Atoms and Molecules
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is 0 (m l = 2, 1, 0, –1, –2 for the five d-shell electrons) so that the ground state
is
7
S 3 .
Example 2
In Sec. 5.4, it was shown that the number of J states for a two-electron system
is the same in LS and in j-j schemes, if one of the electrons has l = 0 or 1. This
result is now extended to l > 1.
Let 1 < l 2 < l 1 for electrons 1 and 2. The allowed values of L are L = l 1 +
l 2 ,..., l 1 – l 2 , while the allowed values of S are S = 0, 1. The corresponding J
values in the LS coupling scheme are:
S = 0 : J = l 1 + l 2 , ..., l 1 – l 2
S = 1 : J = l 1 + l 2 + 1, ..., l 1 – l 2 + 1
(5.73)
J = l 1 + l 2 , ..., l 1 – l 2
J = l 1 + l 2 – 1, ..., l 1 – l 2 – 1
In the j-j coupling scheme, the allowed values of j 1 and j 2 are l 1 ± 1/2 and
l 2 ± 1/2 respectively. Therefore, the allowed J values are:
j 1 = l 1 + 1/2, j 2 = l 2 + 1/2 : J = l 1 + l 2 + 1, ..., l 1 – l 2
j 1 = l 1 + 1/2, j 2 = l 2 – 1/2 : J = l 1 + l 2 , ..., l 1 – l 2 + 1
j 1 = l 1 + 1/2, j 2 = l 2 + 1/2 : J = l 1 + l 2 , ...,l 1 – l 2 – 1
j 1 = l 1 + 1/2, j 2 = l 2 – 1/2 : J = l 1 + l 2 + 1, ..., l 1 – l 2
(5.74)
It is observed that each J is repeated the same number of times in Eq. (5.73),
i.e. the LS coupling scheme, and in Eq. (5.74) i.e. the j-j coupling scheme. For
two inequivalent electrons but with l 1 = l 2 > 1, the allowed J values are those
given in Eqs. (5.73) and (5.74) except that the state with J = l 1 – l 2 – 1 is not
allowed.
For equivalent electrons, l 1 = l 2 > 1, the allowed J values in the LS coupling
scheme are:
S = 0 : J = l 1 + l 2 , l 1 + l 2 – 2, ..., 0
S = 1 : J = l 1 + l 2 , l 1 + l 2 – 2, ...,2
(5.75)
J = l 1 + l 2 – 1, l 1 + l 2 – 3, ..., 1
J = l 1 + l 2 – 2, l 1 + l 2 – 4, ..., 0
In the j-j coupling scheme, the allowed j 1 and j 2 values are j 1 ± 1/2 and
j 2 ± 1/2 respectively. Therefore, the J values are