Group Theory
Symmetry Operations
1. Ê . The identity operation. This operator leaves the object unchanged.
2. Ĉ n . The proper rotation operator. This operator performs a counterclockwise rotation
2
about an axis through the center of mass by an angle of
. For example, Ĉ3 rotates by 120;
n
Ĉ 32 by 240; and Cˆ 31 by 120. Note the operator equalities Cˆ 32  Cˆ 31 and Cˆ 33  Eˆ . The
operator Ĉ rotates by 90 and Ĉ by 180. Therefore, Cˆ 2  Cˆ . If the object has several
4
2
4
2
rotation axes, the axis with the highest order (largest n) is the principal axis. Usually, the center
of mass of the molecule is located at the origin of the Cartesian coordinate system, and the
principal axis is aligned with the z axis. When several C n axes have the same order, the z axis is
taken along the one passing through the greatest number of atoms.
3. iˆ . The inversion operation. This operator projects the object through the center of
symmetry to an equal distance on the other side of the center.
4. ̂ h . The horizontal mirror plane reflection. The mirror plane is perpendicular to a
principal C n axis. This operator reflects the object across the plane of symmetry.
5. ̂ v . The vertical mirror plane reflection. The mirror plane contains a principal C n
axis. This operator reflects the object across the plane of symmetry.
6. ̂ d . The dihedral (or diagonal) mirror plane reflection. The mirror plane contains a
principal C n axis and bisects the angle between two horizontal C 2 axes. (These horizontal C 2
axes are perpendicular to the principal C n axis.) The operator reflects the object across the plane
of symmetry.
7. Ŝ n . The improper rotation operator. This operation performs a counterclockwise
2
rotation about an axis by an angle of
followed by a reflection across a mirror plane
n
perpendicular to that axis. In other words, this is a Ĉ n rotation followed by reflection across a ̂ h
mirror plane.
Sˆ n  ˆ h Cˆ h
If an object has Ĉ n and ̂ h as symmetry elements, then it must also have Ŝ n as
symmetry element. However, it is possible for Ŝ n to be a symmetry element for an object when
Ĉ n and ̂ h are not symmetry elements.
Point Groups
If two symmetry operations are symmetry elements of an object, then their product is also
a symmetry element. Any set of operations for which any product of members of that set is a
member of the set is called a group. Group theory is the study of the mathematical properties of
such a collection of symmetry elements. Groups containing only symmetry elements that leave
one point in the molecule unchanged are called point groups. In addition, crystals also have
symmetries arising from translation through space and are called space groups. However, here
we consider only point groups. Each distinct group of symmetry elements has a name.
We now present the Schoenflies naming system for point groups:
Groups with no proper rotation axis.
C1 . Contains the symmetry element Ê only.
C s . Contains symmetry elements Ê and ̂ .
C i . Contains symmetry elements Ê and iˆ .
S n . Contains symmetry elements Ê and Ŝ n . Note that Sˆ1  ˆ and Sˆ 2  iˆ . Therefore,
these groups are C s and C i respectively.
Groups with one proper rotation axis.
C n . Contains the symmetry elements Ê and Ĉ n only.
C nv . Contains the symmetry elements Ê , Ĉ n , and n vertical mirror planes ̂ v .
C nh . Contains the symmetry elements Ê , Ĉ n , and ̂ h (perpendicular to Ĉ n ).
Dihedral Groups
D n . Contains the symmetry elements Ê , Ĉ n , and n Ĉ 2 axes perpendicular to Ĉ n . Note
that for n = 2, the group has three mutually perpendicular Ĉ 2 axes.
Dnh . Contains the symmetry elements Ê , Ĉ n , n Ĉ 2 axes, and ̂ h (mirror plane is
perpendicular to the principal axis). This group also has n vertical mirror planes and iˆ if n is an
even number.
Dnd . Contains the symmetry elements Ê , Ĉ n , n Ĉ 2 axes, and n vertical mirror planes
̂ d . This group also has an improper axis of double order Sˆ 2n parallel to Ĉ n and iˆ if n is an
 
odd number.
Linear Groups
These groups are special cases of the above groups for n   (i.e. operation about the
axis by any angle is symmetry element).
C v . This group is linear unsymmetrical. For example, HCl and HCN.
D h . This group is linear symmetrical. For example, CO2 .
Cubic groups
Td . This group has the symmetry of a regular tetrahedron. For example, CH 4 .
O h . This group has the symmetry of a regular octahedron.
2
I h . This group has the symmetry of a regular icosahedron. For example, B12 H 12
.
The groups Dnh and Dnd are often difficult to distinguish. Both have Ĉ n , n Ĉ 2 , and n
mirror planes. Dnh has a horizontal mirror plane ̂ h  as well as vertical mirror planes ( ̂ v , each
̂ v plane contains a Ĉ 2 axis). But for the Dnd groups, the vertical mirror planes lie diagonally
between the Ĉ 2 axes and are therefore ̂ d .
In determining Schoenflies point groups, it is useful to examine the molecule
systematically. Mortimer provides a step-by-step chart for classifying molecules according to
their point groups. The following is a different sequence of steps for finding point groups. Please
note that it is unnecessary to use both strategies; you only need to use the one that you find
easiest to understand.
1. Determine if a molecule has Td or O h symmetry. In both of these cubic point groups,
there are four C3 axes along the body diagonals of the cube. If the molecule does not belong to
either of these point groups, but contains more than one threefold axis (i.e. more than one axis of
symmetry greater than a twofold), it is of a point group not discussed here.
2. Determine if the molecule has any C n axes. If so, then go to step 3. If not, then look
for a symmetry plane, in which case it belongs to the point group C s , or a center of symmetry, in
which case it belongs to C i . If no symmetry elements are found, then the molecule belongs to
the point group C1 , which has only the identity element.
3. If at least one C n axis is present, find the principal C n axis (i.e. the C n with the
largest n). [Note that there may not be a unique principal axis. For example, the point group D 2
has three C 2 axes perpendicular to each other.] Next determine if there is an improper S 2 n axis
collinear with the principal C n axis. If there is an S 2 n axis but no other elements of symmetry
(except i), then the molecules belongs to one of the S n (n even) point groups. Otherwise, if there
are no S 2 n axes or if there is an S 2 n axis along with other symmetry elements, go to step 4.
4. Determine if the point group is dihedral ( Dn , Dhn , Dnd ) by observing if n C2 axes
(equally spaced) lie in a plane perpendicular to the principal C n axis. If so, go to step 5. If not,
go to step 6.
5. To distinguish between the point groups Dn , Dhn , Dnd , determine if a horizontal
mirror plane (  h ) is perpendicular to the C n axis. If there is a  h , then the point group is Dnh .
If not, determine if there are vertical mirror planes (  d ) containing the C n axis. [The  d planes
must bisect the angle formed by two horizontal C2 axes that are perpendicular to the principal
C n axis.] If  d planes are present, then the point group is Dnd . If the molecule contains no
mirror planes of any kind, then the point group is D n .
6. If a molecule does not have n horizontal C2 axes perpendicular to the C n axis, then it
belongs to either the C n , C nv , or C nh groups, which contain only one proper rotation axis. If the
C n axis contains a horizontal mirror plane (  h ), then it belongs to the C nh group. If the C n axis
has n vertical mirror planes intersecting the C n axis, then the molecule belongs to the C nv group.
If the molecule contains neither horizontal nor vertical mirror planes, then the point group is C n .
Further discussions of these topics are found in Mortimer pages 651-655, 682-689, 707.