Efficient Implementation of Variation after Projection Generalized Hartree–Fock

Abstract

Projected Hartree-Fock (PHF) theory can restore important symmetries to broken symmetry wave functions. Variation after projection (VAP) implementations make it possible to deliberately break and then restore a given symmetry by directly minimizing the projected energy expression. This technique can be applied to any symmetry that can be broken from relaxing constraints on single Slater determinant wave functions. For instance, generalized Hartree-Fock (GHF) wave functions are eigenfunctions of neither Ŝ nor S. By relaxing these constraints, the wave function can explore a larger variational space and can reach lower energies than more constrained HF solutions. We have implemented spin-projected GHF (SGHF), which retains many of the advantages of breaking symmetry while also being a spin eigenfunction, with some notable improvements over previous implementations. Our new algorithm involves the formation of new intermediate matrices not previously discussed in the literature. Discretization of the necessary integration over the rotation group SO(3) is also accomplished much more efficiently using Lebedev grids. A novel scheme to incrementally build rotated Fock matrices is also introduced and compared with more standard approaches.

Spin-projection for quantum computation: A low-depth approach to strong correlation

“…, N g } for rotationsÛ g = e −iβ gŜy along with accordingly defined weights w g . With this form, one typically requires only a few grid points to achieve Ŝ 2 = s(s + 1) that is accurate to several decimal points, and the convergence is exponentially fast [26,28]. To be precise, in this paper, we propose to evaluate ÔÛ g θ for UCC with a quantum computer and then use a classical adder to evaluate E, instead of constructing a symmetry-projected Ansatz.…”

The practical implementation of Löwdin's method for spin projection

“…As we are going to work with generating functions (polynomials), and the expected values of the Hamiltonian are fractions, see Equations (31) and (32), we prefer to express the optimality (Equation (74)) in terms of variations of the weighted energy;…”

“…The exact (complete) treatment of the spin projection is expected to help in the convergence rate. Li et al, working with the Generalized Hartree‐Fock flavor, realized, by considering different integration grids, that the error in spin symmetry restoration can create a soft bound on the energy's convergence. It is also known that the VAP scheme, with discrete integration, sometimes fails to converge for large molecules when applying the Direct Inversion of Iterative Subspace (DIIS) technique, this is due to the presence of small eigenvalues in the Hessian matrix, this situation often appears in SUHF.…”

Quadratically convergent self-consistent field of projected Hartree–Fock

“…15 The spin-projected unrestricted Hartree-Fock (SUHF) is capable of treating a large amount of static correlation to provide suitable reference for strongly correlated systems. In recent years, post-SUHF methods [16][17][18][19][20][21][22] to include the residual dynamic correlation and some implementations related to variation after projection 23,24 have been developed.…”