Taming the Inverse and Forward Problems in Density Functional Theory

Abstract

The `forward problem' of ground-state density functional theory (DFT) constitutes finding the ground-state density $n(x)$ that minimises a Kohn-Sham total energy functional defined using some exchange-correlation (xc) functional $E_\text{xc}[n]$. Towards this end, the associated Euler-Lagrange equations, i.e. the Kohn-Sham equations, are often solved in practice, which demand a procedure that iterates an initial guess density to a \textit{self-consistent} density (the solution). A new framework is presented for evaluating the performance of self-consistent field methods in Kohn–Sham DFT. The aims of this work are two-fold. First, we explore the properties of Kohn–Sham DFT as it pertains to the convergence of self-consistent field iterations. Sources of inefficiencies and instabilities are identified, and methods to mitigate these difficulties are discussed. Second, we introduce a framework to assess the relative utility of algorithms, comprising a representative benchmark suite of over fifty Kohn–Sham simulation inputs, the \textsc{scf}-$x_n$ suite. This provides a new tool to develop, evaluate and compare new algorithms in a fair, well-defined and transparent manner.

The `inverse problem' of time-dependent (ground-state) DFT constitutes finding the time-(in)dependent Kohn-Sham potential $v_\text{KS}(x,t)$ that yields a given reference density $n(x,t)$ upon solution of the time-(in)dependent Kohn-Sham equations. This inverse map can be unstable, particularly in the presence of low-density regions, and thus methods are designed to alleviate numerical difficulties in the present context. On the other hand, linear response time-dependent DFT centres around the first-order response of the xc potential due to perturbing densities -- the so-called xc kernel $f_\text{xc}(x,x',\omega)$. Computing exact xc kernels represents a linearised version of the previous inverse problem: this state of affairs, whilst still challenging, is more manageable. Methods to ensure the robustness of exact numerical $f_\text{xc}$ computations are set out. In the context of inhomogenous one-dimensional finite systems, these developments permit an improved understanding of $f_\text{xc}$ in itself, and in relation to various applications, such as the optical spectrum and ground-state correlation energies using the adiabatic connection fluctuation-dissipation theorem. We expect that certain key insights derived from this work will assist in the informed development of improved functional approximations.