Symmetry operations are geometrical operations that interchange or transpose the atomic centers of a given molecule such that the newly obtained orientation of the molecule is physically indistuingishable to the original orientation. The following operations should be considered for three dimensional molecular systems:
Symmetry operations
Symbol | Operation |
---|---|
E | identity operation (do nothing) |
σ | reflection through a mirror plane (σv, σh) |
C_{n} | rotation around n-fold axis |
i | inversion through a center of symmetry |
S_{n} | rotation around n-fold axis + reflection in a plane perpendicular to axis of rotation |
Each of these symmetry operations is connected to a symmetry element such as a plane of symmetry, a symmetry axis, a point of inversion, or a rotation-reflection (or improper) symmetry axis. These geometrical entities are denoted with the same symbols as the corresponding symmetry operations, but lack the operator "hat".
A symmetry axis is denoted as C_{n} if clockwise rotation by 360/n degrees leads to an unchanged configuration as described above. The subscript n is called the order of the axis. Multiple applications of the C_{n} operation are denoted with an additional superscript. Twofold rotation around the C_{3} symmetry axis in ammonia (NH_{3}) is therefore described as C_{3}^{2}. In case there is more than one symmetry axis, the highest order axis is referred to as the principal axis.
The description of symmetry planes is modified in the presence of a symmetry axis. The symbol σ_{v} describes a symmetry plane vertical to the principal axis, while a symmetry plane running horizontally through the principal axis is denoted as σ_{h}. In highly symmetric systems one additional type of symmetry plane occurs that divides the angle between two C_{n} axes. This type of plane is denoted σ_{d}.
The complete symmetry properties of molecular systems are described through (symmetry) point groups, here using the Schoenflies symbols.
Point groups (Schoenflies)
Symbol | Symmetry operations | Abelian? |
---|---|---|
C_{1} | E | yes |
C_{s} | E, σ | yes |
C_{i} | E, i | yes |
C_{n} | E, C_{n} | yes |
S_{n} | E, S_{n} (rotary-reflection) | yes |
C_{nv} | E, C_{n}, n*σ_{v} | only C_{2v} |
C_{nh} | E, C_{n}, σ_{h} | yes |
D_{n} | E, C_{n}, n*C_{2}(vertical) | only D_{2} |
D_{nh} | E, C_{n}, n*C_{2}(vertical), σh | only D_{2h} |
D_{nd} | E, C_{n}, n*C_{2}(vertical), n*σ_{v}(to C2 axes) | no |
T_{d} | E, 6*σ_{d}, 4*C_{3}, 3*S_{4} | no |
Oh | E, 6*σ_{d}, 3*σ_{h}, 6*C_{2}, 4*C_{3}, 3*C_{4}, 3*S_{4}, 4*S_{6}, | no |
Point groups are called Abelian if all their symmetry elements commute, that is, if application of the symmetry elements in one or another sequence leads to the same final result.