Question

Consider the Russell- Saunders coupling scheme for two non-equivalent electrons in an excited 2p1 3d1 configuration. For all parts (a) to (c) give your reasoning:

(a) Show that strong coupling between orbital angular momenta (ℓ ℓ coupling) gives rise to P, D and F terms with L magnitudes of 2½ħ, 6½ħ and 12½ħ.

(b) If coupling between spin momenta (s-s coupling) is considered in addition to ℓ ℓ coupling, what six terms arise?

(c) If coupling between the orbit and spin angular momenta is also considered (LS coupling), how many terms arise?

Answer 1

Two non-equivalent electrons in an excited 2p1 3d1 configuration :

(a)   ℓ- ℓ coupling :   2p¹ 3d¹

   ℓ values of electron in p -orbital = 0,1

ℓ values of electron in p -orbital = 0,1,2

= 0, 1, 2, 3     ( we exclude '0' term)

L = 1, 2, 3 terms give rise to P (L=1) , D (L=2), F (L=3) terms.

we have : orbital angular momentum : <L> = (l(l+1)^1/2 ħ

= P -terms : L = 1 -> 2^1/2 ħ

= D -terms : L = 2 -> 6^1/2 ħ

= F -terms : L = 3 -> 12^1/2 ħ

(b)

coupling between spin momenta (s-s coupling) is considered in addition to ℓ ℓ coupling, what six terms arise :

= 1, 2, 3

terms = ^2S+1L

6 terms = ¹P ; ²P ; ¹D ; ²D ; ¹F ; ²F

(c)   LS coupling

= 1, 2, 3

Total terms : 6*5 = 30 terms.