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Point Group Symmetry Character Tables

C1 Cs Ci
C2 C3 C4 C5 C6
C2v C3v C4v C5v C6v
C2h C3h C4h C5h C6h
D2 D3 D4 D5 D6
D2h D3h D4h D5h D6h
D2d D3d D4d D5d D6d
S4 S6 S8 S10
Td Oh Ih
C∞v D∞h

Point Group Symmetry

Point group symmetry is an important property of molecules widely used in some branches of chemistry: spectroscopy, quantum chemistry and crystallography.

An individual point group is represented by a set of symmetry operations:

  • E - the identity operation
  • Cn - rotation by 2π/n angle *
  • Sn - improper rotation (rotation by 2π/n angle and reflection in the plane perpendicular to the axis)
  • σh - horizontal reflection plane (perpendicular to the principal axis) **
  • σv - vertical reflection plane (contains the principal axis)
  • σd - diagonal reflection plane (contains the principal axis and bisect the angle between two C2 axes perpendicular to the principal axis)

* - n is an integer
** - principal axis is a Cn axis with the biggest n.

Molecule belongs to a symmetry point group if it is unchanged under all the symmetry operations of this group.

Certain properties of the molecule (vibrational, electronic and vibronic states, normal vibrational modes, orbitals) may behave the same way or differently under the symmetry operations of the molecule point group. This behavior is described by the irreducible representation(irrep, character). All irreducible representations of the symmetry point group may be found in the corresponding character table. Molecular property belongs to the certain irreducible representation if it changes under symmetry operations exactly as it is specified for this irreducible representation in the character table.

If some molecular property A is a product of other properties B and C, the character of A is a product of B and C characters and may be determined from the character product table.

In general assignment of a character (irriducible representation) to a given molecular property depends on the molecular orientation. To make this assignment unambiguous Mulliken developed conventions for symmetry notations which became widely accepted.

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