quantum number may have several possible values depending on the values
ofand , but now the restriction on negative values of it is lifted:
→ in integral steps (15.17)
For states other than those having 0, the total angular momentum vec-
tor can be moving (“precessing”) about the internuclear axis in two directions,
much the same as the two-dimensional rigid rotor. Therefore, every state with
0 is at least doubly degenerate.
For homonuclear diatomic molecules (which have the point group Dh), an
extra label goes on the term symbol. A homonuclear diatomic molecule has a
center of symmetry, and wavefunctions can be either symmetric with respect
to the center of symmetry, or antisymmetric with respect to the center of sym-
metry. Figure 15.10 illustrates symmetric and antisymmetric molecular wave-
functions. They are analogous to symmetry and antisymmetry labels for atomic
wavefunctions. If a particular electronic state of a homonuclear diatomic mol-
ecule is symmetric with respect to the center of symmetry, the label gerade
(German for “even”) is applied and the letter “g” is added as a right subscript
in the term symbol. If a particular electronic state is antisymmetric with re-
spect to the center of symmetry, the label ungerade(German for “odd”) is ap-
plied and the letter “u” is added to the term symbol. Figure 15.10 has labeled
the example wavefunctions as gerade or ungerade.
Determining term symbols for diatomic molecules follows a procedure sim-
ilar to that for atoms. Consider O 2 as an example. The molecular orbitals for
O 2 , derived from the atomic orbitals of each oxygen atom, are shown in Figure
15.11. In the ground state of the diatomic molecule, the unfilled molecular or-
bitals come from the unfilled subshell of the atoms; in this case, the p^4 elec-
trons. For diatomic oxygen, the unfilled molecular orbitals are the * molec-
ular orbital. As a orbital, this molecular orbital can be assumed to be similar
to a patomic orbital and so would have a single unit of orbital angular mo-
mentum. Using the letter to designate the orbital angular momentum, this
implies that 1 2 1. (Here we are using the subscripts 1 and 2 to indi-
cate the individual electron. It does not matter which is 1 and which is 2.) In
essence, these two electrons have angular momenta that couple just like two
pelectrons, except that now, for molecules, we use lowercase Greek letters to
indicate the term symbols. However, unlike in an atom, we have only two de-
generate orbitals, not three degenerate orbitals (like we have for atomic por-
bitals). In this case, this limits the possible combinations ofto 1
2 and
1 2.
There is a different way to consider this coupling, and it becomes useful for
polyatomic molecules: use symmetry when possible. Each molecular orbital
can be given a symmetry label that is one of the irreducible representations of
the molecular point group. In the case of the homonuclear diatomic, the point
group is Dh. As we might expect for a doubly degenerate molecular orbital,
the label for these * orbitals is , but the Dhpoint group requires a label
of g or u for each irreducible representation. The diagrams of each molecular
orbital in Figure 15.11 show that the * orbitals have gerade symmetry with
respect to the center of symmetry, so that each one can be labeled as g.The
following statement is therefore applicable: the term symbols that label the en-
ergy levels of unpaired electrons in a molecule are determined from the direct
product of the irreducible representations of the molecular orbitals that con-
tain the unpaired electrons. In this case, this means evaluating
gg536 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure
iiAntisymmetric: ungeradeSymmetric: gerade+ – + –Figure 15.10 Wavefunctions for homonuclear
diatomic molecules are labeled gerade or unger-
ade, depending on the behavior of the wavefunc-
tion upon operation by the symmetry element of
the inversion center.