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

Malte Hellmann, Jürgen Gauss

TL;DR

The paper tackles exploiting non-Abelian symmetry in correlated electronic structure calculations by extending the Abelian direct-product decomposition (DPD) approach to the $C_{3v}$ point group. It introduces a dual-representation strategy that uses both reduced and non-reduced representations for four-index quantities, with a dedicated reduction routine to handle transformations, spin-adaptation, and ${O(M^5)}$ contractions in CC methods. A pilot implementation, QUENA, demonstrates substantial computational savings for CCSD in NH$_3$ and PH$_3$ compared with both no symmetry and $C_s$ symmetry, indicating feasibility for large, highly symmetric molecules. These results establish a foundation for general non-Abelian symmetry exploitation in quantum chemistry and outline paths toward broader group coverage and extensions to perturbative triples (CCSD(T)).

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

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

TL;DR

The paper tackles exploiting non-Abelian symmetry in correlated electronic structure calculations by extending the Abelian direct-product decomposition (DPD) approach to the point group. It introduces a dual-representation strategy that uses both reduced and non-reduced representations for four-index quantities, with a dedicated reduction routine to handle transformations, spin-adaptation, and contractions in CC methods. A pilot implementation, QUENA, demonstrates substantial computational savings for CCSD in NH and PH compared with both no symmetry and symmetry, indicating feasibility for large, highly symmetric molecules. These results establish a foundation for general non-Abelian symmetry exploitation in quantum chemistry and outline paths toward broader group coverage and extensions to perturbative triples (CCSD(T)).

Abstract

We demonstrate for the specific case of 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) 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) 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 and PH using different basis sets that significant savings (of more than 20 compared to treatments without symmetry and about 5 compared to treatments using symmetry) are possible and suggest that the exploitation of non-Abelian symmetry would render CC computations on large highly symmetric molecules possible
Paper Structure (12 sections, 28 equations, 2 figures, 2 tables)

This paper contains 12 sections, 28 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Symmetry in Quantum-Chemical Calculations
  3. General Concepts
  4. Direct-Product Decomposition in the Case of Abelian Point Groups
  5. Extension of the Direct-Product Decomposition Scheme to $C_{3v}$ Symmetry
  6. Non-Reduced and Reduced Representations
  7. Resort of Reduced Representations of Four-Index Quantities
  8. Spin-Adaptation of Reduced Representations of Four-Index Quantities
  9. O($M^5$) Contractions in CCD and CCSD Computations
  10. Implementation and Computational Details
  11. Exploratory Computations on NH$_3$ and PH$_3$
  12. Conclusions and Outlook

Figures (2)

  • Figure 1: Slide from one of the author's talk (J.G. together with John F. Stanton, John D. Watts, and Rodney J. Bartlett) at the 31rst Sanibel Symposium 1991 in St. Augustine, (Florida, USA) explaining the block-by-block multiplication in the PPL terms of CCSD within the DPD scheme.
  • Figure 2: Simultaneous usage of non-reduced and reduced representations in CC computations exploiting non-Abelian symmetry.