Studies in Hohenberg-Kohn and Quantal Density Functional theories

For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficent to determine the one-body potential $V(\mathbf{r})$ and vector potential $A(\mathbf{r})$ is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials ${V}_{0}$ and ${\mathbf{A}}_{0}$ . Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and $\mathbf{j}$ , we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficent to determine the one-body potential $V(\mathbf{r})$ and vector potential $A(\mathbf{r})$ is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials ${V}_{0}$ and ${\mathbf{A}}_{0}$. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and $\mathbf{j}$, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.

arXiv:1404.3297Physical Review AMar 23, 201518 citations

Quantal Density Functional Theory

Steven M. Valonez

Journal of the American Chemical SocietyJan 1, 20050 citations

1-Density Operators and Algebraic Version of The Hohenberg-Kohn Theorem

A. I. Panin

ABSTRACT: Interrelation of the Coleman’s representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman’s set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman’s polyhedrons is obtained. Key words: density operators; representability problem; density functional theory I.ABSTRACT: Interrelation of the Coleman’s representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman’s set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman’s polyhedrons is obtained. Key words: density operators; representability problem; density functional theory I.

Jan 1, 20061 citation

For a many-electron system, whether the particle density $\ensuremath{\rho}(\mathbf{r})$ and the total current density

\mathbf{j}(\mathbf{r})

are sufficent to determine the one-body potential

V(\mathbf{r})

and vector potential

A(\mathbf{r})

is still an open question. For the one-electron case, a Hohenberg-Kohn theorem exists formulated with the total current density. Here we show that the generalized Hohenberg-Kohn energy functional ${\mathcal{E}}_{{V}_{0},{\mathbf{A}}_{0}}(\ensuremath{\rho},\mathbf{j})=\ensuremath{\langle}\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j}),H({V}_{0},{\mathbf{A}}_{0})\ensuremath{\psi}(\ensuremath{\rho},\mathbf{j})\ensuremath{\rangle}$ can be minimal for densities that are not the ground-state densities of the fixed potentials

{V}_{0}

and

{\mathbf{A}}_{0}

. Furthermore, for an arbitrary number of electrons and under the assumption that a Hohenberg-Kohn theorem exists formulated with $\ensuremath{\rho}$ and

\mathbf{j}

, we discuss the possibility of a variational principle in total current-density-functional theory such as that of Hohenberg-Kohn.