Effective Nuclear Charge Calculator
Calculate the effective nuclear charge (Z_eff) felt by any electron with Slater's rules — shielding constant, electron configuration and a group-by-group shielding table for elements 1 to 86.
How it works
Z_eff = Z − S · groups: (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)(5d)(6s,6p) · for an ns/np electron: others in the same group 0.35 (0.30 in 1s), electrons in shell n−1: 0.85, shells n−2 and lower: 1.00 · for an nd/nf electron: same group 0.35, every group to its left 1.00 · groups to the right: 0
Inner electrons screen part of the nucleus's charge from an outer electron, so it feels a smaller effective nuclear charge. John Slater's empirical rules (1930) estimate the shielding constant S by grouping orbitals and giving each other electron a fixed weight depending on its group. Z_eff explains periodic trends: it rises across a period (atoms shrink and ionization energy rises) and stays nearly constant down a group. The electron configuration uses the Madelung order with the common exceptions (Cr, Cu, Mo, Ag, Pd, Pt, Au and others). More accurate values come from Hartree–Fock calculations (Clementi–Raimondi).
Worked example
Sodium (Z = 11) is 1s² 2s² 2p⁶ 3s¹. For its 3s electron there are no other electrons in the same group, the 8 electrons of shell 2 shield 0.85 each (6.80) and the 2 electrons of shell 1 shield 1.00 each (2.00), so S = 8.80 and Z_eff = 11 − 8.80 = 2.20.
Sources: Slater, J. C. — Atomic Shielding Constants, Phys. Rev. 36, 57 (1930) · LibreTexts Chemistry — Slater's rules
Frequently asked questions
What is effective nuclear charge?
It is the net positive charge an electron actually feels, the nuclear charge Z minus the screening S by other electrons: Z_eff = Z − S.
How do you use Slater's rules?
Write the configuration in Slater groups, then add up the shielding: 0.35 for each other electron in the same group (0.30 in 1s), 0.85 for each electron one shell lower and 1.00 for deeper shells; for d and f electrons every electron to the left counts 1.00.
How does Z_eff change across the periodic table?
Across a period Z_eff on the valence electrons rises by about 0.65 per element, because each added proton is only partly screened by the added electron in the same shell. Down a group it stays roughly constant.
What is Z_eff for a 4s electron of potassium?
Potassium (Z = 19) is [Ar] 4s¹. Shell 3 has 8 electrons (8 × 0.85 = 6.80) and shells 1–2 have 10 (10 × 1.00), so S = 16.80 and Z_eff = 2.20.
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This calculator is for educational purposes. Double-check important results before acting on them.