Nuclear cusp and critical nuclear charge

The nuclear cusp relations for the wave function and the density are revisited. A quantity called cusp constant is introduced. It is proved that the ground-state energy of a Coulomb system is equal to the cusp constant. It is shown that the cusp constant is the same constant for any nucleus of a Coulomb system. An equation for the critical nuclear charge is derived. It is an implicit equation containing quantities taken at a nucleus, which can be used to express the necessary and sufficient condition for the N-electronic mononegative atom to be stable in terms of the electron density derived quantities defined at the nuclear position.