states in which the totalsand the totalℓare good quantum numbers are best for minimizing the
overlap of electrons, and hence the positive contribution to the energy.
For very heavy atoms, we add the total angular momentum from each electron first then add up
the Js. This is calledj-j coupling. For heavy atoms, electrons are relativistic and the spin-orbit
interaction becomes more important than the effect of electron repulsion. Thus we need to use states
in which the total angular momentum of each electron is a good quantum number.
We can understand Hund’s rules to some extent. The maximum spin state is symmetric under
interchange, requiring an antisymmetric spatial wavefunction which has a lower energy as we showed
for Helium. We have not demonstated it, but, the larger the totalℓthe more lobes there are in
the overall electron wavefunction and the lower the effect of electron repulsion. Now the spin orbit
interaction comes into play. For electrons with their negative charge, largerjincreases the energy.
The reverse is true for holes which have an effective postive charge.
A simpler set of rules has been developed for chemists, who can’t understand addition of angular
momentum. It is based on the same principles. The only way to have a totally antisymmetric state
is to have no two electrons in the same state. We use the same kind oftrick we used to get a feel for
addition of angular momentum; that is, we look at the maximum z component we can get consistent
with the Pauli principle. Make a table with space for each of the differentmℓstates in the outer
shell. We can put two electrons into each space, one with spin up and one with spin down. Fill the
table with the number of valence electrons according to the followingrules.
- Make as many spins as possible parallel, then computemsand call thats.
- Now set the orbital states to make maximummℓ, and call thisℓ, but don’t allow any two
electrons to be in the same state (ofmsandmℓ). - Couple to getjas before.
This method is rather easy to use compared to the other where addition of more than two angular
momenta can make the symmetry hard to determine.
- See Example 26.6.1:The Boron ground State.*
- See Example 26.6.2:The Carbon ground State.*
- See Example 26.6.3:The Nitrogen ground State.*
- See Example 26.6.4:The Oxygen ground State.*
26.4 The Periodic Table
The following table gives the electron configurations for the groundstates of light atoms.