Scalar products and density matrix elements of closed-shell pair geminal wavefunctions are evaluated directly in terms of the pair amplitudes, resulting in an analogue of Wick's theorem for fermions or bosons. This expression is in general intractable, but it is shown how it becomes feasible in three distinct ways for Richardson-Gaudin (RG) states, the antisymmetrized geminal power, and the antisymmetrized product of strongly-orthogonal geminals. Dissociation curves for hydrogen chains are computed with off-shell RG states and the antisymmetrized product of interacting geminals. Both are near exact suggesting that the incorrect results observed with ground state RG states are fixable using a different RG state.
We present a new geminal product wave function ansatz where the geminals are not constrained to be strongly orthogonal nor to be of seniority zero. Instead, we introduce weaker orthogonality constraints between geminals which 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 still to be antisymmetrized according to the Pauli principle to form a \textit{bona fide} electronic wave function.Our geometrical constraints tran…
The antisymmetrized geminal power (AGP) wavefunction has a long history and is known by different names in various chemical and physical problems. There has been recent interest in using AGP as a starting point for strongly correlated electrons. Here, we show that in a seniority-conserving regime, different AGP based correlator representations based on generators of the algebra, killing operators, and geminal replacement operators are all equivalent. We implement one representation that uses number operators as correlators and has linearly independent curvilinear metrics to distinguish the reg…
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 shoul…
Geminal wavefunctions have been employed to model strongly-correlated electrons. These wavefunctions represent products of weakly-correlated pairs of electrons and reasonable approximations are computable with polynomial cost. In particular, Richardson-Gaudin states have recently been employed as a variational ansatz. This contribution serves to explain the Richardson-Gaudin wavefunctions in the conventional language of quantum chemistry.
Simple wavefunctions of low computational cost but which can achieve qualitative accuracy across the whole potential energy surface (PES) are of relevance to many areas of electronic structure as well as to applications to dynamics. Here, we explore a class of simple wavefunctions, the minimal matrix product state (MMPS), that generalizes many simple wavefunctions in common use, such as projected mean-field wavefunctions, geminal wavefunctions, and generalized valence bond states. By examining the performance of MMPSs for PESs of some prototypical systems, we find that they yield good qualitat…
We develop a bivariational principle for an antisymmetric product of nonorthogonal geminals. Special cases reduce to the antisymmetric product of strongly-orthogonal geminals (APSG), the generalized valence bond-perfect pairing (GVB-PP), and the antisymmetrized geminal power (AGP) wavefunctions. The presented method employs wavefunctions of the same type as Richardson-Gaudin (RG) states, but which are not eigenvectors of a model Hamiltonian which would allow for more freedom in the mean-field. The general idea is to work with the same state in a primal picture in terms of pairs, and in a dual …
Ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian are employed as a wavefunction ansatz to model strong electron correlation in quantum chemistry. This wavefunction is a product of weakly-interacting pairs of electrons. While other geminal wavefunctions may only be employed in a projected Schrödinger equation, the present approach may be solved variationally with polynomial cost. The resulting wavefunctions are used to compute expectation values of Coulomb Hamiltionans and we present results for atoms and dissociation curves which are in agreement with doubly-occup…
Seniority-zero geminal wavefunctions are known to capture bond-breaking correlation. Among this class of wavefunctions, Richardson-Gaudin states stand out as they are eigenvectors of a model Hamiltonian. This provides a clear physical picture, clean expressions for reduced density matrix (RDM) elements, and systematic improvement (with a complete set of eigenvectors). Known expressions for the RDM elements require the computation of rapidities, which are obtained by first solving for the so-called eigenvalue based variables (EBV) then root-finding of a Lagrange interpolation polynomial. In thi…
Recently, ground state eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian, Richardson-Gaudin (RG) states, have been employed as a wavefunction ansatz for strong correlation. This wavefunction physically represents a mean-field of pairs of electrons (geminals) with a constant pairing strength. To move beyond the mean-field, one must develop the wavefunction in the basis of all the RG states. This requires both practical expressions for transition density matrices and an idea of which states are most important in the expansion. In this contribution, we present expressions for the …
Eigenvectors of the reduced Bardeen-Cooper-Schrieffer Hamiltonian have recently been employed as a variational wavefunction ansatz in quantum chemistry. This wavefunction is a mean-field of pairs of electrons (geminals). In this contribution we report optimal expressions for their reduced density matrices in both the original physical basis and the basis of the Richardson-Gaudin pairs. Physical basis expressions were originally reported by Gorohovsky and Bettelheim. In each case, the expressions scale like $\mathcal{O}(N^4)$, with the most expensive step the solution of linear equations. Analy…
