Retardation in electron dynamics simulations based on time-dependent density functional theory

When a molecular system is subjected to an external electric perturbation, originating from an electromagnetic wave or from a charged moving particle (in molecule-ion collisions), the electrons are the first particles to respond, on the attosecond timescale. Electron dynamics (ED) can be simulated by so-called real-time time-dependent density functional theory (RT-TDDFT). Within this framework, ED is driven by the electrostatic potential and by exchange-correlation potentials that fluctuate on the attosecond time scale. In vacuum the speed of light approaches 3 as(-1). Therefore, when simulating ED in extended molecular systems the question of retardation in the propagation of the potentials has to be posed. In this contribution we investigate two types of retardation; the first one deals with retardation in the potential created by a collision with a charged projectile. This is done through the Lienard-Wiechert potential (LWP). The second one deals with retardation in the electrostatic interaction between the time-dependent electron density and its environment, here in the context of hybrid schemes coupling RT-TDDFT to polarizable Molecular Mechanics force fields (MMpol). We found that the latter retardation effects can be safely neglected because of the rapid damping vs. distance of the electric fields created by electrostatic dipole moments. This conclusion is also relevant for methodologies, coupling RT-TDDFT to implicit polarizable continuum models. On the other hand, our results recommend the use of the LWP for modelling molecule-ion collisions by first-principles simulations. Remarkably, ionization takes place on faster time scales when relativistic corrections are introduced even for incident kinetic energies of 0.1 MeV.