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Effective Nuclear Charge Calculator

Chemistry

Work out the effective nuclear charge Zeff = Z − σ felt by any electron of an element or ion with Slater's rules, see which electrons do the shielding, and compare a ground state with excited or hypothetical configurations.

Symbol (Fe), name (Iron) or atomic number (26)
Whole number: +2, 2+, -1 or 0

Examples:

Fe — Iron

Z = 2626 electrons[Ar] 3d⁶ 4s²neutral

Zeff felt by a 4s electron of Fe: 3.75

σ = 22.25, so Zeff = 26 − 22.25

Defaults to the outermost electron.
Shielding σ
22.25
Zeff (Slater)
3.75
Orbital exponent ζ
1.014
Zeff / n*, n = 4
Core charge
2
Estimate: Z − inner electrons
Zeff (SCF)
—
No reference value

Slater groups, seen from a 4s electron

(1s)²

fully shields · 1.00

2 × 1.00 = 2.00

(2s,2p)⁸

fully shields · 1.00

8 × 1.00 = 8.00

(3s,3p)⁸

n − 1 · 0.85

8 × 0.85 = 6.80

(3d)⁶

n − 1 · 0.85

6 × 0.85 = 5.10

(4s,4p)²

same group · 0.35

this electron (not counted)

1 × 0.35 = 0.35

Where the nuclear charge goes

0

Z = 26

Fully shielding electrons: 10 × 1.00 = 10.00

n − 1 shell: 14 × 0.85 = 11.90

Same group: 1 × 0.35 = 0.35

Z_eff felt by this electron: 3.75

Of the nucleus's +26, the other electrons cancel 22.25; the 4s electron feels +3.75.

SubshellElectronsSlater groupσZeff (Slater)Zeff (SCF)Core charge
2(1s)0.3025.70——
2(2s,2p)4.1521.85——
6(2s,2p)4.1521.85——
2(3s,3p)11.2514.75——
6(3s,3p)11.2514.75——
6(3d)19.756.25——
outermost
2(4s,4p)22.253.75—2

Slater's rules give ns and np the same value because they share a group; SCF values tell them apart. Click a subshell to follow one of its electrons.

Worked steps

Fe (Z = 26): [Ar] 3d⁶ 4s²
Slater groups: (1s)² (2s,2p)⁸ (3s,3p)⁸ (3d)⁶ (4s,4p)²
Electron of interest: 4s → group (4s,4p)
  same group                   1 × 0.35 =  0.35
  n − 1 = 3: (3s,3p)⁸ (3d)⁶   14 × 0.85 = 11.90
  n ≤ 2: (1s)² (2s,2p)⁸       10 × 1.00 = 10.00
σ = 22.25
Z_eff = 26 − 22.25 = 3.75

Periodic trend (neutral atoms)

Zeff (Slater) of the outermost electron, Z = 1–36

Slater's rules

SCF (Clementi & Raimondi, 1963)

Fe (current element)

Zeff climbs across each period (each new proton is only 35 % cancelled), then drops when a new shell starts. Slater's rules give every alkali metal from Na down exactly 2.20; the SCF values show it actually keeps rising.

Isoelectronic series

Outermost electron: 2p

Ne-like series: Zeff (Slater) of the 2p electron

Same electrons, same shielding (σ is identical across the series: 4.15), one more proton each step: Zeff rises by exactly 1.00, and the ion gets smaller.

Slater's rules are an approximation; SCF values are shown for comparison where available.

About This Tool

Effective Nuclear Charge Calculator – Slater's Rules, Shielding and Trends

An electron in a many-electron atom never feels the full charge of the nucleus. The other electrons get in the way and cancel part of it. The charge that is left is the effective nuclear charge, Z_eff = Z − σ, where σ is the shielding constant. This effective nuclear charge calculator applies Slater's rules to any element or ion, shows which electrons do the shielding, and plots how Zeff changes across the periodic table.

Shielding and penetration

Electrons in inner shells sit between the nucleus and an outer electron almost all the time, so they cancel nearly one unit of charge each. Electrons in the same shell spread over the same region and cancel much less, because an outer electron is often closer to the nucleus than they are. How far an orbital reaches inside the core is called penetration. An s orbital penetrates more than a p orbital of the same shell, and a p more than a d. A more penetrating electron feels a larger effective charge and is held more tightly.

Slater's rules step by step

Write the configuration in Slater groups, in this order: (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d) (4f) (5s,5p) (5d) (5f) (6s,6p) …. Then add up what every other electron contributes:

Electron in Same group n − 1 n − 2 or lower To its right (ns, np) 0.35 (1s: 0.30) 0.85 each, any l 1.00 each 0 (nd), (nf) 0.35 1.00 each: every group to the left 0

Nitrogen 2p. N is (1s)² (2s,2p)⁵. The other four n = 2 electrons give 4 × 0.35 = 1.40 and the two 1s electrons give 2 × 0.85 = 1.70. So σ = 3.10 and Zeff = 7 − 3.10 = 3.90.

Zinc 3d and 4s. Zn is [Ar] 3d¹⁰ 4s². For a 4s electron: the other 4s electron gives 0.35, the eighteen n = 3 electrons (3s, 3p and 3d) give 18 × 0.85 = 15.30, and the ten inner electrons give 10.00. σ = 25.65, so Zeff = 4.35. For a 3d electron, the 4s pair is to the right and does not count. The other nine 3d electrons give 9 × 0.35 = 3.15 and the eighteen electrons to the left give 18.00. σ = 21.15, so Zeff = 8.85. The 3d electrons feel twice the charge of the 4s ones.

Periodic trends in effective nuclear charge

Across a period, each step adds one proton and one same-shell electron that shields only 0.35. Zeff on the valence electrons rises by 0.65 per element (Li 1.30 to Ne 5.85). That rising charge pulls the shell inward, which is why atomic radius falls and ionization energy rises across a period. Down a group, a new shell starts and Zeff collapses (Ne 5.85 to Na 2.20). The outer electron is also much farther out, so the atom gets bigger.

Across the lanthanides the 6s value stays flat at 2.85 or 3.00. Slater counts each new 4f electron as a full 1.00 shield of the 6s electron, cancelling the new proton exactly. Real 4f electrons shield poorly, so the outer electrons do feel a slowly rising charge. That is the cause of the lanthanide contraction, which Slater's rules cannot show.

Potassium: 4s or 3d?

Potassium's 19th electron could go into 4s or 3d. With an argon core, a 4s electron feels Zeff = 2.20: the eight n = 3 electrons shield 0.85 each. A 3d electron feels only 1.00, because all eighteen core electrons shield it fully. The 4s electron is bound more tightly, so the ground state is [Ar] 4s¹. Try both in the Custom configuration tab, together with excited states of your own.

Limitations and SCF comparison

Slater's rules were made for quick hand calculation. They treat ns and np alike, give every alkali metal from sodium down the same 2.20, and use fixed numbers for every element. Self-consistent-field (SCF) values from Clementi and Raimondi (1963) are shown beside them where available. They agree well for chlorine 3p (Slater 6.10, SCF 6.116) but not for sodium 3s (2.20 against 2.507) or potassium 4s (2.20 against 3.495). The SCF values also split boron's 2s (2.576) from its 2p (2.421), which Slater cannot do.

These values are approximations
Slater Zeff values are model estimates for comparing electrons and spotting trends. Do not treat them as measured quantities or use them for precise energies.

Frequently Asked Questions

Is the Effective Nuclear Charge Calculator free?

Yes, Effective Nuclear Charge Calculator is totally free :)

Can I use the Effective Nuclear Charge Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Effective Nuclear Charge Calculator?

Yes, any data related to Effective Nuclear Charge Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What is effective nuclear charge?

Effective nuclear charge (Zeff) is the net positive charge an electron actually feels. The nucleus pulls with +Z, but the other electrons cancel part of that pull, so Zeff = Z − σ, where σ is the shielding constant. A 4s electron of iron feels about +3.75, not +26.

How do Slater's rules work?

Write the configuration in Slater groups: (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d) (4f) and so on. For an s or p electron, each other electron in its group shields 0.35 (0.30 in 1s), each electron one shell down shields 0.85 and each deeper electron 1.00. For a d or f electron, every electron in a group to its left shields 1.00. Electrons in groups to the right shield nothing.

Why do 2s and 2p electrons get the same Zeff?

Slater put ns and np in one group and gave every electron in it the same shielding, so the rules cannot tell them apart. In reality the 2s electron penetrates closer to the nucleus and feels a slightly higher charge, which the SCF reference values in the table show.

Why does Slater give every alkali metal from Na down Zeff = 2.20?

Each alkali atom has one electron over a full noble-gas core, with eight electrons in the shell just below it. Those eight shield 0.85 each and everything deeper shields 1.00, so σ is always Z − 2.20. SCF calculations show the real value keeps rising down the group (Na 2.51, K 3.50).

Why are 4s electrons removed before 3d when iron forms Fe²⁺?

By Slater's rules a 4s electron of iron feels Zeff 3.75 while a 3d electron feels 6.25, so the 4s electrons are the more weakly held and leave first. The 4s electrons also do not shield the 3d ones, which is why the 3d value is the same in Fe and Fe²⁺.

How accurate are Slater's rules compared with SCF values?

They are a quick approximation, usually within about 10–20 % of self-consistent-field values for valence electrons of light atoms, and worse for heavy atoms and for telling s from p. Use them for trends and comparisons, not as measured or exact values.