Solution
The first step is to recognize that the d^8 electron configuration will have the
same term symbols as the d^2 electron configuration, as stated earlier in this
chapter. Therefore, we can use the results from Example 15.7 directly.
According to Hund’s first rule, one of the triplet states will be the ground
state. Hund’s second rule allows us to choose between the^3 P or^3 F term sym-
bol by choosing the state having the larger L: the^3 F. Finally, Hund’s third rule
allows us to choose which value ofJthe ground state will have. Since the d^8
electron configuration represents a subshell that is more than half full, the
higher value ofJwill have the lower energy. Therefore, the J4 term should
be lowest. The term symbol for the ground state of a Ni atom is therefore^3 F 4.Example 15.9
Predict the ground-state term symbol for the tetravalent cation of tech-
netium, Tc^4. Assume that its electron configuration has a d^3 valence sub-
shell. Use Table 15.1 for the partial term symbols.Solution
Of the term symbols for a d^3 electron configuration, the highest multiplici-
ties are 4, for the^4 P and^4 F states (Hund’s first rule). This means that S^32 .
The higher of the values ofLoccurs for the^4 F term, so it will be the ground-
state term (Hund’s second rule). The four values ofJin the^4 F state are
3 ^32 → 3 ^32 or ^92 ,^72 ,^52 , and ^32 . For this less-than-half-filled subshell, the lower
value ofJhas the lower energy (Hund’s third rule). Therefore, the term sym-
bol of the ground state is^4 F3/2. (This example illustrates that atomic ions are
treated the same way neutral atoms are, and also that values ofJcan be half-
integers. In all cases of an odd number of valence subshell electrons,Jwill al-
ways be half-integral.)Finally and briefly, term symbols can also be determined for electronic
states that have more than one unfilled electronic subshell. For example, the
electron configuration 2s^12 p^1 is one possible configuration for an excited state
of an He atom. The individual angular momenta of the two electrons (0,
m0 and 1,m 1 or 0 or 1) combine vectorially to give L1 (the
only possible value for L) and S0 or 1 for^3 P and^1 P terms. Possible values
for Jcan be determined accordingly. In cases like this, however, the Pauli prin-
ciple does not exclude certain combinations of angular momenta, because the
electrons now have different quantum numbers for angular momentum. If the
excited-state electron configuration were 2s^2 , the Pauli principle would elimi-
nate certain term symbols as being impossible.
Because additional labels are necessary to specify electronic states of multi-
electron valence shells, additional selection rules are necessary to indicate al-
lowed transitions between the states. The previous selection rules, equations
15.4 and 15.5, are not strictly applicable because and mare not considered
good quantum numbers. However, there are related (and perhaps not entirely
surprising) selection rules in terms ofLand S, and now one for J:
L0, 1 (15.13)
S 0 (15.14)
J0, 1(but Jinit 0 →Jfinal0 is forbidden) (15.15)532 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure