Generalized symmetric Rayleigh-Ritz procedure applied to the closed-shell Hartree-Fock problem

We present the Generalized Symmetric Rayleigh-Ritz (GSRR) procedure for finding approximate eigenfunctions and corresponding eigenvalues for a linear operator, L, in a finite function space, {φ_i}^N_{i = 1}. GSRR is derived by minimizing the residual in the norm induced by an inner product, (· , ·), under the constraint that the resulting eigenfunctions be mutually orthogonal with respect to another inner product, (· , ·)_a. When L is the closed-shell Fock operator, f, GSRR is a generalization of the Roothaan equations. We apply this method to f with (· , ·) defined by a grid, {r_k}^M_{k = 1}, and (· , ·)_a defined by analytic integration, noting that a grid-defined (· , ·) lends itself to faster operator evaluation (scaling as MN²) and effective parallelization. When a grid is used, GSRR scales as pseudospectral methods do; however, it is in the spirit of conventional spectral methods (e.g., GSRR does not use an inverse transform).