Effective Nuclear Charge and Shielding

Key Ideas

• Actual nuclear charge (Z): equals the atomic number — the number of protons in the nucleus.
• Shielding (screening): the reduction in the nuclear attraction felt by an electron, caused by the repulsion of other electrons, especially inner/core electrons.
• Effective nuclear charge (Zeff): the net positive charge experienced by an electron after shielding is subtracted.
• Penetration: how close an electron's orbital gets to the nucleus. Electrons that penetrate more are shielded less and therefore feel a larger Zeff.
• Penetration order for a given shell (n): s > p > d > f. A 2s electron penetrates more than a 2p electron, so it is shielded less.
• Across a period: Z increases by one per element, but each added electron enters the same valence shell and shields poorly, so Zeff increases steadily.
• Down a group: valence electrons move into higher-n shells farther from the nucleus; the added inner shells shield them almost completely, so the Zeff felt by valence electrons stays roughly constant even as Z rises.

Equations and Variables

• Zeff = Z − S
  • Zeff = effective nuclear charge (positive, unitless)
  • Z = atomic number (number of protons)
  • S = shielding constant (sum of contributions from all other electrons)

Slater's rules (for an electron in an ns or np group):

• Electrons in the same (ns, np) group contribute 0.35 each (the electron being evaluated is excluded).
• Electrons in the (n − 1) shell contribute 0.85 each.
• Electrons in the (n − 2) shell or lower contribute 1.00 each.

How It Works

1. A nucleus with Z protons attracts every electron with a force proportional to its charge.
2. Inner electrons sit between the nucleus and the outer electrons, so they repel the outer electrons while also blocking part of the nuclear attraction.
3. Each electron therefore experiences a reduced, "effective" charge rather than the full Z.
4. Core electrons shield strongly (each contributes close to 1.0 to S); valence electrons shield one another only weakly (0.35 each).
5. Moving across a period, the added electron enters the same shell, so S stays roughly constant while Z climbs by one each step — meaning Zeff rises left to right.
6. The rising Zeff pulls the valence shell inward (smaller atoms) and holds the electrons more tightly (higher ionization energy, higher electronegativity).

Worked Example

Calculate Zeff for the valence 3s electron in Na (Z = 11) and in Mg (Z = 12) using Slater's rules.

Sodium, Na (1s² 2s² 2p⁶ 3s¹):

• Same group (3s): 0 other electrons → 0 × 0.35 = 0
• (n − 1) = 2 shell (2s² 2p⁶): 8 electrons × 0.85 = 6.80
• (n − 2) and lower (1s²): 2 electrons × 1.00 = 2.00
• S = 0 + 6.80 + 2.00 = 8.80
• Zeff = 11 − 8.80 = 2.20

Magnesium, Mg (1s² 2s² 2p⁶ 3s²):

• Same group (3s): 1 other electron × 0.35 = 0.35
• (n − 1) = 2 shell: 8 × 0.85 = 6.80
• (n − 2) and lower: 2 × 1.00 = 2.00
• S = 0.35 + 6.80 + 2.00 = 9.15
• Zeff = 12 − 9.15 = 2.85

The 3s electron in Mg feels a stronger pull (2.85) than in Na (2.20) — which is why Mg's electron is harder to remove and Mg is a smaller atom.

Common Confusions

• "Zeff is the same as Z." — Wrong: Zeff is always smaller than Z in a multi-electron atom because other electrons shield the nucleus.
• "Valence electrons shield each other fully." — Wrong: same-shell electrons contribute only ~0.35 each; the core electrons (0.85–1.00) do most of the shielding.
• "Zeff increases going down a group." — Wrong: down a group, added inner shells shield the new valence shell almost completely, so the valence Zeff stays roughly constant (the shell number n is what increases).
• "Shielding and penetration are the same thing." — Related but distinct: penetration describes how close an orbital gets to the nucleus; shielding is the resulting reduction in nuclear attraction.