  In this paper density functionals for Coulomb systems subjected to electric
and magnetic fields are developed. The density functionals depend on the
particle density, $\rho$, and paramagnetic current density, $j^p$. This
approach is motivated by an adapted version of the Vignale and Rasolt
formulation of Current Density Functional Theory (CDFT), which establishes a
one-to-one correspondence between the non-degenerate ground-state and the
particle and paramagnetic current density. Definition of $N$-representable
density pairs $(\rho,j^p)$ is given and it is proven that the set of
$v$-representable densities constitutes a proper subset of the set of
$N$-representable densities. For a Levy-Lieb type functional $Q(\rho,j^p)$, it
is demonstrated that (i) it is a proper extension of the universal
Hohenberg-Kohn functional, $F_{HK}(\rho,j^p)$, to $N$-representable densities,
(ii) there exists a wavefunction $\psi_0$ such that
$Q(\rho,j^p)=(\psi_0,H_0\psi_0)_{L^2}$, where $H_0$ is the Hamiltonian without
external potential terms, and (iii) it is not convex. Furthermore, a convex and
universal functional $F(\rho,j^p)$ is studied and proven to be equal the convex
envelope of $Q(\rho,j^p)$. For both $Q$ and $F$, we give upper and lower
bounds.
