Effective nuclear charge calculator
Type an element symbol or atomic number. The calculator writes its ground-state electron configuration, applies Slater's rules and shows the effective nuclear charge felt by the outermost electron, the shielding constant, and Zeff for an electron in every occupied group from 1s outwards.
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How to use it
- Type an element symbol (Zn, fe, Cl) or an atomic number (30).
- Read Zeff for the outermost electron and the shielding constant behind it.
- Look down the list for Zeff of an electron in each inner group; it grows towards the 1s shell.
Slater’s rules
Write the configuration in Slater’s groups: (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) (5d) (6s, 6p). For the electron you are interested in:
- s or p electron: 0.35 for each other electron of the same group (0.30 for the other 1s electron), 0.85 for each electron with principal number n − 1, 1.00 for each electron in shells n − 2 and lower.
- d or f electron: 0.35 for each other electron of the same group, 1.00 for every electron in the groups to its left.
- Electrons in groups to the right shield nothing. Then Zeff = Z − S.
Worked example
Iron, Z = 26, configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶ 4s².
- A 4s electron: one other 4s electron (0.35), 14 electrons in shell 3 (14 × 0.85 = 11.9) and 10 in shells 1–2 (10). S = 22.25, Zeff = 3.75.
- A 3d electron: five other 3d electrons (5 × 0.35 = 1.75) and 18 electrons to the left (18). S = 19.75, Zeff = 6.25.
Nitrogen, Z = 7: a 2p electron is shielded by four other n = 2 electrons (1.4) and two 1s electrons (1.7), so Zeff = 7 − 3.1 = 3.9.
Tips
- Same group, same Zeff. In Slater’s scheme 2s and 2p electrons feel the same Zeff; that is a simplification of the rules.
- Approximate by design. Slater’s rules are a quick estimate; Clementi–Raimondi values from calculations differ somewhat.
- Neutral atoms only. The calculator works with neutral atoms; for an ion, remove or add electrons by hand and apply the same rules.
Frequently asked questions
What is effective nuclear charge?
It is the net positive charge an electron actually feels, Zeff = Z − S. The full nuclear charge Z is partly cancelled, or shielded, by the other electrons, mostly by those closer to the nucleus. Zeff explains why atoms shrink across a period and why ionisation energy rises.
What are Slater's rules?
A recipe for the shielding constant S. Group the orbitals as (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) … For an s or p electron, each other electron in its group adds 0.35 (0.30 in 1s), each electron in shell n − 1 adds 0.85, and each deeper electron adds 1.00. For a d or f electron, others in its group add 0.35 and everything to its left adds 1.00. Electrons to the right add nothing.
Why does a 4s electron of zinc feel only about +4.35?
Zinc has Z = 30. The other 4s electron shields 0.35, the 18 electrons of shell 3 shield 0.85 each (15.3), and the 10 electrons of shells 1 and 2 shield fully (10). S = 25.65 and Zeff = 30 − 25.65 = 4.35. Its 3d electrons, being inner, feel 8.85.
Are elements like chromium and copper handled?
Yes. The configurations follow the aufbau order with the known exceptions up to radon, such as Cr 3d⁵ 4s¹, Cu 3d¹⁰ 4s¹, Pd 4d¹⁰, Ag, Pt and Au, and La, Ce and Gd with one 5d electron.