Iterative solution of the Hartree equations

Abstract

An iterative scheme based on eigenpairs of Hilbert–Schmidt operators obtained from a Green’s function representation for solutions of a linearization of the Hartree equations, d2yi(r)/dr2 −[li(li+1)/r2]yi(r)+(2z/r) yi(r) −(2/r) ΣNj=1,j≠iYj(r) yi(r) =λ2iyi(r), Yi(r) =ℱr0y2i(s) ds+rℱ∞0s−1y2i(s) ds, yi(0) =0, yi(∞) =0, ℱ∞0y2i(s) ds=1, i=1,2,...,N, establishes existence of solutions to the Hartree equations and the sequence of eigenpairs generated converge subsequentially to a solution. In the case of the helium atom, for which we show some computational results, sequential convergence is obtained. Due to the Hilbert–Schmidt nature of the operators involved, the iterative method is implementable with Galerkin methods.