Elements of Rayleigh–Schroedinger Perturbation Theory - Methods of Molecular Quantum Mechanics - page 212

3
2 þ

S ð S þ 1 Þ L ð L þ 1 Þ
2J ð J þ 1 Þ

b e J m J ¼ g e ; g e ¼ ð 10 : 124 Þ

Then, the potential energy of the magnetic dipole in the uniform
magnetic field H ¼ k H is

(

H 1 ¼m J H ¼ g e b e HJ z
J z c ¼ M J c ð J M J J Þ

ð 10 :

125 Þ

so that the energy of the (2J þ 1)-sublevels in presence of a field H
will be

E M J ¼ g e b e HM J ð 10 : 126 Þ

The splitting of the Zeeman levels is linear in the strength H of the
magnetic field, and is shown schematically in Figure 10.3.

APPENDIX: EVALUATION OF m AND «

For the STO (10.48), the required integrals are easily calculated in
spherical coordinates.

p

c 5

ð

dxx 2 1

1

drr 4 exp ½ð c þ 1 Þ r m ¼h 2p z j z jc 0 i¼ðc 0 2p z j z Þ¼ 2 p

p

1 0

5

2 c

p

p

c 5

4
3

4 3 2
ð c þ 1 Þ

¼

5 ¼ ð 10 : 127 Þ

c þ 1

Using relations (10.29), we obtain

<

d
dr

½ exp ð cr Þ r ¼ exp ð cr Þð 1 cr Þ

d 2
dr 2

¼ exp ð cr Þð 2c þ c 2 r Þ ½

exp ð cr Þ r

ð 10 : 128 Þ

:

!

2

r 4c þ c 2 r

2
r

r ½ exp ð cr Þ r ¼ exp ð cr Þ