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The Journal of Chemical Physics

Seniority-zero states are mean-field wavefunctions

Marco Martínez González, Daniel F. Calero-Osorio, Michelle Richer, Cassandra Masschelein, Shuoyang Wang, Paul A. Johnson, Stijn De Baerdemacker, Paul W. Ayers

Published Oct 13, 2025• 10 citations
Publisher

Abstract

We use symbolic computation, numerical experiments, and mathematical arguments to establish that every seniority-zero state can be expressed as a (number-symmetry-broken) antisymmetric product of interacting geminals (APIG). Moreover, for bosons, the configuration interaction coefficients are exactly parameterized by matrix permanents. This establishes that bosonic wavefunctions can be exactly described by a mean field, specifically the (number-symmetry-broken) symmetric product of boson orbitals. While exact parameterization is extremely costly, we present numerical evidence showing that the APIG wavefunction approaches the true wavefunction exponentially quickly as the number of interacting geminals increases.

DOI
10.1063/5.0296924

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We use symbolic computation, numerical experiments, and mathematical arguments to establish that every seniority-zero state can be expressed as a (number-symmetry-broken) antisymmetric product of interacting geminals (APIG). Moreover, for bosons, the configuration interaction coefficients are exactly parameterized by matrix permanents. This establishes that bosonic wavefunctions can be exactly described by a mean field, specifically the (number-symmetry-broken) symmetric product of boson orbitals. While exact parameterization is extremely costly, we present numerical evidence showing that the APIG wavefunction approaches the true wavefunction exponentially quickly as the number of interacting geminals increases.
A new truncation scheme for non-orthogonal antisymmetrized products of interacting geminals (APIG) is introduced based on antisymmetrized products of strongly orthogonal geminals (APSG). This wavefunction hierarchy of interacting geminals (IG) allows us to gradually increase the accuracy at which the ground state of a seniority-zero Hamiltonian can be estimated, ranging from APSG up to approximation-free APIG. Mathematical expressions for the lowest four orders (IG0, IG1, IG2, and IG3) are given explicitly and the computational cost to evaluate their transition density matrix is verified to scale only cubically with system size. Exemplary numerical calculations indicate that already a very early truncation level leads to results virtually identical to APIG.
Seniority is a useful way of organizing Hilbert space for strongly correlated systems. The exact zero-seniority wave function, doubly occupied configuration interaction (DOCI), provides accurate results (given the right orbitals) for many strongly correlated electronic systems but has a combinatorial computational cost. In many cases, pair coupled cluster doubles provide a polynomial-cost approximation that closely reproduces the energies of DOCI, but it breaks down in some cases and, as shown herein, it does not provide particularly good density matrices. In this work, we demonstrate that by using the Jordan-Wigner transformation to turn the seniority zero problem back into a Fermionic one, we can provide mean-field variational results of DOCI quality for the Hubbard model and a few small molecular dissociation examples, with polynomial cost, both for the energies and for density matrices, all while being protected from collapse. This success is rooted in the proof we provide, showing that the Hartree-Fock wave function on the Jordan-Wigner-transformed Hamiltonian transforms back to variational coupled cluster doubles in the seniority zero representation, but restricted to have determinant rather than permanent amplitude coefficients, without compromising its overall accuracy.
Abstract We show the transformation from a one-particle basis to a geminal basis, transformations between different geminal bases demonstrate the Lie algebra of a geminal basis. From the basis transformations, we express both the wave function and Hamiltonian in the geminal basis. The necessary and sufficient conditions of the exact wave function expanded in a geminal basis are shown to be a Brillouin theorem of geminals. The variational optimization of the geminals in the antisymmetrized geminal power (AGP), antisymmetrized product of geminals (APG) and the full geminal product (FGP) wave function ansätze are discussed. We show that using a geminal replacement operator to describe geminal rotations introduce both primary and secondary rotations. The secondary rotations rotate two geminals in the reference at the same time due to the composite boson nature of geminals. Due to the completeness of the FGP, where all possible geminal combinations are present, the FGP is exact. The number of parameters in the FGP scale exponentially with the number of particles, like the full configuration interaction (FCI). Truncation in the FGP expansion can give compact representations of the wave function since the reference function in the FGP can be either the AGP or APG wave function.
We propose an approach to the electronic structure problem based on noninteracting electron pairs that has similar computational cost to conventional methods based on noninteracting electrons. In stark contrast to other approaches, the wave function is an antisymmetric product of nonorthogonal geminals, but the geminals are structured so the projected Schrödinger equation can be solved very efficiently. We focus on an approach where, in each geminal, only one of the orbitals in a reference Slater determinant is occupied. The resulting method gives good results for atoms and small molecules. It also performs well for a prototypical example of strongly correlated electronic systems, the hydrogen atom chain.
We present a new geminal product wave function Ansatz where the geminals are not constrained to be strongly orthogonal or to be of seniority-zero. Instead, we introduce weaker orthogonality constraints between geminals that significantly lower the computational effort without sacrificing the indistinguishability of the electrons. That is to say, the electron pairs corresponding to the geminals are not fully distinguishable, and their product has yet to be antisymmetrized according to the Pauli principle to form a bona fide electronic wave function. Our geometrical constraints translate into simple equations involving the traces of products of our geminal matrices. In the simplest non-trivial model, a set of solutions is given by block-diagonal matrices where each block is 2 × 2 and consists of either a Pauli matrix or a normalized diagonal matrix multiplied by a complex parameter to be optimized. With this simplified Ansatz for geminals, the number of terms in the calculation of the matrix elements of quantum observables is considerably reduced. A proof of principle is reported and confirms that the Ansatz is more accurate than strongly orthogonal geminal products while remaining computationally affordable.
ABSTRACT We consider wavefunctions built from antisymmetrized products of two‐electron wavefunctions (geminals), which is arguably the simplest extension of the antisymmetrized product of one‐electron wavefunctions (orbitals) (i.e., a Slater determinant). Extensive use of geminals in wavefunctions has been limited by their high cost stemming from the many combinations of the two‐electron basis functions (orbital pairs) used to build the geminals. When evaluating the overlap of the APG wavefunction with an orthogonal Slater determinant, this cost can be interpreted as the cost of evaluating the permanent, resulting from the symmetry with respect to the interchange of orbital pairs, and the cost of assigning the occupied orbitals to the orbital pairs of the wavefunction. Focusing on the latter, we present a graphical interpretation of the Slater determinant and utilize the maximum weighted matching algorithm to estimate the combination of orbital pairs with the largest contribution to the overlap. Then, the cost due to partitioning the occupied orbitals in the overlap is reduced from to . Computational results show that many of these combinations are not necessary to obtain an accurate solution to the wavefunction. Because the APG wavefunction is the most general of the geminal wavefunctions, this approach can be applied to any of the simpler geminal wavefunction ansätze. In fact, this approach may even be extended to generalized quasiparticle wavefunctions, opening the door to tractable wavefunctions built using components of arbitrary numbers of electrons, not just two electrons.