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Block-Sparse Tensor Algebra for Electronic Structure of Large Molecules Valeev, Edward


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.

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