Tensor representation techniques in post-Hartree–Fock methods: matrix product state tensor format

Abstract

In this proof-of-principle study, we discuss the application of various tensor representation formats and their implications on memory requirements and computational effort for tensor manipulations as they occur in typical post-Hartree–Fock (post-HF) methods. A successive tensor decomposition/rank reduction scheme in the matrix product state (MPS) format for the two-electron integrals in the AO and MO bases and an estimate of the t ₂ amplitudes as obtained from second-order many-body perturbation theory (MP2) are described. Furthermore, the AO–MO integral transformation, the calculation of the MP2 energy and the potential usage of tensors in low-rank MPS representation for the tensor contractions in coupled cluster theory are discussed in detail. We are able to show that the overall scaling of the memory requirements is reduced from the conventional N ⁴ scaling to approximately N ³ and the scaling of computational effort for tensor contractions in post-HF methods can be reduced to roughly N ⁴ while the decomposition itself scales as N ⁵. While efficient algorithms with low prefactor for the tensor decomposition have yet to be devised, this ansatz offers the possibility to find a robust approximation with low-scaling behaviour with system and basis-set size for post-HF ab initio methods.

Keywords:

• electronic structure methods
• post-Hartree–Fock
• configuration interaction
• coupled cluster
• tensor representation
• matrix product state
• canonical polyadic product

Acknowledgements

This paper is dedicated to Professor W. Kutzelnigg on the occasion of his 80th birthday. Professor Kutzelnigg has been a shining example for pure and rigorous scientific work, innovative power and formal elegance. We would like to thank him for many inspiring discussions and wish him all the best for years to come.