Continuity equations for electron charge densities and current densities induced in molecules by electric and magnetic fields.

A series of relationships proving that the stationary current density J(r), induced by a homogeneous time-independent magnetic field B in the electrons of diamagnetic atoms and molecules, is divergenceless are reported, assuming the conventional partition into diamagnetic and paramagnetic contributions and within the representations referred to as CTOCD (continuous translation of the origin of the current density). The continuity equations involving partial time derivatives of the dynamic polarization charge density ρ(r, ω) and divergence of the current density J(r, ω), induced in a molecule by a monochromatic plane wave of frequency ω, obtained by first-order time-dependent quantum mechanical perturbation theory, are investigated supposing that the wavefunctions of the ground and excited states are either real or complex. It is found that these continuity equations are satisfied by the exact eigenfunctions of a model Hamiltonian and by variationally optimal wavefunctions, for which hypervirial theorems are assumed to be valid. They are expected to hold only approximately in calculations using the algebraic approximation, with increasing accuracy for extended high-quality basis sets.