Quantal density-functional theory in the presence of a magnetic field

Abstract

We generalize the quantal density-functional theory (QDFT) of electrons in the presence of an external electrostatic field $\mathcal{E}(\mathbf{r})=\ensuremath{-}\ensuremath{\nabla}v(\mathbf{r})$ to include an external magnetostatic field $\mathbf{B}(\mathbf{r})=\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathbf{A}(\mathbf{r})$, where ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$ are the respective scalar and vector potentials. The generalized QDFT, valid for nondegenerate ground and excited states, is the mapping from the interacting system of electrons to a model of noninteracting fermions with the same density $\ensuremath{\rho}(\mathbf{r})$ and physical current density $\mathbf{j}(\mathbf{r})$, and from which the total energy can be obtained. The properties ${\ensuremath{\rho}(\mathbf{r}),\mathbf{j}(\mathbf{r})}$ constitute the basic quantum-mechanical variables because, as proved previously, for a nondegenerate ground state they uniquely determine the potentials ${v(\mathbf{r}),\mathbf{A}(\mathbf{r})}$. The mapping to the noninteracting system is arbitrary in that the model fermions may be either in their ground or excited state. The theory is explicated by application to a ground state of the exactly solvable (two-dimensional) Hooke's atom in a magnetic field, with the mapping being to a model system also in its ground state. The majority of properties of the model are obtained in closed analytical or semianalytical form. A comparison with the corresponding mapping from a ground state of the (three-dimensional) Hooke's atom in the absence of a magnetic field is also made.