Near-exact treatment of seniority-zero ground and excited states with a Richardson–Gaudin mean-field

Abstract

Eigenvectors of the reduced Bardeen–Cooper–Schrieffer (BCS) 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 an un-physical apparent avoided crossing in ground state dissociation curves for hydrogen chains. In this paper, it is shown that each seniority-zero state of the molecular Coulomb Hamiltonian corresponds directly to an RG state. However, the seniority-zero ground state does not correspond to the ground state of a reduced BCS Hamiltonian. The difficulty is in choosing the correct RG state. The systems studied showed a clear choice, and we expect that it should always be possible to reason physically which state to choose.

Geminal-based electronic structure methods in quantum chemistry. Toward a geminal model chemistry

“…Recently, Johnson and coworkers 69–73 introduced a new geminal wave function ansatz called the antisymmetric product of rank-two geminals (APr2G), based on the solutions of the reduced BCS Hamiltonian 25 In the above equation, the system is defined by a set of single-particle energies { ε p } and a pairing strength g , where n̂ p denotes the particle number operator n̂ p = 2 Ŝ z p + 1,withThe Ŝ p + and Ŝ p − operators in eqn (34) define the creation and annihilation of an electron pair on orbital p , respectively, Ŝ p + = â † p â † p̄ , Ŝ p − = â p̄ â p .The eigenvectors of eqn (34) are employed as a wave function ansatz to describe strong correlation in atoms and molecules. Specifically, these eigenvectors are products of electron pairs,characterized by a complex number u , called rapidities or pairing energies, and where K again indicates the number of orbitals.…”

“…Recently, Johnson and coworkers [69][70][71][72][73] introduced a new geminal wave function ansatz called the antisymmetric product of rank-two geminals (APr2G), based on the solutions of the reduced BCS Hamiltonian 25…”

“…where C p is the APSG geminal coefficient matrix element associate with orbital p. By substituting the above equation into the energy expression of eqn (73) and performing a spinsummation, we obtain…”

2D-Block Geminals: a non 1-orthogonal and non 0-seniority model with reduced computational complexity

“…When these numbers are derived as eigenstate coefficients of reduced Bardeen-Cooper-Schrieffer (BCS) model Hamiltonians 13 , one obtains the so-called Richardson-Gaudin (RG) geminals which are currently under active development. 1,14 C. Inter-geminal constraints…”

“…However, without further restrictions, such a model has still a factorial computational cost with respect to the number of electronic orbitals and its applicability is therefore limited to small systems. Reducing the scaling of the computational cost to a polynomial one is an active research field [1][2][3] . We postpone to the next section a review of the most popular proposals.…”