2
1 Operations
Fig. 1.1 Stereographic view
of the reflection plane. The
point P 1 , indicated by X, is
above the plane of the gray
disc. The reflection operation
in the horizontal plane, ˆ
σ h ,is
the result of the ˆ
C
z
2 rotation
around the center by an angle
of π , followed by inversion
through the center of the
diagram, to reach the position
P 3 below the plane, indicated
by the small circle
the set of coordinate axes that defines the absolute space in a Cartesian way. They
will stay where they are. On the other hand, the structures, which are operated on,
are moving on the scene. To be precise, a symmetry operation ˆ
R will move a point
P 1 with coordinates 1 (x 1 , y 1 , z 1 ) to a new position P 2 with coordinates (x 2 , y 2 , z 2 ):
ˆ
RP 1 = P 2
(1.1)
A pure rotation, ˆ
C n (n>1), around a given axis through an angle 2π/n radians
displaces all the points, except the ones that are lying on the rotation axis itself. A
reflection plane, ˆ
σ h , moves all points except the ones lying in the reflection plane
itself. A rotation–reflection, ˆ
S n (n>2), is a combination in either order of a ˆ
C n
rotation and a reflection through a plane perpendicular to the rotation axis. As a
result, only the point of intersection of the plane with the axis perpendicular to it is
kept. A special case arises for n = 2. The ˆ
S 2 operator corresponds to the inversion
and will be denoted as ˆ
ı. It maps every point onto its antipode. A plane of symmetry
can also be expressed as the result of a rotation through an angle π around an axis
perpendicular to the plane, followed by inversion through the intersection point of
the axis and the plane. A convenient way to present these operations is shown in
Fig. 1.1. Operator products are “right-justified,” so that ˆ
ı ˆ
C
z
2 means that ˆ
C
z
2 is applied
first, and then the inversion acts on the intermediate result:
ˆ
σ h P 1 =ˆ ı ˆ
C
z
2 P 1 =ˆ ıP 2 = P 3
(1.2)
From the mathematical point of view the rotation of a point corresponds to
a transformation of its coordinates. Consider a right-handed Cartesian coordinate
frame and a point P 1 lying in the xy plane. The point is being subjected to a rotation
about the upright z-axis by an angle α. By convention, a positive value of α will
correspond to a counterclockwise direction of rotation. An observer on the pole of
the rotation axis and looking down onto the plane will view this rotation as going
1 The use of upright (roman) symbols for the coordinates is deliberate. Italics will be reserved
for variables, but here x 1 , y 1 ,... refer to fixed values of the coordinates. The importance of this
difference will become clear later (see Eq. (1.15)).
1 Operations
Fig. 1.1 Stereographic view
of the reflection plane. The
point P 1 , indicated by X, is
above the plane of the gray
disc. The reflection operation
in the horizontal plane, ˆ
σ h ,is
the result of the ˆ
C
z
2 rotation
around the center by an angle
of π , followed by inversion
through the center of the
diagram, to reach the position
P 3 below the plane, indicated
by the small circle
the set of coordinate axes that defines the absolute space in a Cartesian way. They
will stay where they are. On the other hand, the structures, which are operated on,
are moving on the scene. To be precise, a symmetry operation ˆ
R will move a point
P 1 with coordinates 1 (x 1 , y 1 , z 1 ) to a new position P 2 with coordinates (x 2 , y 2 , z 2 ):
ˆ
RP 1 = P 2
(1.1)
A pure rotation, ˆ
C n (n>1), around a given axis through an angle 2π/n radians
displaces all the points, except the ones that are lying on the rotation axis itself. A
reflection plane, ˆ
σ h , moves all points except the ones lying in the reflection plane
itself. A rotation–reflection, ˆ
S n (n>2), is a combination in either order of a ˆ
C n
rotation and a reflection through a plane perpendicular to the rotation axis. As a
result, only the point of intersection of the plane with the axis perpendicular to it is
kept. A special case arises for n = 2. The ˆ
S 2 operator corresponds to the inversion
and will be denoted as ˆ
ı. It maps every point onto its antipode. A plane of symmetry
can also be expressed as the result of a rotation through an angle π around an axis
perpendicular to the plane, followed by inversion through the intersection point of
the axis and the plane. A convenient way to present these operations is shown in
Fig. 1.1. Operator products are “right-justified,” so that ˆ
ı ˆ
C
z
2 means that ˆ
C
z
2 is applied
first, and then the inversion acts on the intermediate result:
ˆ
σ h P 1 =ˆ ı ˆ
C
z
2 P 1 =ˆ ıP 2 = P 3
(1.2)
From the mathematical point of view the rotation of a point corresponds to
a transformation of its coordinates. Consider a right-handed Cartesian coordinate
frame and a point P 1 lying in the xy plane. The point is being subjected to a rotation
about the upright z-axis by an angle α. By convention, a positive value of α will
correspond to a counterclockwise direction of rotation. An observer on the pole of
the rotation axis and looking down onto the plane will view this rotation as going
1 The use of upright (roman) symbols for the coordinates is deliberate. Italics will be reserved
for variables, but here x 1 , y 1 ,... refer to fixed values of the coordinates. The importance of this
difference will become clear later (see Eq. (1.15)).