A simple quality test for self-consistent-field atomic orbitals

A quality test is proposed for SCF atomic orbitals, [Formula: see text] approximated as finite linear combinations of suitable basis functions [Formula: see text] The key is in a function, readily derived from the Hartree–Fock equation [Formula: see text] which is identically zero for true Hartree–Fock spin orbitals and not so for approximate orbitals. In this way, our test measures how closely approximate orbital descriptions approach the true Hartree–Fock limit and thus provides a quality ordering of orbital bases with respect to one another and with respect to that limit, in a scale uniquely defined by the latter. Moreover, this analysis also holds for atomic subspaces of our choice, e.g., the valence region. Examples are offered for representative collections of Slater- and Gaussian-type orbital expansions.