Effective Nuclear Charge Calculator

Across a period (left to right), Z increases by 1 per element but shielding increases by only about 0.35 (same-shell electrons shield poorly). So Z_eff increases, making atoms smaller with higher ionization energies. Down a group, although Z increases significantly, n also increases, and the larger orbital radius dominates, making atoms larger with lower ionization energies.

No. Slater's rules are an empirical approximation developed in 1930. More accurate values come from Hartree-Fock self-consistent field calculations or Clementi-Raimondi values (published in 1963), which use variational methods. For example, Slater gives Z_eff = 2.20 for Na 3s, while Clementi-Raimondi gives 2.51. Slater's rules are useful for qualitative understanding and quick estimates.

s electrons have nonzero probability density at the nucleus (they penetrate closer to the nucleus), giving them greater shielding effectiveness. p electrons have a node at the nucleus and penetrate less. The shielding order is s > p > d > f, which is why s electrons in an inner shell contribute more shielding than d or f electrons at similar distances.

Penetration describes how much of an electron's probability density is found close to the nucleus. Electrons with high penetration experience more nuclear charge and are more tightly bound. For the same shell, s orbitals penetrate most, followed by p, d, and f. This explains the subshell energy ordering (s < p < d < f within the same shell) in multi-electron atoms.

Higher Z_eff leads to stronger attraction for electrons, increasing electronegativity and making the atom better at attracting shared electrons in covalent bonds. It also increases ionization energy (harder to remove electrons) and decreases atomic radius. These effects determine bond polarity, molecular geometry, and overall chemical reactivity patterns.