Elements of Rayleigh–Schroedinger Perturbation Theory - Methods of Molecular Quantum Mechanics - page 210

M L + 2 M S

1

+5

1

+4

2

+3

2

+2

3

+1

3

F

3

0

3

− 1

2

− 2

2

− 3

1

− 4

1

− 5

H =0 H ≠ 0

Figure 10.2 Zeeman splitting for a free 3 F ion in a magnetic field H (the residual
degeneracy of each level is shown)

The operator describing spin-orbital LS-coupling is

Z

SO

H

r 3 L S ¼ jð r Þ L S

2
e

¼ g e b ð 10 :

120 Þ

where Z is the nuclear charge of the ion. But:

1

L S ¼

2 ð J 2

L 2 S 2 Þ ð 10 : 121 Þ

Using first-order perturbation theory, we take the expectation value
of H

SO

over the ground-state wavefunction and the relative energy of
levels of given J will be

1
2 J ð J þ 1 Þ L ð L þ 1 Þ S ð S þ 1 Þ

E J ¼ A ½ ð 10 : 122 Þ

whereA is a constant, characteristic of the ion.We then get the relative
energies of the three LS-coupled states occurring for L ¼ 3 (state F)