828 20 The Electronic States of Diatomic Molecules
three-dimensional space, moving that point to a new location. As with other operators,
we denote a symmetry operator by a letter with a caret (̂) over it. For a symmetry
operator̂Owe write
̂OPP′ (20.1-10)or
̂O(x,y,z)(x′,y′,z′) (20.1-11)wherePstands for the point at its original location (x,y,z) andP′stands for the point
at (x′,y′,z′) to which the operator moves the point.
For each symmetry operator there is asymmetry element, which is a point, line, or
plane relative to which the motion is carried out. If a point is located at the symmetry
element of a particular symmetry operator, that symmetry operator does not move the
point.Point symmetry operatorsare symmetry operators that do not move a point if it
is at the origin. The symmetry elements of point symmetry operators always include
the origin. We will discuss only point symmetry operators.
Theinversion operatoris denoted bŷi. It moves a point on a line through the origin
to a location that is at the same distance from the origin as was the original location. The
symmetry element for the inversion operator is the origin. If the Cartesian coordinates of
the original location are (x,y,z) the inversion operator moves the point to (−x,−y,−z).
That is,
̂i(x,y,z)(x′,y′,z′)(−x,−y,−z) (20.1-12)Since there is only one origin, there is only one inversion operator among the point
symmetry operators.
Areflection operatormoves a point on a line perpendicular to a specified plane to
a location on the other side of the plane at the same distance from the plane as the
original location. It is said to reflect the point through the plane, which is the symmetry
element. The reflection operator̂σhreflects through a horizontal plane:̂σh(x,y,z)(x′,y′,z′)(x,y,−z) (20.1-13)There is only one horizontal plane through the origin so there is only onêσhopera-
tor among the point symmetry operators. A symmetry operator that reflects through
a vertical plane is denoted bŷσv. Since there are infinitely many vertical planes
containing the origin, there are infinitely manŷσvoperators among the point sym-
metry operators. We can attach subscripts or other labels to distinguish them from each
other.EXAMPLE20.1
Find the coordinates of the points resulting from the following operations:
a.̂i(6,4,3)
b.̂σh(1, 2,−2)
c.̂σvxz(7,−6, 3) wherêσvxzis the reflection operator that reflects through thexzplane
Solution
a.(−6,−4,−3)
b.(1,2,2)
c.(7,6,3)