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Ryley, Matthew S (2020) Variational Solutions in Orbital Free Density Functional Theory. PhD thesis, University of Nottingham.

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Abstract

This work focusses on testing and developing methods to find solutions to the variational problem at the heart of orbital free density functional theory (OF-DFT). OF-DFT is an inherently linear scaling quantum chemical method which can, in theory, be used to simulate systems with millions of atoms. However, there is a limited choice of OF-DFT codes which are suitable for the development of the theory for chemical systems. In this work we compare and contrast three methods: the Lopez-Acevedo scheme[1]; the Chan, Cohen and Handy Scheme[2,3] (CCH); and the trust-region image method (TRIM) scheme[4]; We find that the scheme developed in this work --- the TRIM scheme --- offers the most efficient methodology for converging the energy for a wide range of functionals in an all electron context for finite chemical systems.

In Chapter 1 some mathematical topics are introduced which are required to understand how the foundations of density functional theory (DFT) built upon convex analysis underpins the variational principles of CCH and TRIM. In Chapter 2 we introduce electronic structure theory and discuss the theoretical foundations of DFT. We also include a discussion on Kohn-Sham DFT [5].

Following on from this in Chapter 3 we discuss OF-DFT and introduce the concept of the orbital free approximation of the non-interacting kinetic energy functional (OF-KEF). In addition, explanations of various forms of OF-KEFs found in the literature are given.

We will then shift focus to discussing some established variational schemes --- Lopez-Acevedo and CCH --- in the literature and discuss how we modified CCH to converge the energy for a wider range of OF-KEFs than had been previously reported. The last section of Chapter 3 will discuss the theory behind the TRIM method we have developed in this work. We will see that the TRIM scheme relies on the fact that the optimisation problem at the heart of OF-DFT is a saddle point optimisation problem.

In Chapter 4 a detailed description of the CCH scheme is given and how we implemented this scheme in order to converge the energy for a wide range of OF-KEFs. It is shown that most functionals predict very poor energies and densities. Furthermore, most OF-KEFs do not predict molecular binding. In Chapter 5 we discuss the form of the potentials generated by the OF-KEFs. It is shown that the sum of the kinetic potential and the effective potential approaches a constant in an oscillating manner, point-wise in space, when using Gaussian basis sets. This means if one wants to compute forces there is a Pulay like term in the equations.

In addition, the balances between the potentials in the Euler equation are examined. This provides an explanation for the small chemical potential values one computes using these OF-KEFs.

In Chapter 6 the TRIM scheme is presented, and our implementation of the scheme is discussed in detail. A demonstration of why the TRIM scheme is more efficient than both the Lopez-Acevedo and CCH schemes is provided. Furthermore, it is highlighted that the importance of the guess density for molecular systems in ensuring convergence is optimal.

In Chapter 7 we examine a variation principle at the interface of wave-function and density functional theories. A recently proposed variation principal[6] is examined and is shown[7] that it can be expressed in terms of the well-known Lieb functional.

The equivalence between the information obtained from the two approaches is illustrated numerically by their implementation in a common framework.

In Chapter 8 a summary of the work is presented and a view on the future of research in OF-DFT is given. These views are supplemented with results from preliminary investigations on these future research directions.

Item Type:	Thesis (University of Nottingham only) (PhD)
Supervisors:	Teale, Andrew M
Keywords:	Orbital Free Density Functional Theory, Optimisation Techniques, Theoretical Chemistry, Density functionals, Atomic orbitals
Subjects:	Q Science > QD Chemistry > QD450 Physical and theoretical chemistry
Faculties/Schools:	UK Campuses > Faculty of Science > School of Chemistry
Item ID:	60023
Depositing User:	Ryley, Matthew
Date Deposited:	16 Jun 2025 14:08
Last Modified:	16 Jun 2025 14:08
URI:	https://eprints.nottingham.ac.uk/id/eprint/60023

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