: The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian. This work was supported in part by Sonderforschungsbereich 288 of the Deutsche Forschungsgemeinschaft...: The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian. This work was supported in part by Sonderforschungsbereich 288 of the Deutsche Forschungsgemeinschaft...

Jan 1, 1994144 citations

Excited States in Hartree-Fock Theory

Antonios Gonis

Computational Materials ScienceJan 1, 20210 citations

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

Thomas M. Henderson, Gustavo E. Scuseria

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.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.

arXiv:1706.01548Physical Review AAug 4, 201714 citations

{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.