Effective Nuclear Charge Calculator
Estimate the shielding constant and effective nuclear charge experienced by a selected electron using Slater's rules.
Atom and electron
Choose an element from hydrogen through xenon.
The shell number of the selected electron.
0 = s, 1 = p, 2 = d, 3 = f.
Results
Shielding contribution breakdown
| Electron group | Electron count | Factor per electron | Contribution to σ |
|---|---|---|---|
| Same n group (3s, 3p), excluding selected electron | 7 | 0.35 | 2.45 |
| n – 1 shell | 8 | 0.85 | 6.80 |
| n – 2 or lower shells | 2 | 1.00 | 2.00 |
The table is generated from the same canonical model used for the headline result and Excel workbook.
How to use the effective nuclear charge calculator
What this calculator does
This calculator estimates the effective nuclear charge, written as Zeff, experienced by one electron in an occupied atomic orbital. It uses Slater's rules to approximate how strongly the other electrons shield the positive nuclear charge. The result is an educational screening estimate, not a full quantum-mechanical calculation of electron density, orbital energy, or ionization energy.
When to use it
Use it to check chemistry homework involving shielding, compare how inner and outer electrons in the same atom experience the nucleus, study periodic trends, or build intuition before using more advanced atomic-structure methods. Slater's rules are especially useful when a problem supplies an electron configuration and asks for a transparent hand calculation.
How to calculate
- The calculator opens with selenium and a 3p electron as a complete demonstration. The displayed result and the Excel workbook are immediately available.
- Choose an Element. Its ground-state electron configuration is generated, and the available shell choices update automatically.
- Choose the Principal quantum number (n), then the Azimuthal quantum number (l). Only occupied orbitals are offered.
- Read Effective nuclear charge (Zeff), Shielding constant (σ), Nuclear charge (Z), Selected electron, and Shielded fraction. The table shows how each electron group contributes to σ.
- Select Download Excel to export the current typed inputs, outputs, and contribution rows. Reset clears the demonstration and disables export until a complete valid selection is entered again.
Input guide
Element is required and uses an atomic number selected from hydrogen through xenon. For example, selenium is atomic number 34. A higher atomic number increases the bare nuclear charge, but it also adds shielding electrons, so Zeff does not rise one-for-one. Principal quantum number (n) is a required integer shell index such as 3. Azimuthal quantum number (l) is also required: 0 means s, 1 means p, 2 means d, and 3 means f. A common mistake is selecting an orbital that is not occupied; this interface prevents that by limiting options to the chosen element's configuration.
Output guide
Effective nuclear charge (Zeff) is the net positive charge felt by the selected electron in units of the elementary charge. Shielding constant (σ) is the summed screening contribution from the other electrons. Nuclear charge (Z) equals the atomic number. Selected electron confirms the orbital used. Shielded fraction is σ divided by Z and is a descriptive ratio, not a probability. The contribution table lists each electron group, its count, Slater factor, and contribution to σ; the contributions sum exactly to the displayed shielding constant.
Worked example
For selenium, Z = 34 and the selected orbital is 3p. Seven other electrons in the 3s/3p group contribute 7 × 0.35 = 2.45. Eight electrons in the n – 1 shell contribute 8 × 0.85 = 6.80. The 1s electrons are handled under the lower-shell rule, contributing 2 × 1.00 = 2.00, so σ = 11.25 and Zeff = 34 – 11.25 = 22.75. This illustrates why the exact grouping rule matters. The calculator applies the same orbital-group logic consistently to every supported element.
For deeper background, review the LibreTexts explanation of electron shielding and the NIST Atomic Spectra Database ionization-energy resource.
How Slater's rules work
Slater's rules arrange orbitals into ordered groups and assign weighting factors to electrons that lie in the same group or in inner groups. For an s or p electron, other electrons in the same principal shell usually contribute 0.35 each, electrons one shell inward contribute 0.85 each, and still deeper electrons contribute 1.00 each. The special 1s pair uses 0.30 for the other electron. For d and f electrons, electrons in groups to the left shield more strongly, while electrons to the right do not shield the selected electron in this approximation.
The core identity is Zeff = Z – σ. Because σ is an approximation, calculated values can differ from results produced by self-consistent-field methods or experimental proxies. The value is still useful for comparing orbitals and explaining why valence electrons are generally easier to remove than core electrons.
Interpretation and limitations
A large Zeff means the selected electron experiences a stronger net attraction to the nucleus under the model. Inner electrons often have high Zeff because relatively few electrons lie between them and the nucleus. Outer electrons are more heavily shielded. Do not interpret Zeff as an orbital energy, a measured force at a fixed radius, or a direct prediction of chemical reactivity. Electron correlation, penetration, relativistic effects, and configuration exceptions can all matter beyond this simplified model.
For periodic-table context, the NIST periodic table of the elements provides authoritative element identities and atomic numbers.