Quantum Numbers in Multielectron Atoms

Atoms with multiple electrons are a complex subject. This is due to the fact that the wavefunction for an atom with more than one electrons is a function of the positions of all of the electrons in the atom. The electrons cannot really be thought of as being indpendent from each other. They are not independent from each other at all. However, this is very very difficult to deal with. As a result, as chemists we try to simplify the problem in a way that we can wrap our head around it. One very important simplfication that we make is that even though we cannot treat each electron separate, we do. This is known as the single electron approximation. And from it we get the very useful idea of "orbitals". We imagine that each electron has its own wavefunction and since wavefunction is an odd cumbersome word we instead use "orbital". To make things even more simple we assume these orbitals are essentially the same as the solutions we had for the hydrogen atom. The only difference being the nuclear charge. So we can use the same quantum numbers that we had for electrons in hydrogen to describe the electrons in a multi-electron atom.

However, it is important to recognize this is an approximation. In fact, whenever we try to discuss one electron in an atom versus another electron in an atom, it is an approximation. Thankfully it is a very good approximation!

To describe the electrons in multi-electron atoms, we use the same wavefunctions found in hydrogen. However, we now must add a new quantum number for electron spin, \( m_s \). There are two values which we call "spin up" or "spin down". These have values of +1/2 and -1/2.  In general for atoms (and molecules) what is important is not the value plus or minus, but whether the electron is paired with another or not.

Another key idea is how many electrons we can place into the same orbital in a multi-electron atom. This idea derives from the Pauli Exclusion Principle. This principle tells us about specific properties that must hold for a wavefunction for a multi-electron system (technically the wave function must be anti-symmetric with respect to exchange of any two electrons). This results in a simple idea. No two electrons can have the same set of quantum numbers. This means that each orbital ( \( n, \ell , m_{\ell}\) ) can have only two electrons in it. One with each value of the spin ( \( m_s \) ).

Now we can simply place the electrons into the orbitals in order of increasing energy. Two electrons into each orbital. Unlike in the hydrogen atom, the energy now depends both on \( n \) and \( \ell \). Generally as the \( \ell \) quantum number increases the energy is slightly higher. As a result 2s and 2p are no longer identical in energy in atoms with more than one electron. Now 2p is slightly higher than 2s. Moreover, as the energy levels get closer and closer together the ordering of the energies with respect to \( n \) can change. However, in general the order stays the same. This is known as the Aufbau Principle.

This video looks at the quantum numbers for multi-electron atoms

The following video looks at the ground state electron configuration of lithium, Li. Note: there is a typo that state it is Be, when the video is about the electron configuration and corresponding quantum numbers for Li.