Guaranteed Convergence of the Kohn-Sham Equations

Abstract

A sufficiently damped iteration of the Kohn-Sham equations with the exact functional is proven to always converge to the true ground-state density, regardless of the initial density or the strength of electron correlation, for finite Coulomb systems. We numerically implement the exact functional for one-dimensional continuum systems and demonstrate convergence of the damped KS algorithm. More strongly correlated systems converge more slowly. The Kohn-Sham (KS) approach employs a fictitious system of non-interacting electrons, defined to have the same density as the interacting system of interest. The potential characterizing this KS system is unique if it exists [9]. Because the KS potential is a functional of the density, in practice one must search for the density and KS potential together using an iterative, self-consistent scheme [10]. The converged density is in principle the ground-state density of the original, interacting system, whose ground-state energy is a functional of this density.Motivated by concerns of convergence and existence, we have been performing KS calculations with the exact functional for one-dimensional (1d) continuum systems [11,12]. Even when correlations are strong, we never find a density whose KS potential does not exist, consistent with the results of Ref. [13]. Nor do we find any system where the KS scheme does not converge, although convergence can slow by orders of magnitude as correlation is increased, just as in approximate calculations [4,5].Exact statements about the unknown density functional inform the construction of all successful DFT approximations [14][15][16][17]. More importantly, they distinguish between what a KS-DFT calculation can possibly do, and what it cannot. The most notorious example is the demonstration that the KS band-gap of a semiconductor does not equal the true charge gap, even when the exact functional is used [11,18]. Our key result is an analytic proof that a simple algorithm guarantees convergence of the KS equations for all systems, weakly or strongly correlated, independent of the starting point. Thus multiple stationary points and failures to converge are artifacts of approximate functionals. Studies of convergence are well-known in applied mathematics; but almost all concern simple approximations, such as LDA [19], , etc., and not those in current use in many calculations.The basic idea lies in a single step of the KS scheme, which proceeds from an input density to produce an output density. For a strongly correlated system as in Fig. 1.a, the output density can differ strongly from the input density, and be further from the true ground-state density. Nevertheless, by proving that the initial slope is always negative as in Fig. 1.b, we show there is always a linear combination of the input and output densities that lowers the energy. By sufficiently damping each KS step, the energy is always reduced each iteration, yielding the ground-state density and energy to within a given arXiv:1305.2967v2 [cond-mat.mtrl-sci]

Electron avoidance: A nonlocal radius for strong correlation

“…(4) [55][56][57][58][59]. The ground-state energy and density are then obtained through a minimization over reasonable densities [54] integrating to a certain desired particle number N :…”

“…The functional derivative of Eq. (7) reveals a gradientdescent procedure for minimizing E v [n] [59], and leads to a set of equations which must be solved self consistently for the electron density n(r):…”

Exact maps in density functional theory for lattice models

“…The wavefunction-to-density map has been fully realized for fixed particle number, i.e. for the twoelectron singlet case by [54,55] and for the more-electron case by [56,57]. In principle, a small system size of the model used in this work allows to graphically illustrate the complete density-to-wavefunction map in the full Fock space of the system, and can be used to show how features such as the intra-system steepening and the inter-system derivative discontinuity of the density-to-potential map appear in the density-to-wavefunction map.…”

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

“…A simpler iterative algorithm could use a fixed t in all iterations, as was done in Ref. 8. To explore this possibility, we fixed a conservative value t = 0.05 for the damping parameter.…”

“…5 Here, both the unknown density of the full system and the effective Kohn-Sham (KS) potential for the non-interacting system are solved for in an iterative manner. Even though the important question of convergence of this procedure has been addressed in several works, [6][7][8][9][10] it has only very recently been answered positively for finite-dimensional settings. 11 The motivation to include current densities and not just the particle density is to obtain a universal functional modelling the internal energy of magnetic systems.…”