Quantal Density Functional Theory (QDFT) in the Presence of an Electromagnetic Field

We derive the QDFT equations of electrons in an external time-dependent field $\cal{E} ({\bf{r}} t) = -$ {\boldmath $\nabla$} $ v ({\bf{r}} t)$ and in the presence of an electromagnetic field characterized by the magnetic induction ${\bf{B}} ({\bf{r}} t) =$ {\boldmath $\nabla$} $\times {\bf{A}} ({\bf{r}} t)$ and electric field ${\bf{E}} ({\bf{r}}t) = -$ {\boldmath $\nabla$} $\Phi ({\bf{r}}t) - (1/c) \partial {\bf{A}} ({\bf{r}}t)/\partial t$. The QDFT is comprised of the mapping from this system to one of noninteracting fermions with the same density $\rho ({\bf{r}}t)$ and physical current density ${\bf{j}} ({\bf{r}}t)$. The mapping is in terms of `classical' fields representative of the different electron correlations that must be accounted for. On deriving the `quantal Newtonian' second law for the interacting and model systems, we obtain the local electron-interaction potential $v_{ee} ({\bf{r}} t)$ of the latter to be the work done in a conservative effective field $\cal{F}^{\mathrm{eff}} ({\bf{r}} t)$. The components of $\cal{F}^{\mathrm{eff}} ({\bf{r}} t)$ are representative of correlations due to the Pauli exclusion principle and Coulomb repulsion and the Correlation-(Kinetic, Current Density, Electric, and Magnetic) effects.