Electron density and its derivatives at the nucleus for spherically confined hydrogen atom

Abstract

It is shown that the energy of a hydrogen-like atom confined inside a spherical cavity of radius, R, and potential barrier, V₀, is quantitatively defined by the ratio [η''_l(0)/η'_l(0)]. Here, the conventional spherical density ρ̅(r) is scaled as η_l(r)=ρ̅(r)/r^2l and the ratio of the second derivative η''_l(r) to η_l(r) is evaluated at the nucleus. Numerical results of the ratios are presented for 1s, 2s, 2p, and 3d states at several values of V₀. For such states, the characteristic radii of confinement leading to the well-defined values of energy are identified.