Generalized Kohn–Sham iteration on Banach spaces

Abstract

A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introduce, in contrast to the common unregularized approach, a well-defined Kohn-Sham iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.

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

“…Yet in all those works the question of a limit density and corresponding KS potential was left open. 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 will be an important ingredient in the convergence proof: A bound on the curvature means the convex functional F ε cannot change from falling to rising too quickly, yielding a secure bound on the possible step length for descent. The regularized F ε is then differentiable and even has a continuous gradient ∇F ε (Fréchet differentiability) [23,Th. 9], something that will also become important in the convergence proof.…”

“…The assumption that E 0 is finite everywhere is trivially fulfilled in a finite-dimensional setting because E 0 is a finite sum. It is still kept here to connect more closely to standard DFT and CDFT, where E 0 is finite even in the infinite-dimensional setting, see Convergence proof.-We refer to the the first part of the proof of Theorem 12 in Laestadius et al [23] to show that the superdifferential ∂E 0 ε (v i+1 ) is everywhere nonempty because of E 0 finite, guaranteeing that the (regularized) ground-state problem in (b) always has at least one solution. The directional derivative of F ε + v at x i in direction x i+1 − x i can be rewritten by (a), [23,Lem.…”

“…Convergence of ∆Ei = ei − Eε(v) for a ring-lattice system with external potential v = cos(2θ) + 0.2 cos(θ) and different ε. The algorithm developed here is labeled "S" while the method "L" chooses the step length maximally [23,27].…”

Unique continuation for the magnetic Schrödinger equation

“…An interesting and recent development is also given in Reference where the existence of generalized Hohenberg‐Kohn theorems is further explored. A different route, where a Hohenberg‐Kohn result comes for free by virtue of the convex‐analytic properties of a regularized energy functional was taken in References . It was specifically implemented for CDFT in Reference and can even be used to prove convergence of the associated Kohn‐Sham iteration Scheme …”