Special invariance properties of the [ N +1/ N ] Padé approximants in Rayleigh-Schrödinger perturbation theory

Abstract

Padé approximants to the electronic energy of atoms and molecules are investigated by using the expansion parameter of Rayleigh-Schrödinger perturbation theory as a formal variable. These problems are characterized by the fact that the exact Hamiltonian is known and, although the Hamiltonian is split into a zero order and a perturbing part, the exact Hamiltonian is recovered when the expansion parameter equals unity. The present study shows that for these problems the sequence of [ N +1/ N ] Padé approximants is special in that, when the expansion parameter is set equal to unity, the numerical value of each of these approximants is invariant to two modifications in the zero-order Hamiltonian; namely, a change of scale and a shift of origin in the zero-order energy spectrum. This suggests that it is the essence of the exact Hamiltonian which produces the final energy result, rather than the arbitrary scaling of the unperturbed Hamiltonian. This formalism is particularly appropriate for ab initio perturbative calculations, where the variational principle cannot be used to determine optimal values for the scale and shift parameters.