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

We describe the clustered low-rank (CLR) framework for block-sparse and block-low-rank tensor representation and computation. The CLR framework exploits the tensor structure revealed by basis clustering; computational savings arise from low-rank compression of tensor blocks and performing block arithmetic in the low-rank form whenever beneficial. The precision is rigorously controlled by two parameters, avoiding ad-hoc heuristics, such as domains: one controls the CLR block rank truncation, and the other controls screening of small contributions in arithmetic operations on CLR tensors to propagate sparsity through expressions. As these parameters approach zero, the CLR representation and arithmetic become exact. As a pilot application, we considered the use of the CLR format for the order-2 and order-3 tensors in the context of the density fitting (DF) evaluation of the Hartree–Fock (exact) exchange (DF-K). Even for small systems and realistic basis sets, CLR-DF-K becomes more efficient than the standard DF-K approach, and it has significantly reduced asymptotic storage and computational complexities relative to the standard O ( N 3 ) and O ( N 4 ) DF-K figures. CLR-DF-K is also significantly more efficientall while negligibly affecting molecular energies and propertiesthan the conventional (non-DF) O ( N ) exchange algorithm for applications to medium-sized systems (on the order of 100 atoms) with diffuse Gaussian basis sets, a necessity for applications to negatively charged species, molecular properties, and high-accuracy correlated wave functions.