Rank reduction methods in electronic structure theory

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Parrish, Robert M.
Brown, Kenneth R.
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Quantum chemistry is plagued by the presence of high-rank quantities, stemming from the N-body nature of the electronic Schrödinger equation. These high-rank quantities present a significant mathematical and computational barrier to the computation of chemical observables, and also drastically complicate the pedagogical understanding of important interactions between particles in a molecular system. The application of physically-motivated rank reduction approaches can help address these to problems. This thesis details recent efforts to apply rank reduction techniques in both of these arenas. With regards to computational tractability, the representation of the 1/r Coulomb repulsion between electrons is a critical stage in the solution of the electronic Schrödinger equation. Typically, this interaction is encapsulated via the order-4 electron repulsion integral (ERI) tensor, which is a major bottleneck in terms of generation, manipulation, and storage. Many rank reduction techniques for the ERI tensor have been proposed to ameliorate this bottleneck, most notably including the order-3 density fitting (DF) and pseudospectral (PS) representations. Here we detail a new and uniquely powerful factorization - tensor hypercontraction (THC). THC decomposes the ERI tensor as a product of five order-2 matrices (the first wholly order-2 compression proposed for the ERI) and offers great flexibility for low-scaling algorithms for the manipulations of the ERI tensor underlying electronic structure theory. THC is shown to be physically-motivated, markedly accurate, and uniquely efficient for some of the most difficult operations encountered in modern quantum chemistry. On the front of chemical understanding of electronic structure theory, we present our recent work in developing robust two-body partitions for ab initio computations of intermolecular interactions. Noncovalent interactions are the critical and delicate forces which govern such important processes as drug-protein docking, enzyme function, crystal packing, and zeolite adsorption. These forces arise as weak residual interactions leftover after the binding of electrons and nuclei into molecule, and, as such, are extremely difficult to accurately quantify or systematically understand. Symmetry-adapted perturbation theory (SAPT) provides an excellent approach to rigorously compute the interaction energy in terms of the physically-motivated components of electrostatics, exchange, induction, and dispersion. For small intermolecular dimers, this breakdown provides great insight into the nature of noncovalent interactions. However, SAPT abstracts away considerable details about the N-body interactions between particles on the two monomers which give rise to the interaction energy components. In the work presented herein, we step back slightly and extract an effective 2-body interaction for each of the N-body SAPT terms, rather than immediately tracing all the way down to the order-0 interaction energy. This effective order-2 representation of the order-N SAPT interaction allows for the robust assignment of interaction energy contributions to pairs of atoms or functional groups (the A-SAPT or F-SAPT partitions), allowing one to discuss the interaction in terms of atom- or functional-group-pairwise interactions. These A-SAPT and F-SAPT partitions can provide deep insight into the origins of complicated noncovalent interactions, e.g., by clearly shedding light on the long-contested question of the nature of the substituent effect in substituted sandwich benzene dimers.
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