The inner electrons of the atom "block the view" (screen) of the nucleus to outer electrons, sensing a lower nuclear charge than the one that the nucleus actually has, called the effective nuclear charge $(Z_ef)$. To calculate the effective nuclear charge, we subtract the actual charge of the nucleus (atomic number, Z) from the shielding produced by the inner electrons (S). \begin{equation} Z_{ef}=ZS \end{equation} The electrons that are in innermost orbitals (penetrate further into the atom), with higher electron density in the vicinity of the nucleus shield external electrons more than electrons They are found in orbitals with an electron density farther from the nucleus. For example, the 1s orbital is the one with the greatest penetration of the atom, its electrons are very close to the nucleus and produce a significant shielding effect on external electrons. The next in order of penetration are the 2s, followed by the 2p...
Orbital penetration order: 1s>2s>2p>3s>3p>3d.
A smaller nuclear charge acts on a shielded electron, assuming an increase in the energy of the electron.
The screening constant, S, can be obtained using Slater's rules. To apply Slater's rules, we write down the electronic configuration of the atom and group the s and p orbitals with the same principal quantum number, irregularities are ignored.
$[1s][2s2p][3s3p][3d][4s4p][4d][4f][5s5p][5d]$
The screening constant for an electron of a certain group is given by the following contributions:
- 0.35 for every other electron in the same group. Except in the case of 1s, where the other electron contributes 0.3
- 0.85 for each electron in groups [sp] with quantum number one unit lower and 1 for each electron with quantum number even lower
- 1 for each electron in groups [d] and [f] with the same or less quantum number.
Let's see some examples of the calculation of the shielding and the effective nuclear charge for several atoms of the first periods.
H: $1s^1$; Z=1; S=0; $Z_{ef}=1$
He: $1s^2$; Z=2; S=0.3; $Z_{ef}=1.7$
Li: $1s^22s^1$; Z=3; S=0.85x2=1.7; $Z_{ef}=1.3$
Be: $1s^22s^2$; Z=4; S=0.35+0.85x2=2.05; $Z_{ef}=1.95$