Choice of basic variables in current-density-functional theory

The selection of basic variables in current-density-functional theory and formal properties of the resulting formulations are critically examined. Focus is placed on the extent to which the Hohenberg-Kohn theorem, constrained-search approach, and Lieb's formulation (in terms of convex and concave conjugation) of standard density-functional theory can be generalized to provide foundations for current-density-functional theory. For the well-known case with the gauge-dependent paramagnetic current density as a basic variable, we find that the resulting total energy functional is not concave. It is shown that a simple redefinition of the scalar potential restores concavity and enables the application of convex analysis and convex (or concave) conjugation. As a result, the solution sets arising in potential-optimization problems can be given a simple characterization. We also review attempts to establish theories with the physical current density as a basic variable. Despite the appealing physical motivation behind this choice of basic variables, we find that the mathematical foundations of the theories proposed to date are unsatisfactory. Moreover, the analogy to standard density-functional theory is substantially weaker as neither the constrained-search approach nor the convex analysis framework carry over to a theory making use of the physical current density.

Figure 1

Schematic illustration of the subdifferential $\underline{\partial }{F}_{\text{VR}}[{\rho }_0,{\mathbf{j}}_{\mathrm{p}0}]$. The set of all slopes [potentials $-(u,\mathbf{A})$] of just-touching tangent planes entirely below the graph at $({\rho }_0,{\mathbf{j}}_{\mathrm{p}0})$. One particular tangent, with slope $-(u_0,{\mathbf{A}}_0)$, is shown, while others are indicated by dashed lines. A similar illustration can be made of the superdifferential $\overline{\partial }\overline{E}[u,\mathbf{A}]$.

Figure 2

Illustration of the considerations in Sec. 3f. Instead of a one-to-one mapping of individual potential pairs and corresponding ground-state density pairs, the subdifferentials and superdifferentials of $F_{\text{VR}}$ and $\overline{E}$, respectively, map (ground-state) densities into convex sets of (ground-state) potentials and potentials into convex sets of densities.

Figure 3

Illustration of the set $\overline{\partial }\overline{E}[u,\mathbf{A}]$ of ground-state densities for potentials $(u,\mathbf{A})$ with degeneracy $G_{\mathrm{d}}=4$. This is a convex set, a three-dimensional simplex, with vertices at $({\rho }_0^i,{\mathbf{j}}_{\mathrm{p},0}^i)$, that is, $\overline{\partial }\overline{E}[u,\mathbf{A}]=\mathrm{co}\left\{({\rho }_0^i,{\mathbf{j}}_{\mathrm{p},0}^i)\right\}.$ In this example, we embed the simplex in ${\mathbb{R}}^3$, but in reality it is a three-dimensional subset of the density space $X_{\mathrm{p}}$. The corresponding set of potentials is also a convex set, but usually with a more complicated structure. See also Fig. 2.