Operations - Group Theory Applied to Chemistry - page 2

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, σ h ,is
the result of the C 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 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 ) :

RP 1 =

P 2 (1.1)

A pure rotation, C n ( n> 1), around a given axis through an angle 2 π/n radians
displaces all the points, except the ones that are lying on the rotation axis itself. A
reflection plane, σ h , moves all points except the ones lying in the reflection plane

itself. A rotation-reflection, S n ( n> 2), is a combination in either order of a 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 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 C 2 means that C 2 is applied
first, and then the inversion acts on the intermediate result:

ı C 2 P 1 =ˆ σ h P 1 =ˆ

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 )).