Effective Nuclear Charge Calculator (Zeff by Slater’s rules, elements 1–108)

Understanding the effective nuclear charge calculator — concept and principle

This calculator estimates the effective nuclear charge Zeff that an electron feels, using Slater's rules, for elements 1 to 108. Chemistry students use it to explain trends in atomic size, ionisation energy and electronegativity across the periodic table.

How it is calculated

Zeff = Z − S

• Z is the atomic number. S is the shielding constant from the other electrons.
• Write the configuration in groups: (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f)(5s,5p)… Electrons in groups to the right add nothing.
• For an s or p electron: each other electron in the same group adds 0.35 (0.30 within 1s), each electron in shell n − 1 adds 0.85, and each deeper electron adds 1.00.
• For a d or f electron: others in the same group add 0.35 and every electron in groups to the left adds 1.00.
• The configurations are the measured ground states.

The principle

Inner electrons sit mostly between the nucleus and an outer electron, so they cancel much of the nuclear charge. Electrons in the same shell are spread around at similar distances and screen each other only a little. Slater turned this picture into simple fixed weights in 1930. The result is an approximation: calculated Hartree–Fock values, such as those of Clementi and Raimondi, differ somewhat, but the trends are the same.

Worked example

Default inputs: chlorine (Z = 17), outermost electron.

1. Configuration 1s² 2s² 2p⁶ 3s² 3p⁵; the valence electron is in the (3s,3p) group, which holds 7 electrons.
2. Same group: 6 other electrons × 0.35 = 2.1.
3. Shell n − 1 (2s, 2p): 8 × 0.85 = 6.8.
4. Deeper (1s): 2 × 1.00 = 2.
5. S = 2.1 + 6.8 + 2 = 10.9, so Zeff = 17 − 10.9 = 6.1. The effective quantum number n* is 3.

How to read the result

A larger Zeff means the electron is held more tightly. Across a period Zeff for the valence electron rises by about 0.65 per element, which is why atoms shrink and ionisation energy rises from left to right. Down a group Zeff changes little while n grows, so atoms get larger. Zeff/n* is a rough measure of how strongly the outer electron is bound.

Common mistakes

• Counting the electron itself in the same-group shielding.
• Using 0.85 for d or f electrons; for them all inner groups count 1.00.
• Grouping 3d with 3s and 3p. Slater puts d and f in separate groups.

Frequently asked questions

Why is Zeff for sodium's 3s electron so low?

Ten inner electrons screen it, eight of them at 0.85 and two at 1.00, leaving Zeff = 2.2.

Do 4s electrons feel a larger Zeff than 3d electrons?

Under Slater's rules they are treated differently, and the 4s electron of a transition metal usually gets the lower value, which fits 4s being removed first in ionisation.

What are Clementi–Raimondi values?

Effective charges derived from self-consistent field wavefunctions published in 1963, often quoted as a more accurate alternative.

Sources: J. C. Slater, "Atomic Shielding Constants", Phys. Rev. 36, 57–64 (1930) · Particle Data Group, Review of Particle Physics 2024, Table 5.1 (NIST ground-state configurations)