On extending and optimising the direct product decomposition

Abstract

The concept of the direct product decomposition (DPD) is extended to arbitrary tensors while maintaining the same theoretical reduction in storage and computation. Additionally, the structure of the DPD as introduced by Gauss and Stanton is shown to be but one of a family of direct product decompositions which may be visualised using graphs. One particular member of this family is also shown to be critically important in relating the DPD and symmetry blocking approaches. Lastly, an implementation of tensor contraction using this extended DPD based on recent work in dense tensor contraction is presented, showing how the particular DPD used to represent the tensors in memory or on disk may be divorced from the optimal DPD used for a particular tensor contraction. The performance of the new algorithm is benchmarked by interfacing with the CFOUR programme suite, where significant speedups for CCSD calculations are observed.

GRAPHICAL ABSTRACT

Keywords:

• Molecular symmetry
• coupled cluster
• tensor contraction
• direct product decomposition

Acknowledgements

Portions of this work were completed while DAM was supported by an Arnold O. Beckman Postdoctoral Fellowship. Work at Southern Methodist University was supported by a generous startup package, and using computational resources provided by the SMU Center for Scientific Computation.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Devin A. Matthews http://orcid.org/0000-0003-2795-5483

Notes

1. The separably-degenerate (cyclic) groups are categorised with the truly degenerate groups here, as they require complex-valued irreps and do not satisfy the self-inverse property. The remaining non-degenerate groups are all isomorphic to ${\mathbb{Z}}_2^n$ for n=0,1,2,3.

2. For ease of indexing the original tensor should be packed in some DPD format, although it does not have to match the DPD partitioning used for the contraction.