The Direct-Product Decomposition Approach for Symmetry Exploitation in Many-Body Methods in Case of Non-Abelian Point Groups

TL;DR

This paper extends the direct-product decomposition scheme for symmetry treatment in coupled-cluster calculations to non-Abelian point groups, enabling significant computational savings for large symmetric molecules.

Contribution

It introduces a method to incorporate non-Abelian symmetry into coupled-cluster calculations using a block structure approach and a mixed representation strategy.

Findings

Achieved over 20% computational savings compared to no symmetry treatment.

Demonstrated the method on NH3 and PH3 molecules with different basis sets.

Confirmed the feasibility of CC calculations on large symmetric molecules with non-Abelian symmetry.

Abstract

We demonstrate for the specific case of $C_{3v}$ how the direct-product decomposition scheme for the treatment of symmetry in coupled-cluster (CC) calculations can be extended to non-Abelian point groups. We show that for the two-electron integrals and CC amplitudes a block structure can be obtained by resolving the reducible products of two irreducible representations into their irreducible representations. To deal with the necessary resorts of the ordering of the two-electron integrals and amplitudes, spin-adaptation, and the O(M$^5$) contractions (with M as the number of basis functions) of a CC calculation, we suggest a strategy that uses both the reduced and non-reduced representation of the corresponding quantities and switches back and forth between them. While the reduced representations are the ones used in the O(M$^6$) contractions, the other steps are better carried out in the non-reduced representation. Our pilot implementation of the CC singles and doubles method confirms in test calculations for NH$_3$ and PH$_3$ using different basis sets that significant savings (of more than 20 compared to treatments without symmetry and about 5 compared to treatments using $C_s$ symmetry) are possible and suggest that the exploitation of non-Abelian symmetry would render CC computations on large highly symmetric molecules possible