Test of the Conventional Quantum Chemistry Methods on the Hydrogen Atom

Abstract

This paper is a numerical study of the reliability and accuracy of some of the conventional methods of quantum chemistry by their application to the problem of the hydrogen atom, a case where the quantities of interest are known exactly. The variation principle, δ<H> = 0, is used to approximate energy eigenvalues and the corresponding eigenfunctions. As basis functions, two simple sets are chosen: a) Gaussians, exp(-kηr²); k = 1, 2,...N, and b) exponentials, exp(-kηr); k = 1, 2...N. N is the order of the basis set and η is the scale factor. Scaled functions φ_k are obtained from the unscaled ones φ_k in the usual way /1/:

$${\bar \phi _k} = {\eta ^{3/2}}{\phi _k}\left( {\eta r} \right)$$

Both sets become complete in the interval 0≤r≤∞ when N→∞ and, for both, all the necessary integrals are elementary.

Sponsored by the Aeronautical Research Laboratories, Wright Air Development Division, Air Research and Development Command, U.S. Air Force; and supported in part by the U.S. Office of Naval Research