dc.contributor.author Woods, Nicholas dc.date.accessioned 2022-02-07T02:18:08Z dc.date.available 2022-02-07T02:18:08Z dc.date.issued 2022-01-28 dc.date.submitted 2021-10-01 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/333701 dc.description.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. dc.rights Attribution 4.0 International (CC BY 4.0) dc.rights.uri https://creativecommons.org/licenses/by/4.0/ dc.subject Density Functional Theory dc.subject Numerical Analysis dc.subject Condensed Matter dc.subject Electronic Structure dc.subject Linear Response dc.subject Excitations dc.title Taming the Inverse and Forward Problems in Density Functional Theory dc.type Thesis dc.type.qualificationlevel Doctoral dc.type.qualificationname Doctor of Philosophy (PhD) dc.publisher.institution University of Cambridge dc.date.updated 2022-01-31T13:25:35Z dc.identifier.doi 10.17863/CAM.81121 rioxxterms.licenseref.uri https://creativecommons.org/licenses/by/4.0/ rioxxterms.type Thesis dc.publisher.college Clare pubs.funder-project-id EPSRC (1819412) cam.supervisor Payne, Mike cam.supervisor Hasnip, Phil cam.supervisor.orcid Payne, Michael [0000-0002-5250-8549] cam.depositDate 2022-01-31 pubs.licence-identifier apollo-deposit-licence-2-1 pubs.licence-display-name Apollo Repository Deposit Licence Agreement
﻿

### This item appears in the following Collection(s)

Except where otherwise noted, this item's licence is described as Attribution 4.0 International (CC BY 4.0)