What this calculator does
Effective nuclear charge (Zeff) is the net positive charge an electron actually experiences, once the pull of the nucleus has been partly cancelled out by the repulsion of the other electrons sitting between it and the nucleus. It is always less than the full atomic number Z, because inner and same-shell electrons shield some of that charge.
Slater's Rules are the standard pencil-and-paper method for estimating that shielding without a full quantum-mechanical calculation. They group an atom's electron configuration into shells, apply a fixed shielding contribution depending on how close each group sits to the electron in question, and subtract the total from the atomic number.
The formula
Electrons are grouped as (1s)(2s,2p)(3s,3p)(3d)(4s,4p)(4d)(4f) and so on. For an electron in an s or p subshell, other electrons in the same group each contribute 0.35 (0.30 within 1s), electrons one shell in (n-1) contribute 0.85 each, and electrons two or more shells in contribute 1.00 each. For an electron in a d or f subshell, other electrons in the same group contribute 0.35, and every electron in a group to the left contributes 1.00, with electrons in groups to the right contributing nothing at all. Zeff is then the atomic number minus this total shielding constant S.
| Term | Meaning |
|---|---|
| Zeff | Effective nuclear charge felt by the electron: Z minus the shielding constant. |
| Z | Atomic number: the number of protons in the nucleus. |
| S | Shielding constant: the total screening contributed by the other electrons, per Slater's Rules. |
The inputs explained
| Field | What to enter |
|---|---|
| Atomic number (Z) | The atomic number of the element (number of protons), from 1 to 118. |
| Subshell of the electron (e.g. 3p) | The subshell holding the electron of interest, written as n and the letter, such as 3p or 4s. |
When to use it
Comparing atomic size trends
Zeff calculated for the outermost electron across a period explains why atomic radius shrinks left to right: the nuclear charge climbs faster than the shielding does.
Explaining ionisation energy patterns
A higher Zeff on the outermost electron means it is held more tightly, which is a large part of why ionisation energy generally rises across a period.
Checking a Slater's Rules worked example by hand
Running the same atomic number and subshell through the calculator checks a hand calculation for Slater's Rules step by step, including the tricky (n-1) versus (n-2) grouping.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How effective nuclear charge changes across period 3 for the outermost 3p or 3s electron
The atomic number rises by one across the period while the shielding constant barely moves, so Zeff climbs steadily.
| Atomic number (element) | Effective nuclear charge (Zeff) | Shielding constant (S) |
|---|---|---|
| 13 | 3.50 | 9.50 |
| 14 | 4.15 | 9.85 |
| 15 | 4.80 | 10.20 |
| 16 | 5.45 | 10.55 |
| 17 | 6.10 | 10.90 |
| 18 | 6.75 | 11.25 |
How Zeff for a 3d electron compares across the first transition series
The 3d electrons shield each other only weakly, and the outer 4s electrons do not shield them at all under Slater's Rules, since 4s sits to the right of 3d in the grouping.
| Atomic number (element) | Effective nuclear charge (Zeff) | Shielding constant (S) |
|---|---|---|
| 21 | 3.00 | 18.00 |
| 23 | 4.30 | 18.70 |
| 25 | 5.60 | 19.40 |
| 26 | 6.25 | 19.75 |
| 28 | 7.55 | 20.45 |
| 30 | 8.85 | 21.15 |
Questions
Why is a 1s electron shielded at 0.30 instead of 0.35?
Slater set this as a special case for the innermost shell: the two 1s electrons are so close to the nucleus and to each other that the general 0.35 same-group value slightly overestimates the shielding, so 0.30 is used instead.
Do outer electrons shield inner ones at all?
No. Slater's Rules state that electrons in groups further out than the one being examined contribute zero shielding, since an outer electron spends essentially no time between the nucleus and an inner electron.
Why does a 4s electron feel more shielding than a 3d electron in the same atom?
A 4s electron is shielded by the entire n=3 shell, including 3s, 3p and 3d, each at 0.85, plus everything below it at 1.00. A 3d electron is shielded only by groups to its left (1s through 3p) at 1.00 each, and the 4s electrons do not shield it at all, since 4s sits to the right of 3d in the Slater grouping.
How accurate is Slater's Rules compared with a full quantum calculation?
Slater's Rules give a useful, hand-calculable approximation and reproduce the correct trends well, but they are not exact. More accurate effective nuclear charge values exist from self-consistent field calculations and differ from the Slater estimate, particularly for heavier atoms.
For the atomic mass side of an element's properties rather than shielding, see the average atomic mass calculator. To check a full electron configuration, see the atom composition calculator.