5.4 Molecular Symmetry
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A symmetry operation is an action that leaves the molecule looking the same after
it has been carried out, i.e., the bond lengths and the bond angles remain unchanged.
Each symmetry operation has a corresponding symmetry element, which consists of
all the points that stay at the same place when the symmetry operation is performed:
axis, plane, or point. There are operations that keep the orientation of a trihedron
(operations of first kind) and operations, which invert it (operations of second kind).
The product of two operations of the first kind is still of the first kind, so that their
set constitutes a subgroup (G 1 ) of the whole group (G). The number of elements of
the second kind is either zero or equal to those of G 1 . It is possible to obtain them all
by associating one of them with G 1 . It is possible to characterize a group G by using
a symbol naming G 1 and, if necessary, a subscript naming the operation of second
kind. To each group G belongs the identity operation E consisting of doing nothing.
5.4.2 Subgroup G 1
In addition to E, the elements of G 1 are rotations:
• C n a n-fold axis of rotation with n an integer. Rotation by 2π /n leaves the molecule
unchanged.
• D n in addition to C n , there are n two-fold rotations about axes perpendicular to
the principal axis (the n-fold axis of rotation).
• T contains the symmetry elements of a regular tetrahedron, i.e., 4 C 3 axes and 3
C 2 axes.
• O contains the symmetry elements of a cube.
5.4.3 Operations of Second Kind
It principally concerns a reflection in a plane of symmetry (usually called σ ) whose
addition to G 1 does not introduce new axes of rotations.
• The plane of symmetry is orthogonal to the principal axis of G 1 . It is marked by
the index h (for horizontal): C nh , D nh , T h , O h .
• The plane of symmetry includes the principal axis. For G 1 = C n , one uses the
index v (vertical). For G 1 = D n , when there is more than one horizontal symmetry
plane, one uses the index d (diagonal), the symmetry plane is the plane bisecting
two binary axes.
• S n Rotation–reflection: rotation of 2π /n about the axis followed by a reflection in
a plane perpendicular to the axis. S 1 is the same as reflection and S 2 is the same
as inversion (through the center of symmetry), also called C i (i for inversion).
111
A symmetry operation is an action that leaves the molecule looking the same after
it has been carried out, i.e., the bond lengths and the bond angles remain unchanged.
Each symmetry operation has a corresponding symmetry element, which consists of
all the points that stay at the same place when the symmetry operation is performed:
axis, plane, or point. There are operations that keep the orientation of a trihedron
(operations of first kind) and operations, which invert it (operations of second kind).
The product of two operations of the first kind is still of the first kind, so that their
set constitutes a subgroup (G 1 ) of the whole group (G). The number of elements of
the second kind is either zero or equal to those of G 1 . It is possible to obtain them all
by associating one of them with G 1 . It is possible to characterize a group G by using
a symbol naming G 1 and, if necessary, a subscript naming the operation of second
kind. To each group G belongs the identity operation E consisting of doing nothing.
5.4.2 Subgroup G 1
In addition to E, the elements of G 1 are rotations:
• C n a n-fold axis of rotation with n an integer. Rotation by 2π /n leaves the molecule
unchanged.
• D n in addition to C n , there are n two-fold rotations about axes perpendicular to
the principal axis (the n-fold axis of rotation).
• T contains the symmetry elements of a regular tetrahedron, i.e., 4 C 3 axes and 3
C 2 axes.
• O contains the symmetry elements of a cube.
5.4.3 Operations of Second Kind
It principally concerns a reflection in a plane of symmetry (usually called σ ) whose
addition to G 1 does not introduce new axes of rotations.
• The plane of symmetry is orthogonal to the principal axis of G 1 . It is marked by
the index h (for horizontal): C nh , D nh , T h , O h .
• The plane of symmetry includes the principal axis. For G 1 = C n , one uses the
index v (vertical). For G 1 = D n , when there is more than one horizontal symmetry
plane, one uses the index d (diagonal), the symmetry plane is the plane bisecting
two binary axes.
• S n Rotation–reflection: rotation of 2π /n about the axis followed by a reflection in
a plane perpendicular to the axis. S 1 is the same as reflection and S 2 is the same
as inversion (through the center of symmetry), also called C i (i for inversion).