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
O p e r a t i o n sa n dP o i n t s ................................
1
1 . 2
O p e r a t i o n sa n dF u n c t i o n s ..............................
4
1 . 3
O p e r a t i o n sa n dO p e r a t o r s ..............................
8
1 . 4
A nA i d eM é m o i r e ..................................
1 0
1 . 5
P r o b l e m s .......................................
1 0
R e f e r e n c e s...........................................
1 0
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 snapshot of a crystal, then the crystal is rotated while the camera is left immobile. A second 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 perspective 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 relative 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 crystal 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 annoying 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 convention in chemistry textbooks. In this script the part of the observer is played by
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_1,
© Springer Science+Business Media Dordrecht 2013
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
O p e r a t i o n sa n dP o i n t s ................................
1
1 . 2
O p e r a t i o n sa n dF u n c t i o n s ..............................
4
1 . 3
O p e r a t i o n sa n dO p e r a t o r s ..............................
8
1 . 4
A nA i d eM é m o i r e ..................................
1 0
1 . 5
P r o b l e m s .......................................
1 0
R e f e r e n c e s...........................................
1 0
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 snapshot of a crystal, then the crystal is rotated while the camera is left immobile. A second 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 perspective 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 relative 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 crystal 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 annoying 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 convention in chemistry textbooks. In this script the part of the observer is played by
A.J. Ceulemans, Group Theory Applied to Chemistry, Theoretical Chemistry and
Computational Modelling, DOI 10.1007/978-94-007-6863-5_1,
© Springer Science+Business Media Dordrecht 2013
1