for a pure sp^3 hybrid orbital). However, in most cases a “pure” hybrid orbital
can be assumed and acts as a good enough approximation.
Hybridization and symmetry are intimately connected because of the rela-
tionships seen above: atomic centers in molecules that have certain hybridiza-
tion have specific shape. Atoms that make sphybrid orbitals impart a linear
shape about that atomic center (which might suggest Cvor Dhpoint groups,
for simple molecules). Atoms that have sp^2 hybrid orbitals make bonds in a
threefold, or trigonal,shape. They also form planar molecules, since the three
bonds formed by the sp^2 hybrid orbitals are in the same plane. Atoms hav-
ing sp^3 hybrid orbitals have a tetrahedral molecular shape about that atom.
Such a correspondence between molecular geometry and hybridization,
though approximate, is a powerful tool in understanding the general shape of
molecules.
In determining the irreducible representation labels for the hybrid orbitals,
one must consider all hybrid orbitals as a set and how that set changes when
the various classes of symmetry operations act on the set. If a hybrid orbital is
moved onto itself, it contributes 1 to the character of that symmetry class. If
a hybrid orbital is negated, it contributes 1 to the character. If a hybrid or-
bital is moved to the position of another hybrid orbital, then it contributes 0
(zero) to the character. Although many classes have more than one individual
symmetry operation represented, all symmetry operations in a class have the
same character (which is how we separated symmetry operations into classes
previously). Therefore only one symmetry operation for each class needs to be
considered, and the usual choice is the easiest symmetry operation to visual-
ize. The contributions of all hybrid orbitals are summed up—not all of the
hybrid orbitals will have the same contributions for any particular symmetry
operation (except E, of course). This set of characters is compared to the irre-
ducible representations of the point group. If necessary, the great orthogonal-
ity theorem is applied. This determines the labels for the hybrid orbitals. Since
only orbitals of the same irreducible representation can interact to make mol-
ecules, such labels are indispensable when considering the fine points of atoms
bonding to make molecules.
The following example illustrates this process.Example 13.16
Determine the irreducible representations for sp^3 hybrid orbitals in the Td
point group.Solution
Figure 13.26 shows the set of four sp^3 orbitals, labeled individually (although
we recognize that we can’t label them in reality). Collectively, they have tetra-
hedral or Tdsymmetry. Each part of Figure 13.26 shows the effect of one
symmetry operation from each of the five classes in the Tdpoint group. For
E, Figure 13.26a shows that each hybrid orbital operates onto itself and so
contributes 1 to the total character; therefore,E4. Figure 13.26b shows
that a C 3 operation keeps one orbital in place (which therefore contributes
1) and three others exchanging positions (which therefore contribute 0
each). Therefore, for the set of four orbitals,C 3 1. Figure 13.26c shows that
for C 2 , all of the orbitals are operated onto different orbitals, so that C 2 0.
Figure 13.26d shows that the S 4 operation has the same effect as C 2 : moving
all orbitals to the positions of different orbitals. Therefore,S 4 0. Finally,
Figure 13.26e shows that a dplane of symmetry reflects two orbitals onto
themselves (for an overall contribution of2) and reflects the two other or-454 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics