Gauge invariance and origin independence of electronic charge density and current density induced by optical fields

Expressions for the first-order polarization charge density ρ ⁽¹⁾ and current density ⁽¹⁾ induced in a molecule by a monochromatic plane wave, obtained by time-dependent quantum mechanical perturbation theory, have been investigated to assess their gauge invariance and independence of the coordinate system in passive and active translations. The conditions arrived at show that, within the (long wavelengths) dipole approximation, only the electric contributions to these densities are needed to rationalize the phenomenology. To the next higher quadrupole approximation, assuming that the magnetic field and the electric field gradient are uniform over the molecular dimensions, corresponding contributions to ρ ⁽¹⁾ and ⁽¹⁾ are considered. It has been found that total densities are independent of the origin, whereas the contributions from electric and magnetic fields are not separately invariant. A magnetic contribution to ⁽¹⁾, which is by itself origin independent, can be defined by means of an approach based on continuous translation of the origin of the coordinate system.