Projected quasiparticle theory for molecular electronic structure

Abstract

We derive and implement symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations and apply them to the molecular electronic structure problem. All symmetries (particle number, spin, spatial, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. We show that the resulting method yields a comprehensive black-box treatment of static correlations with effective one-electron (mean-field) computational cost. The ensuing wave function is of multireference character and permeates the entire Hilbert space of the problem. The energy expression is different from regular HFB theory but remains a functional of an independent quasiparticle density matrix. All reduced density matrices are expressible as an integration of transition density matrices over a gauge grid. We present several proof-of-principle examples demonstrating the compelling power of projected quasiparticle theory for quantum chemistry.

Multireference symmetry-projected variational approaches for ground and excited states of the one-dimensional Hubbard model

“…50 As it is well known, the most general GHF-determinant |D deliberately breaks several symmetries of the original Hamiltonian. 32,[35][36][37][38]40 Typical examples are the rotational (in spin space) and spatial symmetries. To restore the spin quantum numbers in a symmetry-broken GHF-determinant, we explicitly use the full 3D projection operator [35][36][37][38] …”

“…This kind of SR variation-after-projection scheme, has already been applied to the 1D and 2D Hubbard models 35,36 as well as in quantum chemistry within the framework of the Projected Quasiparticle Theory. [37][38][39] One of the main advantages of the symmetry-projected approximations 32,[35][36][37][38][39] is that they offer compact wave functions as well as a systematic way to improve their quality by adopting a multi-reference (MR) approach. In this case, a set of symmetry-broken mean-field states |D i is used to build Goldstone manifoldsR|D i whose superposition can be used to recover the desired symmetries of the Hamiltonian.…”

Bridging Single- and Multireference Domains for Electron Correlation: Spin-Extended Coupled Electron Pair Approximation

“…We have used the integral form of the projection operator, which is computationally more convenient to deal with than Löwdin's operator. 10,11,68 For all the calculations reported in this article, the number of grid points N grid for the numerical integrations was set to either 3 or 4, unless otherwise noted, which was found to be sufficient to obtain the desired Ŝ 2 value with numerical precision of 10 −9 . This means that our methods are effectively multireference; a reference SUHF state is represented by three or four nonorthogonal Slater determinants.…”

Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration

“…To the best of our knowledge, this is the first detailed study of this problem in the quantum chemistry community. Our motivation for embarking in this study is the relevance of zero energy modes in symmetry breaking and restoration, a subject that has recently received much of our attention [19,20]. We begin in Sec.…”

“…The various determinants in the Goldstone manifold are degenerate, non-orthogonal (in finite systems), and can be connected by a rotation with a continuous parameter. Diagonalizing the Hamiltonian in the basis of these determinants restores the symmetry [18][19][20]. If the symmetry that has been broken is instead discrete (such as complex conjugation), then the Hessian matrix of the broken symmetry solution will not have a zeroenergy eigenvector pointing in the direction of symmetry restoration.…”