Kohn–Sham Theory with Paramagnetic Currents: Compatibility and Functional Differentiability

Abstract

Recent work has established Moreau-Yosida regularization as a mathematical tool to achieve rigorous functional differentiability in density-functional theory. In this article, we extend this tool to paramagnetic current-density-functional theory, the most common density-functional framework for magnetic field effects. The extension includes a well-defined Kohn-Sham iteration scheme with a partial convergence result. To this end, we rely on a formulation of Moreau-Yosida regularization for reflexive and strictly convex function spaces. The optimal L p -characterization of the paramagnetic current density L 1 ∩ L 3/2 is derived from the N -representability conditions. A crucial prerequisite for the convex formulation of paramagnetic current-density-functional theory, termed compatibility between function spaces for the particle density and the current density, is pointed out and analyzed. Several results about compatible function spaces are given, including their recursive construction. The regularized, exact functionals are calculated numerically for a Kohn-Sham iteration on a quantum ring, illustrating their performance for different regularization parameters.

Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions

“…On the other hand, the result in Laestadius et al [23] is applicable to not only standard DFT, but to all DFT flavors that fit into the given framework of reflexive Banach spaces. It has already been successfully applied to paramagnetic current DFT (CDFT) [27]. This general approach is also pursued in this Letter.…”

“…A simulation of two electrons on a ring lattice [31] allows us to illustrate the above method. Compared to a previous implementation in a CDFT setting [27], the version given here uses the more conservative damping step that helped to prove convergence. To distinguish the two versions, we denote them "MYKSODA-S" for shorter, conservative steps, and "-L" for the original longer steps [23,27].…”

Unique continuation for the magnetic Schrödinger equation

“…Returning to Equation (18), we set C = 74+3Mr 2 0 h i =3 and use Equation (19) and |rh| ≤ 2/r to conclude for r ≤ r 0…”

Force balance approach for advanced approximations in density functional theories

“…A detailed mathematical analysis of paramagnetic ground-state CDFT including a rigorous setup for constructing a Kohn-Sham iteration scheme that is in fact only possible for the paramagnetic current density was presented in Ref. 41. From Eq.…”

“…This reduced freedom for a choice of A xc when matching paramagnetic currents might also be the reason why we do not arrive at an evolution equation like in the previous case, but at a rigid balance equation like Eq. (41). Note that Eq.…”