Representation of electron-nucleus cusps in Slater and Gaussian basis sets

Prior to spherical averaging, the Coulomb cusp of exact electron densities is described by an equation involving two parameters: the nuclear charge Z and a vector quantifying the cusp anisotropy due to the presence of other nuclei. A similar three-parameter equation describes jump discontinuities of exact kinetic energy densities at nuclear positions. We demonstrate that these same equations, with basis-set-dependent parameters, also describe approximate electron and kinetic energy densities expanded in Slater-type orbitals (STOs) and even cuspless Gaussian-type orbitals (GTOs). Apart from the effective Z values, which are zero for GTOs, the cusp-shape parameters derived from STO and GTO wavefunctions are comparable in magnitude and converge to common complete-basis-set limits. These findings provide a rigorous justification for extending the concepts of effective nuclear charge and density anisotropy vectors to approximate wavefunctions within STO and GTO basis sets. Such extensions are not entailed by the properties of exact solutions to the Schrödinger equation and are not automatically applicable to all basis sets.