So far, we have not considered the Jquantum number in the term symbol,
even though we defined the term symbol as listing a value for J.For each^2 S^1 L
term, the possible values ofJare
JL S→L S in integral steps (15.12)
Jis also limited to positive numbers, and it depends on the values ofLand S.
Equation 15.12 implies that for each combination ofLand S, there are several
possible total angular momenta (as might be expected from the coupling of
quantized angular momentum vectors). Because Jis determined from Land S,
tables of term symbols like Table 15.1 typically leave off the Jfor quantum
number total angular momentum for reasons of clarity.
Now the complete term symbols can be written. For the carbon atom’s p^2
configuration that has^1 S,^1 D, and^3 P states:(^1) S: J 0
0 → 0 0 0 Term symbols: (^1) S 0
(^1) D: J 2
0 → 2 0 2 Term symbols: (^1) D
2
(^3) P: J 1
1 → 1 1 2, 1, 0 Term symbols: (^3) P 2 , (^3) P 1 , and (^3) P 0
Notice that of the original three states, the two that are singlets (that is, their
multiplicity equals 1) have only a single complete state, and the triplet state is
composed of three individual, complete term symbols. Also, recall that there is
a zcomponent of the total angular momentum J, and it has the same possible
values as any zcomponent of an angular momentum, given in equation 15.8.
As such, there are 2J 1 possible MJvalues within each state. Outside of the
presence of a magnetic or an electric field, all of these 2J 1 states are degen-
erate. Therefore, for the p^2 electron configuration we have
(^1) S 0 : degeneracy of 1
(^1) D 2 : degeneracy of 5
(^3) P 2 : degeneracy of 5
(^3) P
1 : degeneracy of 3
(^3) P 0 : degeneracy of 1
Total: 15 separate possible states
Thus, there are 15 individual electronic states just within the p^2 electron
configuration of a carbon atom. Because of the degeneracies, in most cases we
will have only five different energy levels (except in the case of a magnetic or
an electric field). Just as an electron configuration is separated into a group, or
manifold, ofLand Sstates, so the Jquantum number separates each Land S
term symbol into a (potential) manifold of individual states, and under the
proper conditions each Jlevel separates into its 2J 1 different MJstates. This
stepwise separation is illustrated in Figure 15.4.
Example 15.7
Determine the total number of states in an atom having the electron config-
uration d^2 for the valence subshell. Use Table 15.1 for the term symbols.
Solution
For the d^2 electron configuration, the term symbols are^1 S 0 ,^1 D 2 ,^1 G 4 ,^3 P 2 ,^3 P 1 ,
(^3) P 0 , (^3) F 4 , (^3) F 3 , and (^3) F 2. There are 2J 1 values for MJfor each term, so each
state has a degeneracy of 1, 5, 9, 5, 3, 1, 9, 7, and 5, respectively. The total
530 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure
L S J:
5 states
p^2 :
1 state
MJ J to J:
15 states
Energy (not to scale)
Figure 15.4 The identification of electronic
energy levels by Jand ultimately MJquantum
numbers. A p^2 electron configuration suggests
only a single state. However, the combination of
Land Svectors yield 5 different Jstates which,
when separated into MJstates, ultimately yield 15
different states within the p^2 electron configura-
tion. See Figure 15.5 for the term symbols of the
5 states.