Quantal density functional theory (QDFT) in the presence of a magnetic field

We present the QDFT of electrons in an external electrostatic ${\bf{E}}({\bf{r}}) = -$ {\boldmath $\nabla$} $v({\bf{r}})$ and magnetostatic ${\bf{B}}({\bf{r}}) =$ {\boldmath $\nabla$} $\times {\bf{A}} ({\bf{r}})$ field. This is the mapping from the interacting system of electrons to one of noninteracting fermions with the same density $\rho ({\bf{r}})$ and physical current density ${\bf{j}} ({\bf{r}})$. The mapping, based on the `quantal Newtonian' first law, is in terms of `classical' fields and quantal sources, the fields being separately representative of electron correlations due to the Pauli exclusion principle and Coulomb repulsion, and correlation-kinetic and correlation-magnetic effects. The theory is valid for ground and excited states. It is explicated by application to a ground state of the exactly solvable Hooke's atom in the presence of a magnetic field.