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

Abstract

The past several years have seen renewed interest in the use of symmetry-projected HartreeFock 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 Sz 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.

Projected coupled cluster theory: Optimization of cluster amplitudes in the presence of symmetry projection

“…One major avenue of investigation has been to work with a perturbative or coupled-cluster style correction of the PHF wave function where the correlation operators are themselves symmetry adapted. 6,[13][14][15][16][17] These approaches benefit from the observation 14,15,18,19 that the PHF wave function often has a rather simple particle-hole representation in the symmetry-adapted basis, even though its particle-hole expansion in the brokensymmetry basis may be complicated. One may then interpolate between PHF and coupled cluster, 6,16 or attempt to add a symmetry-adapted cluster operator directly atop the PHF wave function.…”

Unrestricted treatment for the direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions

“…It must be emphasized that a ⟨ p Ω⟩ = 0 value implies a value ⟨ p Ŝ2 ⟩ = 0 but it does not necessarily imply a zero value for the ⟨ Ŝ2 ⟩ quantity, what entails the appearance of spin contamination which could be reduced applying procedures dealing with wave functions 41 or reduced density matrices. [42][43][44] The ⟨ Ŝ2…”

Projected coupled cluster theory

“…Neither have we implemented the combination of spin projected coupled cluster based on a generalized Hartree-Fock reference which breaks both S 2 and S z symmetries, which we expect it to be significantly more accurate yet. 25 Thus far we have considered only the ground state energy, and naturally properties, gradients, and excited states can all in principle be accessed by suitable modifications of traditional coupled cluster methods. Nonetheless, we are greatly encouraged by this early foray into the symmetry projection of broken symmetry coupled cluster theory.…”