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Clustered Low Rank (CLR) framework for block-sparse and block-low-rank tensorrepresentation and computation is described. The CLR framework depends on 2parameters that control precision: one controlling the CLR block ranktruncation and another that controls screening of small contributions inarithmetic operations on CLR tensors. As these parameters approach zero CLRrepresentation and arithmetic become exact. There are no other ad-hocheuristics, such as domains. Use of the CLR format for the order-2 and order-3tensors that appear in the context of density fitting (DF) evaluation of theHartree-Fock (exact) exchange significantly reduced the storage andcomputational complexities below their standard $\mathcal{O}(N^3)$ and$\mathcal{O}(N^4)$ figures. Even for relatively small systems and realisticbasis sets CLR-based DF HF becomes more efficient than the standard DFapproach, and significantly more efficient than the conventional non-DF HF,while negligibly affecting molecular energies and properties.

Clustered Low-Rank Tensor Format: Introduction and Application to Fast Construction of Hartree-Fock Exchange