arXiv subjects

Andre Laestadius

Publications and source records attributed to Andre Laestadius.

39 records · Page 3Linked to original sources

arXiv2014

Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory

For a many-electron system, whether the particle density $ρ(\mathbf{r})$ and the total current density $\mathbf{j}(\mathbf{r})$ are sufficient to determine the one-body potential $V(\mathbf{r})$ and vector potential $\mathbf{A}(\mathbf{r})$, 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 $\mathord{\cal E}_{V_0,\mathbf{A}_0}(ρ,\mathbf{j}) = \langle ψ(ρ,\mathbf{j}),H(V_0,\mathbf{A}_0)ψ(ρ,\mathbf{j})\rangle$ 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 $ρ$ and $\mathbf{j}$, we show that a variational principle for Total Current Density Functional Theory as that of Hohenberg-Kohn for Density Functional Theory does not exist. The reason is that the assumed map from densities to the vector potential, written $(ρ,\mathbf{j})\mapsto \mathbf{A}(ρ,\mathbf{j};\mathbf{r})$, enters explicitly in $\mathord{\cal E}_{V_0,\mathbf{A}_0}(ρ,\mathbf{j})$.

quant-ph↗

arXiv2014

Kohn-Sham Theory in the Presence of Magnetic Field

In the well-known Kohn-Sham theory in Density Functional Theory, a fictitious non-interacting system is introduced that has the same particle density as a system of $N$ electrons subjected to mutual Coulomb repulsion and an external electric field. For a long time, the treatment of the kinetic energy was not correct and the theory was not well-defined for $N$-representable particle densities. In the work of [Hadjisavvas and Theophilou, Phys. Rev. A, 1984, 30, 2183], a rigorous Kohn-Sham theory for $N$-representable particle densities was developed using the Levy-Lieb functional. Since a Levy-Lieb-type functional can be defined for Current Density Functional Theory formulated with the paramagnetic current density, we here develop a rigorous $N$-representable Kohn-Sham approach for interacting electrons in magnetic field. Furthermore, in the one-electron case, criteria for $N$-representable particle densities to be $v$-representable are given.

math-ph↗

arXiv2014

Density Functionals in the Presence of Magnetic Field

In this paper density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density, $ρ$, 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 $(ρ,j^p)$ 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 $Q(ρ,j^p)$, it is demonstrated that (i) it is a proper extension of the universal Hohenberg-Kohn functional, $F_{HK}(ρ,j^p)$, to $N$-representable densities, (ii) there exists a wavefunction $ψ_0$ such that $Q(ρ,j^p)=(ψ_0,H_0ψ_0)_{L^2}$, where $H_0$ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional $F(ρ,j^p)$ is studied and proven to be equal the convex envelope of $Q(ρ,j^p)$. For both $Q$ and $F$, we give upper and lower bounds.

math-ph↗