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Li, Kangbo Damle, Anil Kozen, Dexter Scheinberg, Katya DiStasio, Robert 2025-01-14T20:00:00Z 2025-01-14T20:00:00Z 2024-08 ProQuest Submission ID: 14577 ProQuest Publication ID: 31488629 https://hdl.handle.net/1813/116510 https://doi.org/10.7298/9qz4-sr38 16611655 122 pages The goal of this thesis is to document three of the ideas from my PhD study related to quantum Chemistry. These ideas are alternative mathematical foundations to their respective problems. Two of the three ideas come with a working implementation that empirically demonstrates significant advantages over the state of the art with no trade-offs or caveats. The implementation of the other idea has not gathered enough evidence to show a practical advantage, but it appears promising. Chapter one is the theory of combinatory differentiation, which practically brings symbolic differentiation up to speed with algorithmic differentiation and enables the analytic automation of the backpropagation and differential tensor calculus. At the center of this model of differentiation is a serendipitous connection between the combinatory logic and path integrals through a little bit of differential geometry captured in just two equations. This work started as an attempt to automate the differentiation process in quantum mechanics using fundamental concepts in programming language theories. It turned into a theoretical model when the connection between the combinators and the path integral emerged during the first few implementation attempts. Chapter two challenges the self-consistent field (SCF) narrative thatuncorrelated electrons occupy the canonical orbitals, which are the eigenstates of a so-called effective mean-field Hamiltonian. We argue that this pseudo-physical interpretation attached to the SCF is appealing but not physical. In particular, the electron delocalization is a numerical artifact camouflaged as a quantum mechanical phenomenon under the SCF narrative. We show that a manifold HF with localization avoids delocalizing the electrons at all times without any compromise to the energy. This approach points a way to reliably overcome the cubic scaling of independent electron theories through a divide and conquer strategy. Chapter three is a reformulation of the Wannier localization problem with a more consistent Physical model and a more appropriate mathematical optimization framework. This reformulation has lead to a simpler theory that practically accelerates Wannier90 by about $100 \times$ on average when starting from a random initial guess. This project was started as a digression from another project that extends the selected columns of the density matrix (SCDM) algorithm to localize the virtual orbitals. We never returned to writing up the original project even though many questions has been answered. en Attribution 4.0 International https://creativecommons.org/licenses/by/4.0/ combinatory logic differentiation manifold optimization wannier functions THE THEORY OF COMBINATORY DIFFERENTIATION AND LOCALITY IN QUANTUM CHEMISTRY dissertation or thesis https://newcatalog.library.cornell.edu/catalog/16611655 application/pdf Computer Science Cornell University Doctor of Philosophy Ph. D., Computer Science https://hdl.handle.net/1813/59810.2 Publication Cornell Theses and Dissertations Li, Kangbo 5893a6ea-7af3-41d7-abc6-04bcd26ab5df open.access open.access <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="03bc5bfb-51ab-4f1f-872a-ba4028f47eea"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Language>en</Language> <Title>THE THEORY OF COMBINATORY DIFFERENTIATION AND LOCALITY IN QUANTUM CHEMISTRY</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2024-08</PublicationDate> <DOI>https://doi.org/10.7298/9qz4-sr38</DOI> <Authors> <Author> <DisplayName>Li, Kangbo</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <OrgUnit /> </Publisher> </Publishers> <License>https://creativecommons.org/licenses/by/4.0/</License> <Keyword>combinatory logic</Keyword> <Keyword>differentiation</Keyword> <Keyword>manifold optimization</Keyword> <Keyword>wannier functions</Keyword> <Abstract>The goal of this thesis is to document three of the ideas from my PhD study related to quantum Chemistry. These ideas are alternative mathematical foundations to their respective problems. Two of the three ideas come with a working implementation that empirically demonstrates significant advantages over the state of the art with no trade-offs or caveats. The implementation of the other idea has not gathered enough evidence to show a practical advantage, but it appears promising. Chapter one is the theory of combinatory differentiation, which practically brings symbolic differentiation up to speed with algorithmic differentiation and enables the analytic automation of the backpropagation and differential tensor calculus. At the center of this model of differentiation is a serendipitous connection between the combinatory logic and path integrals through a little bit of differential geometry captured in just two equations. This work started as an attempt to automate the differentiation process in quantum mechanics using fundamental concepts in programming language theories. It turned into a theoretical model when the connection between the combinators and the path integral emerged during the first few implementation attempts. Chapter two challenges the self-consistent field (SCF) narrative thatuncorrelated electrons occupy the canonical orbitals, which are the eigenstates of a so-called effective mean-field Hamiltonian. We argue that this pseudo-physical interpretation attached to the SCF is appealing but not physical. In particular, the electron delocalization is a numerical artifact camouflaged as a quantum mechanical phenomenon under the SCF narrative. We show that a manifold HF with localization avoids delocalizing the electrons at all times without any compromise to the energy. This approach points a way to reliably overcome the cubic scaling of independent electron theories through a divide and conquer strategy. Chapter three is a reformulation of the Wannier localization problem with a more consistent Physical model and a more appropriate mathematical optimization framework. This reformulation has lead to a simpler theory that practically accelerates Wannier90 by about $100 \times$ on average when starting from a random initial guess. This project was started as a digression from another project that extends the selected columns of the density matrix (SCDM) algorithm to localize the virtual orbitals. We never returned to writing up the original project even though many questions has been answered.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>