Effective Nuclear Charge Calculator

Find the effective nuclear charge of any electron from an element and its orbital.

Covers all 118 elements with Slater’s rules for s, p, d, and f orbitals.

Updated August 24, 2026

CreatorFrank Zhao

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Introduction

An electron in an atom is pulled toward the nucleus, but it doesn't feel the full nuclear charge. Every other electron sitting between it and the nucleus partially blocks that pull — a screening effect chemists call shielding. The charge an electron actually experiences after this screening is the effective nuclear charge, written $Z_{\mathrm{eff}}$ .

This calculator applies Slater's rules — a classic set of empirical rules for estimating shielding — to any electron in any of the 118 elements. Pick an element, then choose the orbital of the electron you care about, and it returns the total shielding $\sigma$ and the effective nuclear charge $Z_{\mathrm{eff}}$ .

Who is this for? Students working through periodic trends, anyone comparing how tightly different electrons are held in an atom, and learners who want to see Slater's rules applied step by step instead of memorizing the result.

How to use

Pick an element. The calculator shows its electron configuration, which is the map you'll use to choose an electron.

Pick the principal quantum number ( $n$ ) of the shell the electron lives in — for example, 3 for a 3p electron.

Pick the azimuthal quantum number ( $l$ ) — the orbital type s, p, d, or f.

Read the two results: the total shielding and the effective nuclear charge for that electron.

Worked example — a 3p electron in selenium

Selenium has atomic number $Z = 34$ and the configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 3d¹⁰ 4s² 4p⁴. Choose the element, then set $n = 3$ and $l = p$ .

\sigma

=

7 \times 0.35

+

8 \times 0.85

+

2 \times 1.00

=

11.25

Z_{\mathrm{eff}}

=

34 - 11.25

=

22.75

The calculator reports a shielding of 11.25 and an effective nuclear charge of 22.75. A 3p electron in selenium therefore feels a pull equivalent to about 23 protons, not the full 34.

Calculation method

The effective nuclear charge is simply the nuclear charge minus the shielding:

Z_{\mathrm{eff}} = Z - \sigma

The shielding $\sigma$ comes from Slater's rules, which group the electron configuration and assign each group a screening constant. Electrons in orbitals above the chosen one contribute nothing — only electrons at or below it screen the nucleus.

For an ns or np electron:

\sigma

=

0.35\,N_{\mathrm{same}}

+

0.85\,N_{n-1}

+

1.00\,N_{\mathrm{lower}}

The 1s shell is the exception: its two electrons screen each other with 0.30 instead of 0.35.

For an nd or nf electron:

\sigma

=

0.35\,N_{\mathrm{same}}

+

1.00\,N_{\mathrm{lower}}

Where:

$Z$ — atomic number (the number of protons)
$N_{\mathrm{same}}$ — other electrons in the same shell (for ns/np) or the same group (for nd/nf)
$N_{n-1}$ — electrons in the shell one level below (ns/np only)
$N_{\mathrm{lower}}$ — electrons in all lower shells or groups

The two results are read-only: the calculator derives them from your element and orbital choices, so there is no reverse direction to solve for. If you pick a combination that doesn't exist in the element's configuration — for example a 3f orbital, or a shell the atom doesn't reach — the result stays blank and the calculator explains which choice is invalid.

Tips & best practices

Choose an orbital that actually exists

The electron configuration shown under the element is your checklist. If the atom has no 3d electrons, picking n = 3 and l = d will not produce a result — that orbital simply is not occupied.

Outer electrons do not shield

Electrons in higher shells sit outside the one you chose and contribute zero shielding. This is why the calculator only counts groups at or below the chosen orbital.

Compare orbitals within the same atom

A 3d electron and a 4s electron in the same atom feel very different charges. For iron, the calculator gives a 3d electron an effective charge near 6.25 while a 3s electron feels about 14.75 — a useful way to see how inner electrons are held more tightly.

Use the configuration display as a guide

Before choosing n and l, read the configuration the calculator prints. It tells you exactly which shells and orbital types are occupied, so you can pick a valid electron the first time.

If you want to see how an element's configuration is built before choosing an electron, the electron configuration calculator walks through the full configuration for any element.

Frequently asked questions

Why does the calculator show an error for some orbital choices?

The chosen $(n, l)$ pair must match an orbital that is actually occupied in the element's configuration. If the principal quantum number is higher than any shell the atom reaches, or the orbital type isn't present at that shell, there is no real electron to analyze, so the result stays blank with a message explaining which choice is invalid.

Why is the effective nuclear charge different for different orbitals of the same atom?

Electrons in different orbitals are screened by different numbers of inner electrons. A deeper orbital sits closer to the nucleus and is shielded by fewer electrons, so it feels a higher effective charge; a valence orbital is screened by more inner shells and feels a lower one.

Does the effective nuclear charge increase across a period or down a group?

Across a period, the nuclear charge rises while shielding stays roughly similar, so $Z_{\mathrm{eff}}$ increases from left to right. Down a group, a full inner shell is added below the valence electrons, shielding increases, and the outermost electron feels a smaller effective charge.

Is Slater's rules exact?

No. Slater's rules are an empirical approximation that fits observed data reasonably well for many elements, but more detailed methods (such as Hartree–Fock calculations) can give different values. Treat the result as a good estimate for understanding trends, not a precise quantum-mechanical measurement.

Limitations

•Slater's rules are an approximation. They ignore electron–electron correlation, relativistic effects, and the detailed shape of orbitals, so the shielding and effective charge are estimates rather than exact values. Different methods can produce slightly different numbers for the same electron.
•The calculator covers the standard electron configurations used with Slater's rules, including the common transition-metal and lanthanide/actinide exceptions. It is intended for education and estimation — coursework, periodic trends, or building intuition about how tightly electrons are held.
•For research-grade values, consult a quantum-chemistry reference or computational tool rather than relying on this estimate.

Next step: comparing how atoms attract electrons in bonds? The electronegativity calculator is a natural follow-up — electronegativity is closely tied to how strongly an atom's nucleus pulls on bonding electrons.

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