Atomic Ehrenfest forces

Abstract The Ehrenfest force acting on the electrons within an atom of a molecule provides a force-based complement to the energy-centered description of chemical bonding. When integrated over the atomic basins defined by the quantum theory of atoms in molecules, this force can be evaluated either from the divergence of the Pauli kinetic stress tensor or by averaging the electronic force operator over the wavefunction, the latter requiring the pair density. Here we systematically examine how the atomic Ehrenfest force computed by both routes depends on the level of theory using archetypal molecules that represent covalent, polar, ionic, and van der Waals bonding (H $$_{2}$$ , LiH, H $$_{2}$$ O, LiF, He $$_{2}$$ , (NaF) $$_{2}$$ , and CNO $$^{-}$$ ). We find that the force obtained from the pair density is markedly sensitive to basis-set size, can yield a qualitatively incorrect direction for ionic systems with small basis sets, and converges toward the stress-tensor values as the basis set is enlarged, whereas its dependence on the level of theory is comparatively weak. As expected, this behavior parallels that of the Hellmann-Feynman forces and traces to their shared electron-nuclear origin, which also underlies an exact equality between the magnitudes of the total Ehrenfest and Hellmann-Feynman forces.