Author ORCID Identifier

https://orcid.org/0000-0002-8542-8086

Date of Award

Summer 8-14-2026

Document Type

Thesis (Ph.D.)

Department or Program

Mathematics

First Advisor

James Daniel Whitfield

Second Advisor

Dimitrios Giannakis

Third Advisor

Yoonsang Lee

Abstract

Numerically solving the electronic structure problem is a fundamentally difficult problem due to the exponential growth in the dimension of the Hilbert space as the system size increases. In order to solve problems at a chemically relevant accuracy, both the choice of basis set and numerical method are important factors that are intrinsically connected.

In this thesis, we study the discretization and resulting compression of electronic Hamiltonians using diagonal basis sets. A diagonal basis set approximately diagonalizes the matrix and tensor representations of the one- and two-body potentials. This can reduce storage, simplify matrix-vector products, and lower the complexity of tensor network methods. Two important kinds of diagonal basis sets are studied: Discrete variable representation and gausslets.

We first give a background of quantum mechanics, using quantum computing and probability as a motivation. We then set up the electronic structure problem in both first and second quantization. Different ground state numerical methods, along with applications of basis set optimization, are discussed.

For the rest of the thesis, we focus on analysis of diagonal basis sets. We review two frameworks for diagonal approximations: Gaussian quadrature and the completeness, orthogonality, moment, and X-diagonalization (COMX) properties. These frameworks are used to clarify the connection between discrete variable representation (DVR) and gausslets. We analyze and compare separable approximations of the Coulomb potential, giving a simple parameter optimization method for term reduction. We also show how using a separable potential, along with the diagonal approximation, can significantly reduce memory cost for Fock matrix-vector products.

Finally, we end this thesis with a numerical study of both one- and three-dimensional quantum chemistry. One-dimensional chemistry serves as a model system, where we demonstrate a clear advantage for diagonal basis sets over one-dimensional atomic orbitals. We then investigate three-dimensional systems, providing useful benchmarks for even-spaced discretization. We conclude with a discussion of current challenges and potential solutions for scaling diagonal basis sets, which is an early but promising area of research.

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