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 $L^1\cap L^{3/2}$ 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.

References in corpus (20)

• Perspective on density functional theory
• Nonuniqueness and derivative discontinuities in density-functional theories for current-carrying and superconducting systems
• The choice of basic variables in current-density functional theory
• Arbitrary Choice of Basic Variables in Density Functional Theory. I. Formalism
• Kohn-Sham calculations with the exact functional
• Differentiable but exact formulation of density-functional theory
• Time-dependent current density functional theory on a lattice
• Guaranteed convergence of the Kohn-Sham equations
• U(1)$\times$SU(2) Gauge Invariance Made Simple for Density Functional Approximations
• Current Densities in Density Functional Theory
• Guaranteed Convergence of a Regularized Kohn-Sham Iteration in Finite Dimensions
• One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
• Uniform magnetic fields in density-functional theory
• Fermion $N$-representability for prescribed density and paramagnetic current density
• Generalized Kohn-Sham iteration on Banach spaces
• Non-existence of a Hohenberg-Kohn Variational Principle in Total Current Density Functional Theory
• Ground-State Quantum-Electrodynamical Density-Functional Theory
• Density-functional theory for internal magnetic fields
• Density-Wavefunction Mapping in Degenerate Current-Density-Functional Theory
• A bivariate view of Kohn-Sham iteration and the case for potential mixing

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• The structure of the density-potential mapping. Part II: Including magnetic fields
• Exchange energies with forces in density-functional theory
• Density-potential inversion from Moreau-Yosida regularization
• Geometry of Degeneracy in Potential and Density Space
• Unique Continuation for the Magnetic Schrödinger Equation
• Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer
• Kohn-Sham inversion with mathematical guarantees
• Revisiting density-functional theory of the total current density
• Adaptation of Moreau-Yosida regularization to the modulus of convexity
• Physical spin torques from exactly constrained exchange-correlation torques
• Regularised density-potential inversion for periodic systems: application to exact exchange in one dimension
• Perspective on Moreau-Yosida Regularization in Density-Functional Theory