# What is the effective nuclear charge?

Finds the effective nuclear charge Zeff = Z − S felt by an electron of any element, with the shielding constant S worked out by Slater’s rules from the element’s ground-state electron configuration.

- Page: https://www.acalculator.org/chemistry/effective-nuclear-charge-calculator
- JSON spec: https://www.acalculator.org/chemistry/effective-nuclear-charge-calculator.json
- Version: da1abed9eef1

## Default answer

Example with the default inputs (Element 26 Fe (Iron), Electron in Outermost electron): By Slater’s rules, a 4s electron of iron (Fe) feels Zeff = 3.75.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| z | Element | The element, by atomic number, symbol and name. |
| e | Electron in | Which electron: the outermost one, or one in a chosen Slater group. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| zeff | Effective nuclear charge (Zeff) | Z minus the shielding constant, felt by the chosen electron. |
| shielding | Shielding constant (S) | The sum of the other electrons’ shielding by Slater’s rules. |
| z | Nuclear charge (Z) | The atomic number. |
| working | Working | The shielding sum by kind of electron, then Z − S. |
| groups | Slater groups | The ground-state configuration written in Slater’s groups. |
| which | Electron | The group of the chosen electron and its element. |

## Method

Zeff = Z − S. Group the electrons [1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s,5p] [5d] [5f] [6s,6p] [6d] [7s,7p]. For an s or p electron: 0.35 for each other electron in its group (0.30 in 1s), 0.85 for each in shell n − 1, 1.00 for each further in. For a d or f electron: 0.35 for each other in its group, 1.00 for each in a group to its left. Groups to the right add nothing.

## Assumptions

- Neutral atoms in their ground state: NIST’s configuration for Z up to 108, the Madelung filling order beyond.
- Slater’s rules are an approximation; they give one Zeff for s and p electrons of the same shell.
- The outermost electron is one in the last occupied group of the list (4s for iron, 4d for palladium).

## Worked examples

1. z = 26, e = outer gives shielding = 22.25, zeff = 3.75, working = S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25; Zeff = 26 − 22.25 = 3.75. Source: Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 4s, S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25, Zeff = 3.75.
2. z = 26, e = 3d gives shielding = 19.75, zeff = 6.25. Source: Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 3d, S = 0.35 × 5 + 1.00 × 18 = 19.75, Zeff = 6.25.
3. z = 26, e = 3sp gives shielding = 11.25, zeff = 14.75. Source: Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 3s and 3p, S = 0.35 × 7 + 0.85 × 8 + 1.00 × 2 = 11.25, Zeff = 14.75.
4. z = 26, e = 1s gives shielding = 0.3, zeff = 25.7. Source: Wikipedia, Slater’s rules (Slater, Phys. Rev. 36, 57 (1930), https://doi.org/10.1103/PhysRev.36.57; the iron example), https://en.wikipedia.org/wiki/Slater%27s_rules: Fe 1s, S = 0.30, Zeff = 25.70.
5. z = 7, e = outer gives shielding = 3.1, zeff = 3.9. Source: LibreTexts, McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322), https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules: N 2p, S = 0.85(2) + 0.35(4) = 3.10.
6. z = 29, e = 3d gives shielding = 21.15, zeff = 7.85. Source: LibreTexts, McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322), https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules: Cu 3d, S = 21.15, Zeff = 7.85.

## FAQ

### What is effective nuclear charge?

The net pull of the nucleus on one electron after the other electrons screen part of it: Zeff = Z − S, where Z is the atomic number and S the shielding (screening) constant. For hydrogen, with one electron, Zeff = Z = 1.

### How do I calculate Zeff with Slater’s rules?

Write the configuration in groups (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)… and pick your electron. Ignore groups to its right. For an s or p electron, add 0.35 for each other electron in its group (0.30 in 1s), 0.85 for each in the next shell in, and 1.00 for each further in. For a d or f electron, add 0.35 for each other in its group and 1.00 for each to its left. Then Zeff = Z − S.

### What is the effective nuclear charge of a valence electron of iron?

Iron is (1s²)(2s² 2p⁶)(3s² 3p⁶)(3d⁶)(4s²). For a 4s electron, S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25, so Zeff = 26 − 22.25 = 3.75. A 3d electron of iron feels much more: S = 0.35 × 5 + 1.00 × 18 = 19.75 and Zeff = 6.25.

### Do 3d electrons shield a 4s electron by 0.85 or 1.00?

By 0.85. In Slater’s rules every electron in the shell below (n − 1) shields an s or p electron by 0.85, whether it is in s, p, d or f, so all 14 electrons with n = 3 in iron (3s² 3p⁶ 3d⁶) count 0.85 for a 4s electron. A 3d electron itself is shielded by 1.00 from every electron to its left.

### Why does Zeff explain periodic trends?

Across a period the nuclear charge grows by 1 each step while shielding grows only slightly, so the outer electrons feel a larger Zeff and the atoms get smaller. Down a group, new shells add core electrons that shield well, so the outer electrons are held less tightly.

### How accurate are Slater’s rules?

They are a quick approximation from 1930: they give the same Zeff to s and p electrons of one shell, and later screening constants worked out from wave functions (Clementi and Raimondi, 1963) differ from them. They are good for comparing atoms and electrons, not for exact energies.

## Sources

- J. C. Slater, Atomic Shielding Constants, Physical Review 36, 57–64 (1930): the original paper, at the publisher. https://doi.org/10.1103/PhysRev.36.57
- Wikipedia, Slater’s rules: the groups, the 0.35 (0.30 for 1s), 0.85 and 1.00 contributions, and the iron example (4s 3.75, 3d 6.25, 3s and 3p 14.75, 2s and 2p 21.85, 1s 25.70). https://en.wikipedia.org/wiki/Slater%27s_rules (retrieved 2026-10-19)
- LibreTexts, Brett McCollum, 1.3.4 Slater’s Rules (Ursinus College CHEM322: Inorganic Chemistry, CC BY-SA): Zeff = Z − S, the nitrogen 2p example (S = 3.10) and the copper 3d example (Zeff = 7.85). https://chem.libretexts.org/Courses/Ursinus_College/CHEM322%3A_Inorganic_Chemistry/01%3A_Atomic_Structure/1.03%3A_Multi-Electron_Atoms/1.3.04%3A_Slater%27s_Rules (retrieved 2026-10-19)
- OpenStax, Chemistry 2e, §6.5 Periodic Variations in Element Properties: Zeff = Z − shielding, and hydrogen’s Zeff equals Z. https://openstax.org/books/chemistry-2e/pages/6-5-periodic-variations-in-element-properties (retrieved 2026-10-19)
- NIST Standard Reference Database 111, Ground Levels and Ionization Energies for the Neutral Atoms: ground-state configurations for Z = 1 to 108, in its HTML table in the NIST Atomic Spectra Database. https://www.nist.gov/pml/ground-levels-and-ionization-energies-neutral-atoms (retrieved 2026-10-19)