Taming the Inverse and Forward Problems in Density Functional Theory

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<sub>\</sub>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<sub>n</sub> 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',ω). 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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2022

Author dc:creator

• Woods, Nicholas

Advisors dc:contributor.advisor

• Payne, Mike
• Hasnip, Phil

• Density Functional Theory
• Numerical Analysis
• Condensed Matter
• Electronic Structure
• Linear Response
• Excitations

Licence

• https://creativecommons.org/licenses/by/4.0/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.81121

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/333701