Density Functionals in the Presence of Magnetic Field

Andre Laestadius

Published 2014-04-03Version 1

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.