Success of quantum mechanical approximations for molecular geometries and electron–nuclear attraction expectation values: Gift of the Coulomb potential?

A perturbation theory is derived for the study of molecular structure. For a large class of approximate formulations (including Hartree–Fock and higher-level schemes and exact exchange-only density-functional theory), it is proved that the change in the error of the energy, upon geometry change, is zero through the second-order perturbation term, thus helping to explain why approximate energy curves often closely parallel exact curves and give accurate geometries. In contrast, for each non-Coulomb potential considered, the error change is not zero through second order, suggesting that accurate geometries are due, in part, to a special quality of the Coulomb potential. Comparable results are obtained for atomic electron–nuclear attraction expectation values (isoelectronic energy changes), which are exact through second order in the Coulomb cases. The conclusions are supported by a nonperturbative argument and by a numerical example.