Momentum conservation for ab initio calculations with pseudopotentials via a phase space electronic structure formulation

We demonstrate how to build a robust phase space electronic Hamiltonian ${\widehat{H}}_{\text{PS}}(\mathit{R},\mathit{P})$ that depends on nuclear position $\mathit{R}$ and nuclear momentum $\mathit{P}$ in the presence of ionic pseudopotentials. In particular, we show that in order to account for the non-local component of a pseudopotential while also conserving linear and angular momentum, one is forced to introduce a non-local one-electron operator ${\widehat{\mathbf{\Gamma }}}^{\text{nl}}$ that couples electronic and nuclear momenta together. To test the validity of our ansatz, we derive a generalized form of Nafie’s exact equality (which relates electronic momentum and electronic position) so as to include the presence of pseudopotentials. We then benchmark our proposed phase space electronic structure theory against this generalized Nafie-like result for a diverse set of diatomic molecules spanning a wide range of elements. We find excellent agreement across this ab initio test set, highlighting the strength of our method in capturing meaningful electronic momentum beyond the Born-Oppenheimer framework.