What is the effective nuclear charge?
Pick an element and which electron you mean. The calculator writes the element’s ground-state configuration in Slater’s groups, adds up the shielding constant S, and gives the effective nuclear charge Zeff = Z − S.
- Effective nuclear charge (Zeff)
- 3.75
By Slater’s rules, a 4s electron of iron (Fe) feels Zeff = 3.75.
- Shielding constant (S)
- 22.25
- Nuclear charge (Z)
- 26
- Working
- S = 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25; Zeff = 26 − 22.25 = 3.75
- Slater groups
- (1s²) (2s² 2p⁶) (3s² 3p⁶) (3d⁶) (4s²)
- Electron
- a 4s electron of iron (Fe)
Effective nuclear charge (Zeff): 3.75. By Slater’s rules, a 4s electron of iron (Fe) feels Zeff = 3.75.
How to calculate
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.
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.
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.
- 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
Each example is checked against the calculator on every build.
- Element 26 Fe (Iron), Electron in Outermost electron gives Shielding constant (S) 22.25, Effective nuclear charge (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
- Element 26 Fe (Iron), Electron in 3d gives Shielding constant (S) 19.75, Effective nuclear charge (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
- Element 26 Fe (Iron), Electron in 3s or 3p gives Shielding constant (S) 11.25, Effective nuclear charge (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
- Element 26 Fe (Iron), Electron in 1s gives Shielding constant (S) 0.3, Effective nuclear charge (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
- Element 7 N (Nitrogen), Electron in Outermost electron gives Shielding constant (S) 3.1, Effective nuclear charge (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
- Element 29 Cu (Copper), Electron in 3d gives Shielding constant (S) 21.15, Effective nuclear charge (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
How it works
Formula. Zeff = Z − S, where Z is the atomic number and S the shielding constant of the chosen electron.
Configuration. The neutral atom’s ground state: subshells filled in the Madelung order (1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p, holding 2, 6, 10 and 14 electrons for s, p, d and f), replaced by NIST’s ground state for the 20 elements where it differs (the noble-gas core, then the subshells listed). Elements 109 to 118 use the filling order.
| Z | Element | NIST ground state |
|---|---|---|
| 24 | Cr | [Ar] 3d⁵ 4s¹ |
| 29 | Cu | [Ar] 3d¹⁰ 4s¹ |
| 41 | Nb | [Kr] 4d⁴ 5s¹ |
| 42 | Mo | [Kr] 4d⁵ 5s¹ |
| 44 | Ru | [Kr] 4d⁷ 5s¹ |
| 45 | Rh | [Kr] 4d⁸ 5s¹ |
| 46 | Pd | [Kr] 4d¹⁰ |
| 47 | Ag | [Kr] 4d¹⁰ 5s¹ |
| 57 | La | [Xe] 5d¹ 6s² |
| 58 | Ce | [Xe] 4f¹ 5d¹ 6s² |
| 64 | Gd | [Xe] 4f⁷ 5d¹ 6s² |
| 78 | Pt | [Xe] 4f¹⁴ 5d⁹ 6s¹ |
| 79 | Au | [Xe] 4f¹⁴ 5d¹⁰ 6s¹ |
| 89 | Ac | [Rn] 6d¹ 7s² |
| 90 | Th | [Rn] 6d² 7s² |
| 91 | Pa | [Rn] 5f² 6d¹ 7s² |
| 92 | U | [Rn] 5f³ 6d¹ 7s² |
| 93 | Np | [Rn] 5f⁴ 6d¹ 7s² |
| 96 | Cm | [Rn] 5f⁷ 6d¹ 7s² |
| 103 | Lr | [Rn] 5f¹⁴ 7s² 7p¹ |
Slater’s groups, in order. [1s] [2s, 2p] [3s, 3p] [3d] [4s, 4p] [4d] [4f] [5s, 5p] [5d] [5f] [6s, 6p] [6d] [7s, 7p]. The s and p subshells of a shell form one group.
Shielding S of the chosen electron (sum over every other electron):
| Other electron | Chosen electron in an s or p group | Chosen electron in a d or f group |
|---|---|---|
| In the same group | 0.35 each (0.30 if the group is 1s) | 0.35 each |
| In a group to the left, in shell n − 1 (any of s, p, d, f) | 0.85 each | 1.00 each |
| In a group to the left, in shell n − 2 or lower | 1.00 each | 1.00 each |
| In a group to the right | 0 | 0 |
Here n is the chosen electron’s principal quantum number. S is a whole number of hundredths, so Zeff is shown exactly to 2 decimals.
Which electron. “Outermost” is an electron in the last occupied group of the list: 4s for iron, 2p for nitrogen, 4d for palladium (which has no 5s electrons). Or pick a group; a group the element has no electrons in gives no answer.
Worked examples by hand
Iron, 4s (Z = 26). Groups (1s²)(2s² 2p⁶)(3s² 3p⁶)(3d⁶)(4s²). Same group: 1 other × 0.35. Shell 3 (3s² 3p⁶ 3d⁶): 14 × 0.85. Shells 1 and 2: 10 × 1.00. S = 0.35 + 11.90 + 10.00 = 22.25; Zeff = 26 − 22.25 = 3.75.
Iron, 3d. Same group 5 × 0.35 = 1.75; the 18 electrons to the left × 1.00. S = 19.75, Zeff = 6.25.
Iron, 3s or 3p. Same group 7 × 0.35 = 2.45; shell 2: 8 × 0.85 = 6.80; 1s: 2 × 1.00. The 3d and 4s groups are to the right. S = 11.25, Zeff = 14.75.
Iron, 1s. One other 1s electron × 0.30: S = 0.30, Zeff = 25.70.
Nitrogen, 2p (Z = 7). (1s²)(2s² 2p³): S = 0.85 × 2 + 0.35 × 4 = 3.10; Zeff = 3.90.
Copper, 3d (Z = 29). (1s²)(2s² 2p⁶)(3s² 3p⁶)(3d¹⁰)(4s¹): S = 1.00 × 18 + 0.35 × 9 = 21.15; Zeff = 7.85.
Other questions people ask
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.