900 21 The Electronic Structure of Polyatomic Molecules
C 2 axis(^) xzplane
Ha
(c)
(a)
(b)
Hb
Hb Ha
z
C 2 axis
(^) yzplane
y
x^ xzplane
z
O
O
O
Hb Ha
Figure 21.11 The H 2 O Molecule.(a) The molecule in the standard location in the coordi-
nate system. The symmetry elements C 2 ,σxz, andσyzare shown. (b) The action of the C 2
operation on the nuclei. (c) The action of theσxzoperation on the nuclei.
by saying that it hasC 2 vsymmetry. The NH 3 molecule belongs to theC 3 vpoint group,
which contains the operationsE,C 3 ,C^23 , and threeσvoperations.
There are many different point groups. Table 21.4 shows some common point groups
with their Schoenflies symbols, a list of the symmetry operations in each group, and a
representative molecule belonging to each group. A rotation axis of order greater than 2
implies the presence in the group of more than one operation for that axis. For example,
aC 3 operation corresponds to a counterclockwise rotation of 120◦. The group must
include its inverse, which is a rotation of 120◦in the clockwise direction. This inverse is
equivalent toC 32 , corresponding to a rotation of 240◦in the counterclockwise direction,
soC^23 must be a member of any group containingC 3. These two operations are listed
as 2C 3. Benzene has a six-fold axis so that in order to include all the operations and
their inverses for this axis, the group must includeC 6 ,C^26 ,C^36 ,C^46 , andC^56. These are
listed as 2C 6 ,2C 3 , andC 2 , where 2C 6 indicatesC 6 andC^56 ,2C 3 indicatesC^26 (same as
C 3 ) andC^46 (same asC^23 ), and whereC 2 indicatesC^36.
The main part of the Schoenflies symbol is a capital letter. There is a capitalC
if there are no two-fold axes perpendicular to the principal rotation axis (the axis of
largest value ofn) or a capitalDif there are such two-fold axes. There is a subscript that
denotes the value ofnfor the highest-order rotation axis. A subscriptvindicates the
presence of vertical mirror plans, and a subscripthdenotes the presence of a horizontal