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 <i>C</i>_3<i>v</i>, 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 re-sorts of the ordering of the two-electron integrals and amplitudes, spin adaptation, and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> contractions (with <i>M</i> as the number of basis functions) of a CC calculation, we suggest a strategy that uses both the reduced and nonreduced representations of the corresponding quantities and switches back and forth between them. While the reduced representations are the ones used in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> contractions, the other steps are better carried out in the nonreduced representation. Our pilot implementation of the CC singles and doubles method confirms in test calculations for NH₃ and PH₃ using different basis sets that significant savings (of more than 20 compared to treatments without symmetry and about 5 compared to treatments using <i>C</i>_s symmetry) are possible and these findings suggest that the exploitation of non-Abelian symmetry would render CC computations on large, highly symmetric molecules possible.