This repulsion between electrons in many electron species which opposes the attraction of the electrons towards the nucleus, and which causes the energies of the atomic orbitals in many electron species to be lower in the hydrogenic species is known as electron shielding.

In the hydrogenic species, the energy of an atomic orbital is given by: , where n is the principal quantum number of the orbital and Z is the nuclear charge.

In the many electron species, we can take into account the effects of the inter-electron repulsion by substituting the nuclear charge by an effective nuclear charge, thus

Zeff = Z – σFor a hydrogenic ion, Zeff = Z. In many electron species, the difference between the nuclear charge and the effective nuclear charge is known as the shielding parameter, σ:

Some values of Zeff and σ for electrons in the 1s orbital are given in the table below.

Effective nuclear charge and shielding parameters
Hydrogenic species Many electron species
Zeff σ Zeff σ
H 1 0 H 0.24 0.76
He+ 2 0 He 1.34 0.66
Li2+ 3 0 Li+ 2.36 0.64

Values of Zeff and σ for the electron in higher energy orbitals in He, obtained from spectroscopic studies of excited states are shown in the table below.

Orbital energies and effective nuclear charges in 1s1nl1 excited state configurations of He
n ns orbitals np orbitals nd orbitals
-ε (eV) Zeff -ε (eV) Zeff -ε (eV) Zeff
2 4.570 1.160 3.561 1.023
3 1.819 1.098 1.561 1.017 1.515 1.002
4 0.9744 1.071 0.8718 1.013 0.8521 1.002
5 0.6065 1.056 0.5556 1.011 0.5455 1.002

We should note from the above values:

The highly effective shielding in these states: σ ~ 1 for the excited electron, implying that repulsion by the 1s electron almost cancels the effect of one entire unit of nuclear charge.

ns, np, and nd orbitals have different orbital energies, s < p < d, unlike the situation in hydrogenic species.

we can understand these facts by considering the radial distribution functions for the different orbitals. These are a measure of the probability of finding an electron in a given orbital at a given distance from the nucleus.

When we look at the radial distribution functions, RDFs, we note the general increase in the size of the orbitals with increasing n. A useful measure of this is the value of rmax, the distance at which the RDF has its maximum value. rmax also depends on the value of l, eg. rmax(3s) > rmax(3p) > rmax(3d).

Although there is the above trend in the values of rmax, it should be noted that in the one-electron cases, the attraction of an electron to the nucleus, and hence the energy of the orbital, is the same for the 3s, 3p, and 3d orbitals due to the compensating effect of the inner parts of the RDFs for the 3s and 3p orbitals (ie. the peaks at lower r mean that the electron spends appreciable time close to the nucleus and experiences increased electrostatic attraction).

If we consider an electronic configuration like 1s23d1, we see that the RDF for the 3d orbital lies almost entirely outside that for 1s. The repulsion between the two electrons is nearly the same as if the 1s electron were concentrated at the nucleus. Thus the total electrostatic attraction experienced by the 3d electron towards the nucleus is the same as if the the nuclear charge had been reduced by one unit.

This means that the shielding by the 1s electron is almost perfect.

The tendency of an electron to spend much of its time at distances corresponding to peaks in the RDF means that the electrons in the 3d, 3p, and 3s orbitals tend to spend progressively more time close to the nucleus. They also spend progressively more time closer to the nucleus than the electron in the 1s orbital: these orbitals are said to penetrate the 1s orbital.

Repulsion between the two electrons is lower in the 1s13p1 and 1s13s1 configurations than the 1s13d1 configuration, as the electrons tend to be further apart more of the time, and the screening of the outer electron is less as they are able to get closer to the nucleus and experience a higher nuclear charge.

These effects can be summarized as:

  • Amount of penetration: ns > np > nd for a given n.
  • More penetration means less screening,
  • which means higher effective nuclear charge, Zeff,
  • And therefore a lower, more negative, orbital energy