Example data table
| Case | Model | Z | n | Zeff | Energy (eV) | Energy (kJ/mol) |
|---|---|---|---|---|---|---|
| Hydrogen 1s | hydrogenic | 1 | 1 | 1 | -13.6 | -1312.2 |
| He+ 2p | hydrogenic | 2 | 2 | 2 | -13.6 | -1312.2 |
| Na 3s (estimated) | zeff | 11 | 3 | 2.2 | -7.313778 | -705.67 |
| Fe 3d (shielding) | slater | 26 | 3 | computed | computed | computed |
Formula used
- En = −13.6 × (Zeff2 / n2) in eV.
- Zeff = Z for hydrogenic ions (one electron).
- Zeff = Z − S when shielding is applied.
- S uses simplified shielding inputs:
- ns/np: same group × 0.35 (1s uses 0.30), (n−1) × 0.85, (n−2 or lower) × 1.00
- nd/nf: same group × 0.35, lower shells × 1.00
- Optional reduced-mass correction multiplies energy by μ/me ≈ 1 / (1 + (me / (A mp))).
How to use this calculator
- Choose a model: hydrogenic, manual effective charge, or shielding inputs.
- Enter Z and n.
- If using manual effective charge, enter Zeff (≤ Z).
- If using shielding inputs, select orbital family and electron counts.
- Optionally enable reduced-mass correction and supply A.
- Press Calculate, then export CSV or PDF.
Article
Hydrogenic energy scale in electronvolts
Single‑electron ions follow En = −13.6·Z²/n². For Z=1, n=1 gives −13.6 eV, while n=2 gives −3.40 eV, a 4× reduction. For He⁺ (Z=2), the n=1 level is −54.4 eV. The calculator reproduces these values instantly and converts them to kJ/mol.
Effective nuclear charge and screening
Multi‑electron atoms deviate because inner electrons shield the nucleus. Using Zeff replaces Z in the same equation, so energy depends on Zeff². For a sodium 3s estimate with Zeff≈2.2 and n=3, En≈−7.31 eV (≈−706 kJ/mol). Small Zeff changes matter; increasing Zeff by 0.1 raises |En| by about 9%.
Simplified shielding inputs for quick estimates
When you choose shielding inputs, the page computes S and sets Zeff = Z − S. For ns/np, same‑group electrons contribute 0.35 each (1s uses 0.30), (n−1) contributes 0.85 each, and lower shells contribute 1.00 each. For nd/nf, same‑group uses 0.35 and lower shells use 1.00. A breakdown table shows each contribution.
Reduced‑mass correction for isotope awareness
Orbital energies scale with the reduced mass μ, not exactly the electron mass. The optional factor μ/me ≈ 1/(1+me/(A·mp)) slightly lowers the magnitude for light nuclei. For hydrogen A=1, the factor is about 0.99946; for heavier nuclei it approaches 1.00000. The calculator applies this factor consistently across the graph and exports.
Energy trends and the Plotly curve
The Plotly chart plots En versus n at fixed Z and Zeff. Because En ∝ −1/n², the curve rises toward zero quickly as n increases. This helps compare excited states, estimate ionization thresholds, and visualize how shielding shifts the entire curve upward. The selected n is highlighted so your input matches the plotted point.
Reporting, validation, and repeatable outputs
Chemistry workflows often need transparent assumptions. The tool validates Z and n ranges, enforces Zeff ≤ Z, and blocks negative shielding counts. After calculation, you can export a CSV for spreadsheets or a compact PDF for lab notes. Keeping the same inputs reproduces identical numbers, supporting audit trails for coursework, tutorials, and quick checks with clear units and context.
FAQs
What does a negative orbital energy mean?
It means the electron is bound to the nucleus. The more negative the value, the more tightly bound the orbital is, and the more energy is required to ionize the electron.
Which model should I choose for atoms versus ions?
Use hydrogenic for single‑electron ions like H, He⁺, or Li²⁺. Use manual Zeff when you already have an estimate from data. Use shielding inputs for quick classroom‑style Zeff approximations.
Can I use this for multi‑electron excited states?
It gives an effective one‑electron estimate, not a full many‑electron calculation. It is useful for trends, rough comparisons, and teaching. For precise spectroscopy, use quantum‑chemistry or experimental term values.
Why is Zeff limited to be less than or equal to Z?
Effective nuclear charge represents the net attraction felt by an electron. Screening can reduce that attraction, but it cannot exceed the full nuclear charge Z for the same nucleus in this simplified model.
How should I set the shielding electron counts?
Enter electrons in the same n and family group excluding the target electron, then count electrons in the (n−1) shell and lower shells. The breakdown table shows how each group contributes to S.
What do the CSV and PDF exports include?
They include the selected model, Z, n, Zeff, optional S, the reduced‑mass factor, and energies in eV, joules, and kJ/mol. Use CSV for analysis and PDF for quick documentation.