Spin-projected generalized Hartree-Fock method as a polynomial of particle-hole excitations

The past several years have seen renewed interest in the use of symmetry-projected Hartree-Fock for the description of strong correlations. Unfortunately, these symmetry-projected mean-field methods do not adequately account for dynamic correlation. Presumably, this shortcoming could be addressed if one could combine symmetry-projected Hartree-Fock with a many-body method such as coupled-cluster theory, but this is by no means straightforward because the two techniques are formulated in very different ways. However, we have recently shown that the singlet $S^2$-projected unrestricted Hartree-Fock wave function can in fact be written in a coupled-cluster-like wave function. That is, the spin-projected unrestricted Hartree-Fock wave function can be written as a polynomial of a double-excitation operator acting on some closed-shell reference determinant. Here, we extend this result and show that the spin-projected generalized Hartree-Fock wave function (which has both $S^2$ and $S_z$ projection) is likewise a polynomial of low-order excitation operators acting on a closed-shell determinant and provide a closed-form expression for the resulting polynomial coefficients. The spin projection of the generalized Hartree-Fock wave function introduces connected triple and quadruple excitations which are absent when spin-projecting an unrestricted Hartree-Fock determinant. We include a few preliminary applications of the combination of this spin-projected Hartree-Fock and coupled-cluster theory to the Hubbard Hamiltonian and comment on generalizations of the methodology. Results here are not for production level, but a similarity-transformed theory that combines the two offers the promise of being accurate for both weak and strong correlation, and may offer significant improvements in the intermediate correlation regime where neither projected Hartree-Fock nor coupled cluster is particularly accurate.

Figure 1

Energies in the periodic one-dimensional six-site Hubbard lattice at half-filling. Left panel: Total energies per electron. We have excluded the spin-projected methods as they are all very close to the exact result. Right panel: Errors per electron with respect to the exact result. Note the enormous improvement in going from SUHF and its SUVCC derivatives to SGHF and its SGVCC derivatives.

Figure 2

Energies in the periodic one-dimensional eight-site Hubbard lattice with six electrons. Left panel: Total energies per electron. We have not shown SUVCC or SGVCC, which are virtually indistinguishable from the variational coupled-cluster results. Right panel: Errors per electron with respect to the exact result. We have excluded SUHF and SGHF, which are clearly much poorer than VCCSD for this system.