with all other operations of the group. That is, ifArepresents the iden-
tity operation and Brepresents any other operation, then the combina-
tion ABhas the same effect as the combination BAwhich has the same
effect of the Boperation by itself.
•An inverse operationmust be present for every operation in the point
group that reverses the action of each operation. Some operations are
their own inverse.- In combinations of more than one operation, the associative lawapplies.
This means that if you have three symmetry operations labeled A,B, and
C, the combinations (AB)Cand A(BC) must yield the same overall effect. - Every possible combination of more than one operation in the group
must be equivalent to a single operation of the group. This property is
called closure.
We have already defined the identity operation E, which is a member of all
point groups. Consider the inverse operation requirement. Assume that one
can keep track of the individual identity of the hydrogens in the water mole-
cule shown in Figure 13.9. Operation on the molecule by the vshown switches
the positions of the hydrogens. A second operation on the molecule moves the
hydrogens back to their original positions. This shows that the vis its own
inverse. All planes of symmetry are their own inverse.
Application ofC 2 to the water molecule also returns the atoms to their orig-
inal starting points, so C 2 is also its own inverse. However, consider NH 3
(Figure 13.10). It has a threefold axis C 3 passing through the N, but applica-
tion of the C 3 operation does not return the atoms to their original positions,
as shown. There are two choices. Either the molecule can be rotated by 120° in
the opposite direction, which would be labeled C 3 ^1. Or, the molecule can be
rotated by 240°, or two-thirds of a complete circle, which would return the
atoms to their original points. This operation would be labeled C^23 .Each op-
eration is equivalent. It is customary to consider that all rotations about the
same axis are performed in the same direction, so the operation of choice
would be C^23. What this means is that the NH 3 molecule has just one threefold
axis,but two different rotational symmetry operationsabout that axis. Both
symmetry operations must be defined in order for the symmetry operations to
constitute a mathematical group.
The association requirement and the closure requirement both refer to sit-
uations in which more than one symmetry operation is performed sequen-
424 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics
OHHOHb HavO vHa HbOHa HbvFigure 13.9 The symmetry operation vis its own inverse.