Spinorbit coupling
The spinorbit coupling may be treated in a similar fashion as for oneelectron atoms. The total angular momentum quantum number J = L + S will of course be used and proper eigenfunctions are those of L^{2}, S^{2}, J^{2} and J_{z}. We call these, which are the correct zero order functions,
The first order correction to the energy E_{so} is then given by
J may take all integer values between L+S and LS which means that the spinorbit coupling splits a term into either 2L+1 or 2S+1 fine structure levels. Since the energy is independent of M, every level will have (2J+1)fold degeneracy.
As for oneelectron atoms, the spinorbit coupling may be expressed in terms of the different angular momenta using
where A(L,S) is called the interval factor. The distance between adjacent fine structure levels J and J1 is obtained as
This is the Landé interval rule.
Use the result from first order perturbation theory to determine the energy of the fine structure levels in ^{3}Pstate from a p^{2}configuration.
We have the following values of the quantum numbers: L=1, S=1, J=0,1,2
Inserting L and S gives the formula
with the following energies:
J 

2 
A 
1 
A 
0 
2A 
The spinorbit splitting has been exaggerated in order to make a legible figure. Of course, spinorbit splittings may be of this size for heavy atoms, but the perturbation treatment may then be less appropriate.
These energy levels are: E(^{3}P_{0}) = 0 eV; E(^{3}P_{1}) = 0,00203 eV; E(^{3}P_{2}) = 0,00539 eV;E(^{1}D_{2}) = 1,26386 eV; E(^{1}S_{0}) = 2,68406eV;