The Hellmann--Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upwards of 10 eV/Ang in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forwards for the accurate and efficient simulation of large systems using ML densities and the HF theorem.

The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, affording an approach to calculating forces of large systems with machine learning (ML) models that predict electron density. The primary issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upwards of 10 eV/Ang in forces. In this work, we construct specialized atom-centered Gaussian basis sets to reduce the Pulay force, and demonstrate the basis sets' effectiveness in computing accurate HF forces. We find that HF forces computed using the σNZHF (N = Single, Double, Triple) basis sets developed in this work yield comparable accuracy to forces computed with the Pulay term using size matched cc-pVNZ [1] and pcseg-N [2] basis sets for water clusters and pcseg-N and aug-pcseg-N basis sets for DNA fragments. Our results illustrate that the σNZHF basis sets yield HF forces with state-of-the-art accuracy, paving a clear path forwards for accurate and efficient calculations of forces for large systems using the HF theorem and ML densities.