and reliable alternative to the direct use of the Hellmann–Feynman theorem for force computations in DFT with discontinuous basis sets, particularly in cases where wavelet-based methods are employed. In addition, we integrate this approach with machine learning techniques, demonstrating that the forces obtained through surface integrals are sufficiently accurate to be used as training data for machine-learned potentials. This stands in contrast to forces calculated using the Hellmann–Feynman theorem, which do not offer this level of accuracy.