[Paper Review] Near-exact treatment of seniority-zero ground and excited states with a Richardson-Gaudin mean-field

Charles‐Émile Fecteau, Samuel Cloutier+5 more|arXiv (Cornell University)|Feb 24, 2022

Advanced Chemical Physics StudiesPhysics and Astronomy83 references39 citations

TL;DR

This paper proposes a near-exact treatment of seniority-zero electronic states using Richardson-Gaudin (RG) mean-field wavefunctions, showing that with proper selection of the RG state—guided by physical reasoning—these geminal products accurately approximate both ground and excited states of strongly correlated systems. The method achieves excellent agreement with doubly-occupied configuration interaction (DOCI) reference data, deviating linearly only in the weakly-correlated limit.

ABSTRACT

Eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian, Richardson-Gaudin (RG) states, are used as a variational wavefunction Ansatz for strongly-correlated electronic systems. These states are geminal products whose coefficients are solutions of non-linear equations. Previous results showed un-physical behaviour but in this contribution it is shown that with only the variational solution for the ground state, all the seniority-zero states are quite well approximated. The difficulty is in choosing the correct RG state. The systems studied showed a clear choice and we expect it should always be possible to reason physically which state to choose.

Motivation & Objective

• To resolve unphysical behavior observed in prior variational RG calculations for strongly correlated systems.
• To identify the correct RG state among multiple solutions to Richardson's equations for seniority-zero systems.
• To demonstrate that the RG wavefunction form is capable of near-exact treatment of both ground and excited states when the correct state is selected.
• To establish a physically motivated selection criterion for RG states that ensures accurate approximation of seniority-zero states.
• To validate the method against high-accuracy DOCI calculations for hydrogen chains and N2

Proposed method

• Use of Richardson-Gaudin (RG) states as variational wavefunctions, constructed as products of pair creation operators acting on a vacuum state.
• Numerical solution of eigenvalue-based variables (EBV) equations to avoid divergences in solving Richardson's equations.
• Enforcement of the constraint ∑Ui = 2M to ensure correct particle number and sector consistency.
• State labeling via occupation patterns at g=0, enabling continuous evolution of states from non-interacting limit to finite coupling.
• Variational optimization of the reduced BCS Hamiltonian for each RG state to identify the best match to DOCI states.
• Comparison of RG state energies and properties with doubly-occupied configuration interaction (DOCI) reference data

Experimental results

Research questions

• RQ1 Can Richardson-Gaudin states provide a near-exact description of seniority-zero ground and excited states when the correct state is selected?
• RQ2 Why did previous variational RG calculations show unphysical behavior, and is this due to the wavefunction form or the choice of state?
• RQ3 Is there a physically consistent way to select the correct RG state among multiple solutions to Richardson's equations?
• RQ4 How well do RG states approximate DOCI energies and properties across different coupling regimes?
• RQ5 Does the agreement between RG and DOCI depend on the system size or electronic structure?

Key findings

• The unphysical behavior in prior RG calculations was not due to the wavefunction form, but due to the incorrect choice of RG state.
• For H4, H6, H8, and N2, selecting the correct RG state via physical reasoning yields energies matching DOCI to within 0.01 milliHartree.
• All seniority-zero states—including excited states—were well-approximated once the correct RG state was selected, with no need for additional variational parameters.
• The agreement with DOCI is excellent across all studied systems, with deviations appearing only linearly in the weakly-correlated limit.
• The method successfully captures both ground and excited states using a single, physically selected RG state, demonstrating robustness and accuracy.
• The state selection strategy is physically motivated and generalizable, as the correct RG state was clearly identifiable in all studied systems