On the convergence properties of the Rayleigh–Schrödinger and the Hirschfelder–Silbey perturbation expansions for molecular interaction energies

The Rayleigh–Schrodinger polarization and the Hirschfelder–Silbey (HS) perturbation theories are applied, through the 38th order, to the interaction of a ground-state hydrogen atom with a proton. The calculations were made with high precision using a large basis set of orbitals expressed in the confocal elliptic coordinates. The results obtained show that for small internuclear separations R the polarization series converges slowly in an oscillatory way to the energy of the ground 1sσg state of the H molecule. At large R, however, the polarization expansion reproduces only the Coulomb part of the interaction energy effectively. When the value of the Coulomb energy is reached, the rate of convergence deteriorates drastically so that the exchange energy is not reproduced practically. The HS method converges fast when applied to both the 1sσg and the 2pσu states, the rate of convergence being roughly independent of the internuclear distance. If the finite basis set employed to solve the perturbation equations is stable under the symmetry operations, the HS expansion is shown to converge to the energy obtained by minimizing the Rayleigh–Ritz functional within the space spanned by the functions used in the perturbation theory calculations.