# Effective Nuclear Charge Calculator

URL: https://speedycalculator.com/calculators/chemistry/effective-charge/
Last updated: 2026-10-30

> The effective nuclear charge calculator applies Slater’s rules to any element or ion: pick the electron’s orbital and get the shielding constant σ, the effective nuclear charge Z_eff = Z − σ, a table of every shielding contribution and the Z_eff felt by electrons in each occupied subshell.

## Inputs

- Element (symbol, name or Z) — Type a symbol (Fe), a name (iron) or an atomic number (26).
- Principal quantum number n of the electron
- Subshell (azimuthal quantum number l)
- Charge (0 for a neutral atom)

## Results

- Effective nuclear charge (Z_eff)
- For an electron in
- Shielding constant (σ)
- Nuclear charge (Z)
- Slater effective quantum number n*
- Slater orbital exponent ζ = Z_eff/n*
- Electron configuration used
- Shielding contributions
- Z_eff for every occupied subshell
- Calculation

## Formula

```
Z_eff = Z − σ
σ for an ns/np electron: 0.35 per other electron in the same (ns, np) group (0.30 for 1s), 0.85 per electron in shell n − 1, 1.00 per electron in shells n − 2 and lower
σ for an nd/nf electron: 0.35 per other electron in the same group, 1.00 per electron in every group to its left
```

## Explanation

An electron in a many-electron atom does not feel the full charge of the nucleus. The other electrons, especially those closer to the nucleus, screen part of it. The charge that is left over is the effective nuclear charge, Z_eff = Z − σ, where σ is the shielding (screening) constant. It explains most periodic trends: across a period Z rises by one per element but σ only by about 0.35, so Z_eff grows, atoms shrink and ionisation energies increase.

In 1930 John C. Slater proposed simple rules to estimate σ. Write the configuration in groups – (1s)(2s, 2p)(3s, 3p)(3d)(4s, 4p)(4d)(4f)(5s, 5p)… – and add up the contributions of all electrons except the one you are looking at. Electrons to the right of its group do not shield it; electrons in the same group shield 0.35 each; for s and p electrons the shell just below shields 0.85 each and deeper shells 1.00, while d and f electrons are shielded fully (1.00) by everything to their left.

The calculator builds the ground-state configuration (with the real exceptions such as chromium and copper, and for ions as well), applies the rules to the orbital you choose and also lists Z_eff for every occupied subshell and Slater’s orbital exponent ζ = Z_eff/n*.

### How to use the Effective Nuclear Charge Calculator

- **Element (symbol, name or Z)** — Type a symbol (Fe), a name (iron) or an atomic number (26).
- **Principal quantum number n of the electron**
- **Subshell (azimuthal quantum number l)**
- **Charge (0 for a neutral atom)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The element’s ground-state configuration (or the ion’s, removing electrons from the outermost shell first) is divided into Slater groups. For the chosen electron, every other electron is assigned a factor of 0, 0.35 (0.30 in 1s), 0.85 or 1.00 according to its group, the products are summed to σ, and Z_eff = Z − σ. If the chosen orbital is empty, the outermost electron is used.

### Worked example

Selenium (Z = 34) is [Ar] 3d¹⁰ 4s² 4p⁴, which in Slater groups is (1s)²(2s, 2p)⁸(3s, 3p)⁸(3d)¹⁰(4s, 4p)⁶. For a **3p** electron: the seven other electrons in (3s, 3p) shield 7 × 0.35 = 2.45; the eight electrons of shell n − 1 = 2 shield 8 × 0.85 = 6.80; the two 1s electrons (n − 2) shield 2 × 1.00 = 2.00; the 3d and 4s/4p electrons lie to the right and do not count. σ = 2.45 + 6.80 + 2.00 = 11.25, so Z_eff = 34 − 11.25 = **22.75**.

For the outermost **4p** electron: 5 × 0.35 + 18 × 0.85 + 10 × 1.00 = 1.75 + 15.30 + 10.00 = 27.05, so Z_eff = 6.95.

### Slater’s shielding factors

Electron being shieldedSame groupShell n − 1Shell n − 2 and lowerGroups to the right1s0.30––0ns or np0.350.851.000nd or nf0.351.001.000

### Tips

- Type formulas with normal capitals and numbers, e.g. Ca(OH)2 or CuSO4·5H2O – brackets and hydrates work.
- Atomic masses are IUPAC standard atomic weights; radioactive elements use the mass number of their longest-lived isotope.
- Electronegativity values are on the Pauling scale.
- Everything is calculated in your browser.

### Common mistakes to avoid

- Writing CO (carbon monoxide) when you mean Co (cobalt) – capitals matter.
- Forgetting to multiply the bracket contents by the subscript after the bracket.
- Mixing up atomic number (protons) and mass number (protons + neutrons).

## Worked example

Inputs — Element (symbol, name or Z): Se; Principal quantum number n of the electron: 3; Subshell (azimuthal quantum number l): p (l = 1); Charge (0 for a neutral atom): 0.

Results — **Effective nuclear charge (Z_eff) = 22.75**; For an electron in = 3p electron of selenium (Se); Shielding constant (σ) = 11.25; Nuclear charge (Z) = 34; Slater effective quantum number n* = 3; Slater orbital exponent ζ = Z_eff/n* = 7.583.

## Key terms

- **Shielding (screening) constant σ**: The part of the nuclear charge cancelled by the other electrons.
- **Effective nuclear charge Z_eff**: The net charge felt by one electron, Z − σ.
- **Slater group**: One of the brackets (1s)(2s, 2p)(3s, 3p)(3d)(4s, 4p)… used to assign shielding factors.
- **Penetration**: The ability of an orbital (s > p > d > f) to reach close to the nucleus and escape shielding.

## FAQ

**What is effective nuclear charge?**
The net positive charge an electron actually feels: the nuclear charge Z minus the shielding σ of the other electrons. A 2p electron of neon feels about +5.85, not +10.

**How do I calculate Z_eff with Slater’s rules?**
For neon’s 2p electron: 7 other electrons in (2s, 2p) shield 0.35 each = 2.45, and two 1s electrons shield 0.85 each = 1.70, so σ = 4.15 and Z_eff = 10 − 4.15 = 5.85.

**How does Z_eff change across a period?**
It increases by about 0.65 per element (one more proton, but the new electron in the same shell shields only 0.35). From Li to Ne the outer-electron Z_eff rises from 1.30 to 5.85.

**How does Z_eff change down a group?**
For the outermost electron it stays nearly constant or grows only slightly, while the electron sits in a larger shell – so atoms get bigger and lose electrons more easily down a group.

**Why do 4s electrons leave before 3d electrons in ions?**
In iron, a 4s electron feels Z_eff = 3.75 by Slater’s rules while a 3d electron feels 6.25. The 4s electrons are held less tightly, so they are removed first.

**How accurate are Slater’s rules?**
They are an approximation. Clementi and Raimondi (1963) derived values from Hartree–Fock wavefunctions that are often somewhat larger, but Slater’s rules reproduce the trends well and are the standard textbook method.

**Do the rules work for ions?**
Yes – use the ion’s configuration. Na⁺ has a 2p electron with Z_eff = 11 − 4.15 = 6.85, much higher than the 2.20 felt by the 3s electron of neutral sodium.

**What formula does the Effective Nuclear Charge Calculator use?**
Z_eff = Z − σ.


## Sources

- [J. C. Slater – Atomic Shielding Constants, Physical Review 36, 57 (1930)](https://doi.org/10.1103/PhysRev.36.57)
- [OpenStax – Chemistry: Atoms First 2e](https://openstax.org/details/books/chemistry-atoms-first-2e)
- [LibreTexts Chemistry](https://chem.libretexts.org/)