Symmetry Essentials
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) |
| Cn | rotation around n-fold axis |
| i | inversion through a center of symmetry |
| Sn | 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 Cn 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 Cn operation are denoted with an additional superscript. Twofold rotation around the C3 symmetry axis in ammonia (NH3) is therefore described as C32. 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 σv describes a symmetry plane vertical to the principal axis, while a symmetry plane running horizontally through the principal axis is denoted as σh. In highly symmetric systems one additional type of symmetry plane occurs that divides the angle between two Cn axes. This type of plane is denoted σd.
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? |
|---|---|---|
| C1 | E | yes |
| Cs | E, σ | yes |
| Ci | E, i | yes |
| Cn | E, Cn | yes |
| Sn | E, Sn (rotary-reflection) | yes |
| Cnv | E, Cn, n*σv | only C2v |
| Cnh | E, Cn, σh | yes |
| Dn | E, Cn, n*C2(vertical) | only D2 |
| Dnh | E, Cn, n*C2(vertical), σh | only D2h |
| Dnd | E, Cn, n*C2(vertical), n*σv(to C2 axes) | no |
| Td | E, 6*σd, 4*C3, 3*S4 | no |
| Oh | E, 6*σd, 3*σh, 6*C2, 4*C3, 3*C4, 3*S4, 4*S6, | 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.