Operations - Group Theory Applied to Chemistry - page 1

Chapter 1
Operations

Abstract In this chapter we examine the precise meaning of the statement that a
symmetry operation acts on a point in space, on a function, and on an operator. The
difference between active and passive views of symmetry is explained, and a few
practical conventions are introduced.

Contents

1.1 OperationsandPoints ................................ 1

1.2 OperationsandFunctions .............................. 4

1.3 OperationsandOperators .............................. 8

1.4 AnAideMémoire .................................. 10

1.5 Problems ....................................... 10

References........................................... 10

1.1 Operations and Points

In the usual crystallographic sense, symmetry operations are defined as rotations
and reflections that turn a body into a congruent position. This can be realized in
two ways. The active view of a rotation is the following. An observer takes a snap-
shot of a crystal, then the crystal is rotated while the camera is left immobile. A sec-
ond snapshot is taken. If the two snapshots are identical, then we have performed a
symmetry operation. In the passive view, the camera takes a snapshot of the crystal,
then the camera is displaced while the crystal is left immobile. From a new perspec-
tive a second snapshot is taken. If this is the same as the first one, we have found
a symmetry-related position. Both points of view are equivalent as far as the rela-
tive positions of the observer and the crystal are concerned. However, viewed in the
frame of absolute space, there is an important difference: if the rotation of the crys-
tal in the active view is taken to be counterclockwise, the rotation of the observer in
the passive alternative will be clockwise. Hence, the transformation from active to
passive involves a change of the sign of the rotation angle. In order to avoid annoy-
ing sign problems, only one choice of definition should be adhered to. In the present
monograph we shall consistently adopt the active view , in line with the usual con-
vention in chemistry textbooks. In this script the part of the observer is played by