Effective Nuclear Charge Calculator — Slater's Rules (Z_eff)
Enter an atomic number and select an electron orbital to calculate Z_eff — the net positive charge experienced by that electron after inner electrons partially screen (shield) the full nuclear charge. Uses Slater's rules.
Electron orbital
Net nuclear charge felt by an electron in 4s (Slater's rules)
- 1
Shielding constant S (Slater's rules)
17.15Sum of shielding contributions: 0.35 per same-group electron, 0.85 per (n−1)-shell electron, 1.00 per deeper-shell electron. - 2
Effective nuclear charge Z_eff = Z − S
20 − 17.15 = 2.85
How does this calculator work?
Z_eff = Z − S where S is the shielding constant from Slater's rules. For s/p electrons: same-group neighbours shield 0.35 each; (n−1)-shell electrons 0.85 each; deeper shells 1.00 each. For d/f electrons: all electrons below in the Aufbau order shield 1.00 each. Z_eff rises across a period and falls down a group.
Formula
How this is calculated
In a multi-electron atom, each electron feels not the full nuclear charge Z (the atomic number) but a reduced effective charge Z_eff = Z − S, where S is the shielding constant: the cumulative screening effect of the other electrons. Slater's rules provide a simple empirical scheme to estimate S for any orbital.
For an electron in an **s or p orbital** (principal quantum number n): electrons in the same [ns, np] group each contribute 0.35 (or 0.30 for the 1s group); electrons in the (n−1) shell contribute 0.85 each; electrons in shells n−2 and lower contribute 1.00 each. For an electron in a **d or f orbital**: same-group electrons contribute 0.35; all electrons lower in the Aufbau filling order contribute 1.00 each.
Slater's rules are a useful teaching approximation — they explain periodic trends in atomic radius, ionisation energy and electronegativity. They are not highly accurate for excited states, heavy elements (Z > 86), or orbitals with multiple filled d/f shells. Real density-functional calculations give more accurate Z_eff values. Exceptions to the ideal Aufbau order (e.g. Cr, Cu and several 4d/5d transition metals) are not accounted for; the calculator uses the ideal filling sequence.
Frequently asked questions
Moving across a period, each element gains one proton (Z increases by 1) but the new electron goes into the same principal shell. Electrons in the same shell shield each other poorly (σ = 0.35), so Z increases faster than S — the net effective charge Z_eff rises, making atoms smaller and harder to ionise from left to right in the same period.
Slater derived the shielding coefficients by fitting to Hartree–Fock orbital energies for a limited set of atoms. The grouping of orbitals (e.g. treating all [ns, np] electrons as a single block) is a simplification. Actual electron density distributions are more complex. For many purposes Slater's Z_eff underestimates shielding for d orbitals and overestimates it for heavier atoms. Use Clementi–Raimondi or DFT values for more accurate work.
Z_eff and Z* are used interchangeably in most textbooks and refer to the same quantity: the effective nuclear charge after subtracting the shielding constant. Some authors reserve Z* for more accurate Hartree–Fock-derived values (Clementi–Raimondi tables) and Z_eff for Slater's approximation, but the notation is not standardised.
Also known as
TG we-Calculate Editorial Team. (2026). Effective Nuclear Charge Calculator — Slater's Rules (Z_eff) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/effective-charge-calculator
TG we-Calculate Editorial Team. "Effective Nuclear Charge Calculator — Slater's Rules (Z_eff)." TG we-Calculate. 2026. https://we-calculate.com/calculator/effective-charge-calculator.
TG we-Calculate Editorial Team, "Effective Nuclear Charge Calculator — Slater's Rules (Z_eff)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/effective-charge-calculator
@misc{wecalculate_effective_charge_calculator, title = {Effective Nuclear Charge Calculator — Slater's Rules (Z_eff)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/effective-charge-calculator}}, year = {2026}, note = {TG we-Calculate} }
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