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Block-Sparse Tensor Algebra for Electronic Structure of Large Molecules

Banff International Research Station for Mathematical Innovation and Discovery

Exploitable structure of operators and functions of interest to electronic structure can be roughly classiffed into two classes: factorizability into components of lower rank and/or order (rank sparsity) and small magnitude when expressed in an appropriate basis (element sparsity). Recent evidence suggests that in addition to useful global rank-sparsity (e.g. coupled-cluster style factorization) the local rank-sparsity plays a key role in eliminating the gap between the dense (small molecule) and element-sparse (very large molecule) regimes. We introduce a simple tensor format (Clustered Low Rank, CLR) that can exploit both types of sparsity effectively. The format is a novel combination of several simple ideas: basis clustering, tensor-product tiling, and low-rank approximation to tensor tiles. The format is not specific to electronic structure, but it can be constrained to a surprising number of seemingly unrelated numerical approximations in electronic structure that recently appeared in the literature. To demonstrate practical computation in this format we present a competitive implementation of Hartree-Fock method with density-fitting-style factorization of Coulomb two- electron integrals that is free of ad hoc models of sparsity (e.g., domains) and avoids numerical artifacts of atomic density fitting. Use of the CLR format for the order-2 and order-3 tensors that appear in the context of density fitting (DF) evaluation of the Hartree-Fock (HF) exchange operator significantly reduced the storage and computational complexities below their standard O(N3) and O(N4) figures. Even for relatively small systems and realistic basis sets CLR-based DF HF becomes more efficient than the standard DF approach, and significantly more efficient than the conventional non-DF HF, while negligibly affecting molecular energies and properties.

Mathematics; Quantum theory; Partial differential equations; Applied computer science

video/mp4

Author affiliation: Virginia Tech

2016-07-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58536

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/