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Richardson-Gaudin states of non-zero seniority I: matrix elements

Paul A. Johnson

TL;DR

This work extends the RG-state framework beyond seniority zero by deriving the matrix elements of a general two-body (Coulomb) operator between RG states of seniorities zero, two, and four. All couplings are shown to be expressible via cofactors of the effective overlap matrix $J$ (and its subblocks like $J(ab)$, $J(abcd)$), with norms and overlaps handled through eta factors and EBV-based determinants; this yields numerically stable and parallelizable expressions. Proof-of-principle calculations demonstrate that a single-reference CI built on RG states delivers results comparable to seniority-based CI at substantially reduced cost, suggesting a viable path to excitation-based RGCI. The study also sets the stage for developing Slater-Condon rules in the next manuscript and discusses the potential for extending this framework to spin-changing operators via Wigner-Eckart theory, with H$_6$ (and limited H$_8$) benchmarks supporting the method's validity.

Abstract

Seniority-zero wavefunctions describe bond-breaking processes qualitatively. As eigenvectors of a model Hamiltonian, Richardson-Gaudin states provide a clear physical picture and allow for systematic improvement via standard single reference approaches. Until now, this treatment has been done in the seniority-zero channel. In this manuscript, the corresponding states with higher seniorities are identified, and their couplings through the Coulomb Hamiltonian are computed. In every case, the couplings between the states are computed from the cofactors of their effective overlap matrix. Proof of principle calculations demonstrate that a single reference configuration interaction is comparable with seniority-based configuration interaction computations at a substantially reduced cost. The next manuscript in this series will identify the corresponding Slater-Condon rules and make the computations feasible.

Richardson-Gaudin states of non-zero seniority I: matrix elements

TL;DR

This work extends the RG-state framework beyond seniority zero by deriving the matrix elements of a general two-body (Coulomb) operator between RG states of seniorities zero, two, and four. All couplings are shown to be expressible via cofactors of the effective overlap matrix (and its subblocks like , ), with norms and overlaps handled through eta factors and EBV-based determinants; this yields numerically stable and parallelizable expressions. Proof-of-principle calculations demonstrate that a single-reference CI built on RG states delivers results comparable to seniority-based CI at substantially reduced cost, suggesting a viable path to excitation-based RGCI. The study also sets the stage for developing Slater-Condon rules in the next manuscript and discusses the potential for extending this framework to spin-changing operators via Wigner-Eckart theory, with H (and limited H) benchmarks supporting the method's validity.

Abstract

Seniority-zero wavefunctions describe bond-breaking processes qualitatively. As eigenvectors of a model Hamiltonian, Richardson-Gaudin states provide a clear physical picture and allow for systematic improvement via standard single reference approaches. Until now, this treatment has been done in the seniority-zero channel. In this manuscript, the corresponding states with higher seniorities are identified, and their couplings through the Coulomb Hamiltonian are computed. In every case, the couplings between the states are computed from the cofactors of their effective overlap matrix. Proof of principle calculations demonstrate that a single reference configuration interaction is comparable with seniority-based configuration interaction computations at a substantially reduced cost. The next manuscript in this series will identify the corresponding Slater-Condon rules and make the computations feasible.
Paper Structure (34 sections, 2 theorems, 227 equations, 7 figures, 10 tables)

This paper contains 34 sections, 2 theorems, 227 equations, 7 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Electron pairs
  4. su(2)
  5. sp(N)
  6. RG states
  7. Norms and scalar products
  8. Seniority-zero
  9. Seniority-two
  10. 0 - 2 coupling
  11. 2 - 2 coupling
  12. 2 shared blocked levels
  13. 1 shared blocked level
  14. No shared blocked levels
  15. Seniority-four
  16. ...and 19 more sections

Key Result

Lemma 3.1

For $z$ an arbitrary complex number distinct from $\{\varepsilon\}$, the diagonal rank-N update to the determinant of $J$ is equivalent to the rank-1 update in terms of the vectors

Figures (7)

  • Figure 1: Cyclic labelling of seniority-four states: for $\omega[(ab)(cd)] = 1$ or $\omega[(ab)(cd)] = 3$clockwise if $(a < b \;\text{and}\; c<d)$ or $(a > b \;\text{and}\; c>d)$, otherwise counterclockwise. For $\omega[(ab)(cd)] = 2$counterclockwise if $(a < b \;\text{and}\; c<d)$ or $(a > b \;\text{and}\; c>d)$, otherwise clockwise.
  • Figure 2: Seniority-based CI curves for the symmetric bond dissociation of linear H$_6$. Slater determinant and RG state seniority-based CI results are indiscernible. Results are computed in the optimal orbitals for seniority-zero Slater determinant CI obtained in ref. johnson:2020 in the STO-6G basis.
  • Figure 3: Errors of RGCISD in $\Omega=0,2$ and $\Omega=0,2,4$ channels for the symmetric bond dissociation of linear H$_6$. Results are computed in the optimal orbitals for seniority-zero Slater determinant CI obtained in ref. johnson:2020 in the STO-6G basis.
  • Figure 4: Dyck paths for 4-electron singlets.
  • Figure 5: Dyck paths for 6-electron singlets.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Lemma 3.1
  • Lemma 3.2