Energy levels of multi-electron atomic orbits (1)

For single-electron systems, the energy of electrons or orbitals is only related to the main quantum number n; orbitals with the same n have the same energy, and the larger the n, the higher the energy
.

In a multi-electron system, the electron is not only subjected to the gravitational force of the nucleus, but also to the repulsion of other electrons, and the energy of the orbit is not only related to the principal quantum number n
.

5.
3.
1 Energy levels of multi-electron atomic orbitals

1.
Shielding effect

For a multi-electron atom, the nucleus and other electrons other than the electron in question can be regarded as a whole, and the gravitational force between this whole and the electron in question can be studied, that is, the nucleus and the electron in question after other electrons have offset part of the positive charge.
Gravity
.

The relationship between the effective nuclear charge Z* and the nuclear charge Z after offsetting part of the positive charge is

Z*=Z-σ

In the formula, σ is the shielding constant, and its size is related to the quantum numbers n and l
.

The energy of the orbital or electron in the multi-electron system is

which is

In the multi-electron system, because the inner electrons offset or neutralize part of the positive charge, the gravitational pull of the electron in question by the nucleus decreases and the energy increases.

This phenomenon is called the shielding effect of other electrons on the electron in question

E _ns <E _np <E _nd <E _nf

Atomic orbitals with the same n but different l have different energies, which is called energy level splitting
.

2.
Drill-through effect

The energy level splitting of atomic orbitals with the same n but different l can be attributed to the different radial distribution of the electron cloud
.

That is, the ability of electrons to pass through the inner layer and penetrate to the vicinity of the core to avoid the shielding of other electrons is different, so that their energy is different

From the radial probability distribution diagram in Fig.
5-7, it can be seen that there are 2 probability peaks in 3s that have penetrated to the vicinity of the core, 3p has 1 probability peaks that have penetrated to the vicinity of the core, and 3d has no probability peaks that have penetrated to the vicinity of the core
.

That is, the penetration capability of each track is ns>np>nd>nf, therefore, the order of the energy magnitude of the track is E _ns <E _np <E _nd <E _nf

The phenomenon in which electrons penetrate the inner orbit to near the nucleus and reduce their energy is called the drill-through effect
.

The drill-through effect can explain the energy-level splitting phenomenon, and it can also explain the energy-level interleaving phenomenon

In a multi-electron atom, when the main quantum number n and the angular quantum number l are both different, the energy of the orbital with a large main quantum number n is lower than the energy of the orbital with a small main quantum number n, which is called energy level interleaving, such as 4s orbitals.
The energy is lower than the 3d orbit
.

The 4s orbital has 3 probability peaks in the inner layer, while the 3d orbital has no probability peaks in the inner layer (Figure 5-10)

Figure 5-10 Radial probability distribution diagram of 3d orbit and 4s orbit

3.

Approximate energy level diagram of atomic orbital

American chemist Pauling proposed an approximate energy level diagram of the atomic orbital of a multi-electron atom based on spectral data and theoretical calculation results

Pauling divided the atomic orbitals into 7 energy level groups.

Table 5-1 The division of energy level groups