Effective Nuclear Charge and Electron Shielding

Effective nuclear charge (Z_eff) is the net positive charge felt by an electron in an atom, equal to the actual nuclear charge (Z) minus the shielding (S) by other electrons, so Z_eff = Z − S. In brief: inner electrons shield the outer electrons very well, while electrons in the same shell shield each other poorly, so Z_eff increases from left to right across a period and changes little down a group; for sodium, the valence electron feels a Z_eff of about 2.2, not 11.

The concept of effective nuclear charge explains periodic trends in atomic size, ionization energy and electronegativity. This article describes shielding and penetration, shows how to estimate Z_eff with a simple rule and with Slater’s rules, presents the values across periods 2 and 3 in tables, works through calculations and explains how Z_eff accounts for atomic trends.

What is electron shielding?

In an atom with more than one electron, each electron is attracted by the nucleus and repelled by the other electrons. The repulsion cancels part of the attraction, as if the other electrons “screened” or “shielded” the nuclear charge. An electron in the outer shell therefore feels a smaller positive charge than the full nuclear charge Z. Inner-shell electrons are between the nucleus and the outer electron and are the most effective in shielding. Electrons in the same shell are at about the same distance, and shield each other only weakly.

What is penetration?

Orbitals of the same shell differ in how close they come to the nucleus. An s electron has some probability of being very close to the nucleus, which is called penetration, while a p electron penetrates less, and a d electron less again. Better penetration means less shielding and a stronger attraction. That is why, in a many-electron atom, the energy order within a shell is s < p < d < f. This is also why the 4s orbital is lower in energy than the 3d for the first elements of the fourth period.

How can you estimate Z_eff simply?

A simple approximation treats all inner (core) electrons as shielding the nuclear charge completely and the other valence electrons as not shielding: Z_eff ≈ Z − number of core electrons, also called the core charge. For sodium, 11 − 10 = +1. For chlorine, 17 − 10 = +7. The core charge equals the number of valence electrons for main-group elements. This overestimates Z_eff, but shows the direction of the trends.

How do Slater’s rules work?

Slater’s rules, proposed by John C. Slater in 1930, give a more realistic estimate. Electrons are grouped as (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f), and so on. For an electron in an ns or np group:

• Each other electron in the same group contributes 0.35 (0.30 for 1s).
• Each electron in the shell n − 1 contributes 0.85.
• Each electron in shell n − 2 or lower contributes 1.00.
• Electrons in higher shells contribute 0.

For an electron in an nd or nf group, each electron in the same group contributes 0.35 and every electron in a group to the left contributes 1.00. Then S is the sum of the contributions and Z_eff = Z − S.

Z_eff across periods 2 and 3

Z_eff rises by 0.65 from one element to the next in period 2 and by 0.65 in period 3. Going down group 1, Slater’s rules give about 2.20 for sodium, potassium, rubidium and cesium alike, which shows that the increase in nuclear charge is nearly cancelled by the extra shielding.

Worked examples

1. Sodium (Z = 11). Configuration 1s² 2s² 2p⁶ 3s¹. For the 3s electron: no other electron in the 3s, 3p group (0 × 0.35); the 8 electrons in the second shell contribute 8 × 0.85 = 6.8; the 2 electrons in the first shell contribute 2 × 1.00 = 2.0. S = 8.8 and Z_eff = 11 − 8.8 = 2.2.
2. Sulfur (Z = 16). Configuration [Ne] 3s² 3p⁴. For a 3p electron: 5 other electrons in the 3s, 3p group: 5 × 0.35 = 1.75; 8 in the second shell: 6.8; 2 in the first shell: 2.0. S = 10.55, so Z_eff = 16 − 10.55 = 5.45.
3. Iron 4s electron (Z = 26). Configuration [Ar] 3d⁶ 4s². For a 4s electron: 1 other 4s electron: 0.35; 14 electrons in the third shell (3s, 3p, 3d): 14 × 0.85 = 11.9; 10 electrons in shells 1 and 2: 10.0. S = 22.25, so Z_eff = 26 − 22.25 = 3.75.
4. Compare with the simple approximation. The core charge of sulfur is 16 − 10 = 6, compared with Slater’s 5.45. The simple model overestimates, since the other valence electrons also shield a little.
5. Increase across a period. From Na (2.20) to Cl (6.10) the Z_eff increases by a factor of 6.10 ÷ 2.20 = 2.8, and the atomic radius decreases from 166 pm to 102 pm, which shows that the valence electrons are held much more tightly.

How does Z_eff explain periodic trends?

Across a period the electrons are added to the same shell, so shielding does not increase much, but the nuclear charge grows. This is why atoms get smaller across a period. Down a group a new shell is added at each step, which increases the distance of the outer electron and shields the nucleus almost completely, so the outer electrons are held more loosely.

Where does the concept come from?

The idea of shielding was used by physicists in the early days of quantum theory to explain X-ray spectra. Moseley’s law of 1913 included a screening constant. In 1930 John C. Slater published simple rules that allow quick calculation, and these are still taught, although modern computational methods, such as those of Clementi and Raimondi (1963), give more accurate values.

Which mistakes are common?

Mistake 1: Using the full nuclear charge for the valence electrons

A valence electron in sodium feels about +2.2, not +11.

Mistake 2: Saying that electrons in the same shell shield completely

Their shielding is weak, which is why Z_eff increases across a period.

Mistake 3: Thinking that Z_eff increases down a group

It stays about the same, but the distance from the nucleus increases.

Mistake 4: Applying Slater’s rules to inner electrons without checking the group

The rules differ for s and p electrons and for d and f electrons.