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

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