Atomic-charge monotonicity and cusp-type inequalities: Applications to heliumlike systems.

Abstract

An integral representation for the spherically averaged electron density \ensuremath{\rho}(r) of atoms and molecules is proposed. Only the order k of the monotonicity of \ensuremath{\rho}(r) was assumed to be known. This representation is used to bound the mth derivative ${\mathrm{\ensuremath{\rho}}}^{(\mathit{m})}$(r), m>=2, by means of the values of the density \ensuremath{\rho}(r) and its first derivative \ensuremath{\rho}'(r). Particular consequences of this general result are (i) lower bounds of the curvature \ensuremath{\rho}''(r) at any value of r and (ii) atomic cusp-type inequalities for the derivative ${\mathrm{\ensuremath{\rho}}}^{(\mathit{m})}$(0) of any order m [i.e., inequalities which allow one to bound such derivatives at the nucleus by means of the nuclear charge Z and the value of the density \ensuremath{\rho}(0)]. For example, it is obtained that the value (2Z${)}^{2}$\ensuremath{\rho}(0)(k-2)/(k-1) is a lower bound to the curvature of the electron density at the nucleus. Some of these results are compared with other recent findings of similar nature and are numerically analyzed by means of the highly accurate 204-term Hylleraas wave functions in several heliumlike atoms.