Non-adiabatic Ehrenfest dynamics with norm-conserving and ultra-soft pseudo-potentials with nuclear velocity corrections on the atomic orbitals within the Projector Augmented Wave Method framework

We derive the first-principles Ehrenfest molecular dynamics describing non-adiabatic processes with the inclusion of the nuclear-velocity-dependent phases (also known as electron-translation factors) on the atomic-orbital basis. These phases, appearing when nuclei are treated dynamically, affect effective Hamiltonians constructed from localised orbitals. In this work, we focus on the effects in the first-principles pseudo-potential Hamiltonian, both for the norm-conserving and ultra-soft cases, derived within the Projector-Augmented-Wave (PAW) method framework. Peierls-like phases depending on the nuclear velocities appear in the non-local part of the potential, while additional nuclear velocity and acceleration-dependent corrections appear in the ultra-soft pseudo-potential case. The use of velocity-including atomic orbital basis enables a Galilean-invariant description of the non-adiabatic Ehrenfest molecular dynamics, removing spurious non-adiabatic couplings that arise from neglecting the nuclear velocity phases in the atomic orbitals.