{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/333701"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/333701","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"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.","abstract_html":"The `forward problem&#x27; 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 <span class=\"etd-inline-math\">E<sub>\\</sub>text{xc}[n]</span>. 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}-<span class=\"etd-inline-math\">x<sub>n</sub></span> suite. This provides a new tool to develop, evaluate and compare new algorithms in a fair, well-defined and transparent manner. The `inverse problem&#x27; of time-dependent (ground-state) DFT constitutes finding the time-(in)dependent Kohn-Sham potential <span class=\"etd-inline-math\">v<sub>\\</sub>text{KS}(x,t)</span> 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 <span class=\"etd-inline-math\">f<sub>\\</sub>text{xc}(x,x&#x27;,&omega;)</span>. 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 <span class=\"etd-inline-math\">f<sub>\\</sub>text{xc}</span> computations are set out. In the context of inhomogenous one-dimensional finite systems, these developments permit an improved understanding of <span class=\"etd-inline-math\">f<sub>\\</sub>text{xc}</span> 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.","abstract_has_math":true,"creators":["Woods, Nicholas"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Payne, Mike","Hasnip, Phil"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-28","date_published":"2022-01-28","updated_at":"2026-07-22T22:24:01Z","subjects":["Density Functional Theory","Numerical Analysis","Condensed Matter","Electronic Structure","Linear Response","Excitations"],"languages":["eng"],"rights":[],"rights_urls":["https://creativecommons.org/licenses/by/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.81121","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Payne, Mike","Hasnip, Phil"]},{"key":"dc:creator","label":"Author","values":["Woods, Nicholas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2022-01-28"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/333701"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Density Functional Theory","Numerical Analysis","Condensed Matter","Electronic Structure","Linear Response","Excitations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://creativecommons.org/licenses/by/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.81121"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/795e70a3-d6e3-482a-ba98-265107797c18/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["8ca5ede3f032969dba35e8e2c4bad4ad"]},{"key":"dc:title","label":"Title","values":["Taming the Inverse and Forward Problems in Density Functional Theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Payne, Mike","Hasnip, Phil"],"dc:creator":["Woods, Nicholas"],"dc:date.issued":["2022-01-28"],"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:format.checksum.md5":["8ca5ede3f032969dba35e8e2c4bad4ad"],"dc:identifier.doi":["10.17863/CAM.81121"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/795e70a3-d6e3-482a-ba98-265107797c18/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/333701"],"dc:rights":["https://creativecommons.org/licenses/by/4.0/"],"dc:subject":["Density Functional Theory","Numerical Analysis","Condensed Matter","Electronic Structure","Linear Response","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)"]},"updated_at":"2026-07-22T22:24:01Z"}