All calculatorsEffective Nuclear Charge (Z*) Calculator — Slater’s Rules
Compute the effective nuclear charge Z* for any electron using Slater’s rules. Use Element Mode to auto-load ground-state configurations, or Manual Mode to paste an electron configuration. Includes step-by-step shielding (S) breakdown.
Background
Effective nuclear charge estimates the net positive charge “felt” by an electron: Z* = Z − S, where Z is atomic number and S is the shielding from other electrons. Slater’s rules group orbitals and assign shielding weights depending on the electron of interest (valence ns/np vs. nd/nf).
How to use this calculator
- Element Mode: Choose an element to auto-load its ground-state configuration; then pick the orbital (e.g., 3p) and calculate.
- Manual Mode: Enter Z, paste a configuration (bracket notation OK), and specify the orbital of interest.
- Output: We show S by groups per Slater’s rules and the final .
This tool uses the standard Slater grouping: (1s), (2s,2p), (3s,3p), (3d), (4s,4p), (4d), (4f), (5s,5p), … MO anomalies and strong correlation effects are beyond Slater’s scope.
Formula & Equation Used
Effective charge:
Slater’s rules (summary):
- Group orbitals as: (1s), (2s,2p), (3s,3p), (3d), (4s,4p), (4d), (4f), (5s,5p), …
- For an electron in ns/np of principal quantum number n: same-group electrons (ns/np, same n) shield 0.35 each (except 1s → 0.30); electrons in n−1 shell shield 0.85 each; electrons in n−2 or lower shells shield 1.00 each.
- For an electron in nd/nf (same n): same-group (nd/nf) shield 0.35 each; all electrons in groups to the left (including ns/np) and all lower shells shield 1.00 each.
Example Problems & Step-by-Step Solutions
Example 1 — Cl 3p electron
[Ne] 3s² 3p⁵, Z = 17. For a 3p electron: same-group 3s/3p → 0.35 each (6 others), n−1 (2s/2p) → 0.85 each, n−2 or lower → 1.00 each. We compute S and Z* step-by-step.
Example 2 — Fe 3d electron
[Ar] 3d⁶ 4s², Z = 26. For a 3d electron: same-group 3d → 0.35 each (others), all groups to the left (including 3s/3p and all lower shells) → 1.00 each. We compute S and Z*.
Frequently Asked Questions
Q: Does Z* depend on which electron I choose?
Yes. A 3p electron and a 3d electron in the same atom generally see different shielding and hence different Z*.
Q: Why is 1s treated specially (0.30)?
Slater’s rules assign 0.30 for the other 1s electron instead of 0.35 used for other same-group s/p cases.
Q: Are Slater’s rules exact?
They’re an empirical approximation—great for trends and problems—but high-level quantum methods give more precise values.
