The fundamental significance of the components of the electronic Kohn-Sham potential evaluated at the nucleus is highlighted via the numerical studies on atoms He-Lu which suggest their formally similar power-law relationship in expressing the associated components of total electronic energy. Similar studies on the isoelectronic series of closed shell atoms lead to the linear correlations. The proposed static exchange-correlation charge density concept [S. Liu, P. A. Ayers, and R. G. Parr, J. Chem. Phys. 111, 6197 (1999)] is used to interpret these relationships. The maxima in the static integrated radial exchange-correlation charge density function, Q(xc)(r), in atoms are shown to reflect the shell boundaries. The quantum Monte Carlo density derived exchange-correlation potentials for Be and Ne are used to obtain Q(xc)(r) that can be used as standards to directly assess the quality of approximate exchange-correlation potentials. For the negative ions, Q(xc)(r) displays a characterstic outer minimum as a consequence of the Sen-Politzer theorem [K. D. Sen and P. Politzer, J. Chem. Phys. 90, 4370 (1989)]. This minimum is found to be related with the stability of negative ions.