Because the Dhpoint group has a formal order of, other methods must
be used to determine how this direct product reduces. It does reduce tog^ g^ (^) g
(One way to rationalize this in the absence of the great orthogonality theo-
rem is that there are three possible ways to write the two electrons in two mo-
lecular orbitals: (1) separate orbitals, spins same direction, (2) separate or-
bitals, spins opposite directions, and (3) same orbital, spins opposite directions
(they cannot be the same direction due to the Pauli principle). The first two
states are singly degenerate. How many ways can you put indistinguishable
electrons in the orbitals in the same or different spins? (Recall that we cannot
differentiate between “spin up” and “spin down” without a magnetic field.)
One way for each spin, therefore two individual (degeneracy 1) states.
However, how many ways can you put the two electrons in a single orbital with
different spins? Two ways, because there are twodifferent orbitals. Therefore,
a doubly degenerate state is needed. The total number of ways? Four, the
same as the character of the Esymmetry element for a direct product.
The Pauli principle limits the possible spins for the above term symbols.
This is due strictly to the antisymmetry requirement of the Pauli principle. The
and labels on the sigma () states in the direct sum above indicate sym-
metric and antisymmetric spatial symmetry, respectively (specifically, with re-
spect to the vertical reflection planes). Similarly, the term symbol represents
15.6 Electronic Spectra of Diatomic Molecules 537
O O 2 O
g
u
2 s 2 s
g
u
(^) u (^) u
(^) g (^) g
2 p 2 p
g
u
1 s 1 s
Figure 15.11 Molecular orbitals of O 2. Simple diagrams like those in Figure 15.10 make it
easy to determine which orbitals are gerade and which are ungerade.