How to Determine Screening Constant and Effective Nuclear Charge

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In many atoms, each electron is said to experience less than the actual nuclear charge because of shielding or screening by the other electrons. For each electron in an atom, Slater's rules provide a value for the screening constant, denoted by σ.

The effective nuclear charge may be defined as the actual nuclear charge (Z) minus the screening effect caused by the electrons intervening between the nucleus and valence electron.[1]
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Effective nuclear charge, Z* = Z - σ
Where, Z= Atomic number, σ = Shielding or screening constant.

To calculate the effective nuclear charge (Z*) we need the value of screening constant (σ) which can be calculated by using following rules.

Steps

Write down the electronic configuration of the element as shown below. (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) (5d)… Fill the electrons according to Aufbau principle. Any electrons to the right of the electron of interest do not contribute to shielding constant. The shielding constant for each group is formed as the sum of the following contributions: Each other electron in the same group as the electron of interest shield to an extent of 0.35 nuclear charge units except 1s group, in which the other electron contributes only 0.30. If the group (n) is of [s, p] type, an amount of 0.85 from each electron in (n-1)th shell and an amount of 1.00 for each electron from (n-2) and lower shells is added to the shielding constant. If the group is of [d] or [f] type, an amount of 1.00 for each electron from all lying left to that orbital.

For example: (a) Calculate effective nuclear charge in Nitrogen for 2p electron. Electronic configuration- (1s) (2s, 2p). Screening constant, σ = (0.35 × 4) + (0.85 × 2) = 3.10 Effective nuclear charge, Z* = Z – σ = 7 – 3.10 = 3.90

(b) Calculate effective nuclear charge and screening constant seen in 3p electron in Silicon. Electronic configuration- (1s) (2s, 2p)(3s, 3p). σ = (0.35 × 3) + (0.85 × 8) + (1 × 2) = 9.85 Z* = Z – σ = 14 – 9.85 = 4.15

(c) Calculate effective nuclear charge in Zinc for 4s electron & for 3d electron. Electronic configuration- (1s) (2s, 2p)(3s, 3p)(3d)(4s). For 4s electron, σ = (0.35 × 1) + (0.85 × 18) + (1 × 10) = 25.65 Z* = Z – σ = 30 – 25.65 = 4.35 For 3d electron, σ = (0.35 × 9) + (1 × 18) = 21.15 Z* = Z – σ = 30 – 21.15 = 8.85

(d) Calculate effective nuclear charge on one of 6s electron in tungsten. (At. No. =74) Electronic configuration- (1s) (2s, 2p)(3s, 3p)(4s, 4p) (3d) (4f) (5s, 5p)(5d), (6s) σ = (0.35 × 1) + (0.85 × 12) + (1 × 60) = 70.55 Z* = Z – σ = 74 – 70.55 =3.45