Degenerate Orbitals

What Are Degenerate Orbitals?

A hydrogen atom contains only one electron, so the energy of that electron is only dependent on the value of the principal quantum number ‘n’ (which determines the energy of the shell). The energy of an electron increases with an increase in the principal quantum number. However, it is only for a single-electron system, which only the hydrogen atom has. The reason for it is that the only electrical interaction present is the attraction between the single electron and the nucleus.

The order of energies would be: 

1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f < …

Therefore, degenerate orbitals are those orbitals that have the same energy. As given in the above representation, 2s and 2p orbitals are degenerate; 3s, 3p and 3d orbitals are degenerate, and so on. The degeneracy of the orbitals breaks down in the case of multi-electron systems. This is because of the shielding effect by the electrons present in the orbitals of s, p, d, f and so on. The attraction of the nuclear force experienced by any electron would be shielded by other electrons. The sequence of shielding effects of orbitals is s > p > d > f. 

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Aufbau Principle

The increasing order of the energy of different orbitals with different main energy levels was given by the Aufbau principle. “Aufbau” is a German term that means to construct or build up. The Aufbau principle states that “electrons are added one by one into the various orbitals in order of their increasing energy starting with the lowest energy orbital”.

According to the Aufbau principle, electrons first occupy the lowest energy orbital available to them, followed by the filling of higher energy orbitals only when the lowest energy orbital gets completely filled up. For multi-electron atoms, the following sequence represents the increasing order of the energy of the orbitals. 

1s, 2s, 2p, 3s, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s

In order to remember the above sequence, a few things need to be kept in mind:

1. The number of subshells in a shell is equal to the principal quantum number ‘n’ of the shell. For example, a shell with n=1 has only one subshell, 1s, and a shell with n=2 has two subshells, 2s and 2p.
2. The sequence of the energy of the orbitals is according to Bohr-Bury’s rule.

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Bohr-Bury’s Rule:

The energy of an orbital is determined by the (n+l) rule. Here, n=principal quantum number and l=azimuthal quantum number (determines the orbital angular momentum).

• The orbital with the lower (n+l) value has lower energy. Example: 

In 4s-orbital, n=4 and l=0. So, n+l=4.

In 3d-orbital, n=3 and l=2. So, n+l=5.

Thus, 4s-orbital has lower energy than 3d-orbital, hence would be filled first.

• When two orbitals have the same (n+l) value, then the orbital having a lower ‘n’ value has lower energy. Example:

In 2p-orbital, n=2 and l=1. So, n+l=3.

In 3s-orbital, n=3 and l=0. So, n+l=3.

Thus, 2p-orbital has lower energy than 3s-orbital.

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Conclusion:

In this article, we saw an overview of degenerate orbitals along with some examples. Degenerate orbitals are those orbitals that have the same energy. The energy of an electron increases with an increase in the principal quantum number. However, it is only for a single-electron system, which only the hydrogen atom has. However, the degeneracy of the orbitals breaks down in the case of multi-electron systems. In this case, the shielding effect comes into play. 

The sequence of shielding effects of orbitals is s > p > d > f. We also discussed the filling of the orbitals according to their energy levels by the Aufbau principle. It states that electrons are added one by one into the various orbitals in order of their increasing energy, starting with the lowest energy orbital. 

According to the Aufbau principle, electrons first occupy the lowest energy orbital available to them, followed by the filling of higher energy orbitals only when the lowest energy orbital gets completely filled up. The Aufbau principle follows the Bohr-Bury’s rule, or simply (n+l) rule for the filling up of the energy orbitals.