Article — Effective Nuclear Charge Calculator (Z_eff)
Effective nuclear charge calculator (Z_eff via Slater rules)
Effective nuclear charge Z_eff is the apparent positive charge felt by an outer electron after inner electrons partially block (screen) the full nuclear pull. Slater's 1930 rules give Z_eff = Z − σ in a few seconds: each same-shell electron contributes 0.35 to σ, each n−1 shell electron 0.85, and each deeper electron 1.00.
Z_eff is the workhorse parameter behind almost every periodic trend. Atomic radius shrinks across a period because Z_eff rises. First ionization energy rises across a period because Z_eff rises. Electronegativity correlates with Z_eff. Slater's rules let you estimate Z_eff from electron configuration alone, with no quantum mechanics required.
What is effective nuclear charge?
An atom has Z protons in its nucleus. A naive picture says every electron feels the full +Z pull. Reality is different: each electron is repelled by all the other electrons, and especially blocked from the nucleus by those closer in. The net inward force on a given electron is therefore less than Z, by an amount called the shielding (or screening) constant σ.
Z_eff captures the apparent nuclear charge after shielding. For hydrogen with one electron, Z_eff = Z = 1. For sodium's 3s electron, Z = 11 but Z_eff ≈ 2.2 because the ten inner-shell electrons block most of the pull. This single number controls how tightly the outer electron is held, which in turn determines ionization energy, electron affinity, and ultimately reactivity.
John Slater published the rules that bear his name in Physical Review in 1930, at age 30. He was trying to make quantum-mechanical results accessible to chemists who didn't want to solve the Schrödinger equation. The rules survive almost a century later because they are both fast and roughly right.
Slater rules for shielding
The rules assign a shielding contribution to each electron based on its orbital and proximity to the target:
Same group (ns + np) 0.35 each (1s → 0.30)Shell n − 1 0.85 eachShell n − 2 or deeper 1.00 eachSame n, d or f 1.00 eachFor nd or nf target electrons, the rules differ slightly: same-group mates still contribute 0.35, but all electrons in groups closer to the nucleus shield fully at 1.00. The intuition: a 3d electron stays farther from the nucleus than 3s or 3p in the same shell because the d radial function peaks farther out, so the inner-shell screening is more complete.
The effective nuclear charge formula
Z_eff = Z − σ. Sum up the shielding contributions from every other electron, subtract from Z, and you have Z_eff. The recipe for any element:
- Write the electron configuration using Aufbau order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s,...
- Group electrons into Slater groups: (1s), (2s 2p), (3s 3p), (3d), (4s 4p), (4d), (4f), and so on.
- Pick the target electron — usually the valence (outermost) electron.
- Apply the shielding rules based on which group each other electron is in.
- Subtract from Z to get Z_eff.
Worked example for nitrogen (Z = 7), configuration 1s² 2s² 2p³. Target: a 2p electron. Same-group electrons: 2 (in 2s) + 2 (other 2p) = 4 × 0.35 = 1.40. Shell n−1 (1s): 2 × 0.85 = 1.70. Total σ = 3.10. Z_eff = 7 − 3.10 = 3.90.
Z_eff across the periodic table
Z_eff rises systematically across each period. Each step adds 1 proton and 1 electron, but the new electron sits in the same shell and shields by only 0.35, so net Z_eff rises by 0.65 per step. Down a group, Z_eff stays nearly flat: each new shell adds full inner-shell shielding for the new outermost electron.
- Li (Z=3) 2s electron: Z_eff = 1.30.
- Be (Z=4) 2s electron: Z_eff = 1.95.
- C (Z=6) 2p electron: Z_eff = 3.25.
- N (Z=7) 2p electron: Z_eff = 3.90.
- O (Z=8) 2p electron: Z_eff = 4.55.
- F (Z=9) 2p electron: Z_eff = 5.20.
- Ne (Z=10) 2p electron: Z_eff = 5.85.
Z_eff and atomic radius
Atomic radius scales inversely with Z_eff: r ∝ n² / Z_eff. Higher apparent nuclear charge pulls electrons closer, shrinking the atom. The periodic-table trend across a row is dominated by Z_eff (rises → radius falls). The trend down a group is dominated by n (rises → radius rises despite roughly constant Z_eff).
The contrast between Li (152 pm) and F (71 pm) illustrates the same-period effect. Both have one shell of inner electrons, but F's Z_eff of 5.20 yanks its valence electrons in to less than half Li's 1.30. Down group 1, Li (152 pm) → Na (186 pm) → K (227 pm): the radii grow because the principal quantum number n keeps rising while Z_eff stays near 1.3–2.2.
Z_eff vs. ionization energy
The hydrogen-like formula gives a first estimate: IE = 13.6 · (Z_eff)² / n² eV. For sodium (n = 3, Z_eff = 2.20): IE ≈ 13.6 × 4.84 / 9 ≈ 7.3 eV. Real value: 5.14 eV. Slater overestimates because it ignores electron correlation and assumes a hydrogen-like radial function, but the trend is right.
The same arithmetic works across the period. C: IE ≈ 13.6 × 10.56 / 4 ≈ 35.9 eV. Real: 11.26 eV. F: IE ≈ 13.6 × 27.04 / 4 ≈ 91.9 eV. Real: 17.42 eV. The absolute numbers diverge but the ratio C: F follows roughly (3.25 / 5.20)² = 0.39 in Slater, against the experimental 11.26 / 17.42 = 0.65. The qualitative trend is preserved.
Slater's rules predict trends correctly but the absolute Z_eff values are systematically low. For more accurate Z_eff, look up Clementi-Raimondi tabulated values (1963), which were computed self-consistently and improve on Slater by 10–20% across the table.
Effective charge pitfalls
Four common errors trip up first-time Slater users:
- Forgetting to subtract the target electron from its own group — only "other" electrons contribute to σ.
- Mis-grouping d and f electrons — 3d is its own Slater group separate from 3s/3p.
- Using 0.35 for 1s — the special rule says 0.30 for 1s-to-1s shielding.
- Skipping Slater anomalies — Cr is 3d⁵ 4s¹, not 3d⁴ 4s². Cu, Mo, Ag also break the pattern.
For 3d and 4d series, Slater's approximations lose accuracy by 15–30% versus Hartree-Fock. The d-orbital radial functions are spatially distinct from s and p, and Slater's coefficients don't capture this well. Use Clementi-Raimondi or modern computational tabulations for transition-metal work.
Slater's 1930 paper was only 4 pages long. The rules he proposed remain in every general-chemistry textbook nearly a century later — a remarkable longevity for a back-of-envelope approximation that bypassed quantum mechanics in an era when QM was less than a decade old.