Connections between Richardson-Gaudin States, Perfect-Pairing, and Pair Coupled-Cluster Theory

TL;DR

This work establishes that perfect-pairing (PP) can be understood as an eigenvector of a simplified reduced BCS Hamiltonian expressed in bonding/antibonding orbital pairs, with valence-bond gaps $\omega_\alpha$ governing the PP state and its density matrices. The complementary eigenvectors provide a principled route to incorporate weak correlation perturbatively, enabling second-order ENPT (EN2) corrections on PP that closely match pair CC doubles (pCCD) results. The authors show that, in the single-pair (seniority-zero) sector, PP and pCCD are effectively equivalent, unifying orbital- and geminal-based descriptions of correlation. Numerical results on hydrogen chains demonstrate that EN2 on PP nearly reproduces DOCI and that OO-pCCD tracks DOCI closely, supporting the potential of hybrid PP/pCCD/RG approaches for efficient, accurate treatment of both static and dynamic correlation. The framework offers a clear pathway to extend to RG states with non-zero seniority, paving the way for systematic improvements over conventional seniority-zero methods.

Abstract

Slater determinants underpin most electronic structure methods, but orbital-based approaches often struggle to describe strong correlation efficiently. Geminal-based theories, by contrast, naturally capture static correlation in bond-breaking and multireference problems, though at the expense of implementation complexity and limited treatment of dynamic effects. In this work, we examine the interplay between orbital and geminal frameworks, focusing on perfect-pairing (PP) wavefunctions and their relation to pair coupled-cluster doubles (pCCD) and Richardson-Gaudin (RG) states. We show that PP arises as an eigenvector of a simplified reduced Bardeen-Cooper-Schrieffer (BCS) Hamiltonian expressed in bonding/antibonding orbital pairs, with the complementary eigenvectors enabling a systematic treatment of weak correlation. Second-order Epstein-Nesbet perturbation theory on top of PP is found to yield energies nearly equivalent to pCCD. These results clarify the role of pair-based ansätze and open avenues for hybrid approaches that combine the strengths of orbital- and geminal-based methods.

Figures (8)

• Figure 1: Schematic representation of the orbital diagram in PP: reference PP wave function (top) and the three types of excited configurations (bottom). Here, we have represented two VBS, $\{ \alpha_{0}, \alpha_{1} \}$ and $\{ \beta_{0}, \beta_{1} \}$, characterized by VBS gaps $\omega_\alpha = \xi_{\alpha_{1}} - \xi_{\alpha_{0}}$ and $\omega_\beta = \xi_{\beta_{1}} - \xi_{\beta_{0}}$, respectively.
• Figure 2: Symmetric dissociation of H4 chain in STO-6G. Orbitals were optimized separately for PP, pCCD, and DOCI. Left: Reduced energies for RHF, OO-DOCI, and FCI. Right: Absolute errors of OO-PP, OO-PP-EN2, and OO-pCCD with respect to OO-DOCI plotted logarithmically. OO-PP-EN2 is above OO-DOCI until $R_{\ce{H-H}}=3.1\bohr$, where it goes below and overcorrelates the rest of the way to dissociation. OO-pCCD always overcorrelates relative to OO-DOCI.
• Figure 3: Symmetric dissociation of H10 chain in STO-6G. Orbitals were optimized separately for PP, pCCD, and DOCI. Left: Reduced energies for RHF, OO-DOCI, and FCI. Right: Absolute errors of OO-PP, OO-PP-EN2, and OO-pCCD with respect to OO-DOCI plotted logarithmically. OO-PP-EN2 is above OO-DOCI until $R_{\ce{H-H}}=3.2\bohr$ where it goes below and overcorrelates the rest of the way to dissociation. OO-pCCD always overcorrelates relative to OO-DOCI.
• Figure 4: Symmetric dissociation of H10 chain in STO-6G. Left: 1-RDM elements $\{n_{\alpha}\}$ for H10 optimized with OO-PP/STO-6G. Right: VBS gaps $\{\omega_{\alpha}\}$ for H10 optimized with OO-PP/STO-6G plotted logarithmically.
• Figure 5: Progressive dimerization due to Peierls instability of H10: $R_0$ is kept fixed at 3.6 $a_0$ while the distance $R$ is varied.