  Clustered Low Rank (CLR) framework for block-sparse and block-low-rank tensor
representation and computation is described. The CLR framework depends on 2
parameters that control precision: one controlling the CLR block rank
truncation and another that controls screening of small contributions in
arithmetic operations on CLR tensors. As these parameters approach zero CLR
representation and arithmetic become exact. There are no other ad-hoc
heuristics, such as domains. 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 (exact) exchange significantly reduced the storage and
computational complexities below their standard $\mathcal{O}(N^3)$ and
$\mathcal{O}(N^4)$ 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.
