Can someone explain the idea of degeneracy of orbitals?

[eccles1214]

This is a confusing topic. What does it mean to be degenerate?
How does this apply to atomic orbitals versus molecular orbitals?
Can you point me to a good reference book on the subject?

 

[G01]

The concept of degeneracy is related to the energies of the different quantum states of a system. A given system is said to be degenerate if more than one eigenstate has the same energy as an eigenvalue. Take the hydrogen atom as an example:

In the H atom there are different energy levels and within each energy level you have an orbital corresponding to different allowed values for the electron's angular momentum. These orbitals are the eigenstates for the electron bound to the proton.

There is only one orbital(eigenstate) which has the ground energy as an eigenvalue. This is the S orbital for n=1. Thus, the n=1 energy is non degenerate since there is only one eigenstate the electron can be in (the s orbital) and have that energy.

Now, consider the n=2 energy for hydrogen. Any electron in the s orbital state or one of the p-orbital states will have this energy. Thus, the n=2 energy is said to be degenerate since an electron can have this energy and be in a multitude of different eigenstates of the system.

That is an example of degeneracy in an atomic system. As for a molecular system, I think it is best if i left that to someone else. I have not yet covered that in any class, so I would not be the person to ask.

 

[denian]

The concept of degeneracy of orbitals refers to the fact that within a given energy level or shell, there can be multiple orbitals that have the same energy. This means that these orbitals are essentially equal in terms of their energy and do not have any preference for being occupied by electrons.

To understand degeneracy, it is important to first understand the electronic structure of atoms. An atom's electrons occupy different energy levels or shells, and within each shell, there are different subshells (s, p, d, f) that can hold a certain number of electrons. Each subshell is made up of one or more orbitals, which are regions of space where an electron is most likely to be found.

Now, within a given subshell, there can be multiple orbitals that have the same energy. For example, in the p subshell, there are three orbitals (px, py, pz) that have the same energy. This is known as degeneracy. It essentially means that these orbitals are equivalent in terms of their energy and have no preference for being occupied by electrons.

This concept also applies to molecular orbitals, which are formed when two or more atoms come together to form a molecule. Just like atomic orbitals, molecular orbitals can also be degenerate. For example, in the bonding molecular orbital of a diatomic molecule, the two electrons are distributed equally between the two atoms, resulting in a degenerate orbital.

A good reference book on this topic would be "Molecular Orbitals and Organic Chemical Reactions" by Ian Fleming. It provides a comprehensive explanation of degeneracy and its application in atomic and molecular orbitals.