a symmetric spatial electronic state (check its character!). Therefore, the sym-
metric spatial states must be paired with antisymmetric spin states, and anti-
symmetric spatial states can be paired with symmetric spin states. When this
is done, the following term symbols are possible:(^1) g (^) , (^3) g (^) , and (^1) g
Hund’s rules are applicable to molecular electronic states as well as atomic
electronic states (which is partly why they are so useful). Therefore, the highest-
multiplicity electronic state, the^3 g^ state, is predicted to be the ground state.
(It is, as determined experimentally by various means.) The^1 g^ and^1 gelec-
tronic states are excited states within the ()^2 electronic configuration of the
diatomic molecule (see Figure 15.11).
Example 15.10
Predict the term symbol(s) of the ground electronic configuration of an ex-
cited state of oxygen that has one of its electrons in the antibonding or-
bital. See Figure 15.11 for assistance in determining the symmetry label of the
excited electron. Note that you need not know all of the characters of the ir-
reducible representation to determine the necessary characters of the direct
product.
Solution
The excited electron in the antibonding orbital is in a molecular orbital
that has ^ usymmetry, and the electron still in the orbital has usym-
metry. The direct product of these two symmetries is simply u. (Verify this.)
The and electrons can have either the same direction spin or different
direction spin (rather, the zcomponents of their spin), so multiplicities of 3
or 1 are possible. Therefore, the term symbols for this excited-state electron
configuration are^3 uand^1 u.states are doubly degenerate. In this case,
there are two possible, degenerate * molecular orbitals for the unexcited
electron. Complete term symbols would have values for included. For the
triplet state,can be 2, 1, or 0. For the singlet state,can only be 1. Unlike
atomic term symbols, it is relatively uncommon to see the values listed ex-
plicitly in the term symbols of diatomic molecules. One would see^3 uto rep-
resent all three individual states, rather than^3 u,2,^3 u,1, and u,0.
Now we need to consider selection rules. The following rules are applicable
to diatomic molecules only. For allowed electronic transitions:
0, 1 (15.18)
S 0 (15.19)
0, 1 (15.20)
g ←→u (for homonuclear diatomics) (15.21)
For states, ←→+, ←→ ,but not ←→ (15.22)
where in equations 15.21 and 15.22, an arrow means that states having this
change in label (ungerade to gerade, or gerade to ungerade) are allowed. These
selection rules are qualitatively similar to the selection rules for atoms. Note
once again a restriction on the allowed change in the Squantum number: no
change is allowed. This is the case for diatomic molecules having small atoms.
538 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure