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POINT GROUP SYMMETRY CHARACTER TABLES


 Hint: click on a point group symmetry symbol to get irreducible representations, character and product tables
Nonaxial
C1
Cs
Ci
Cn
C2
C3
C4
C5
C6
Cnv
C2v
C3v
C4v
C5v
C6v
Cnh
C2h
C3h
C4h
C5h
C6h
Dn
D2
D3
D4
D5
D6
Dnh
D2h
D3h
D4h
D5h
D6h
Dnd
D2d
D3d
D4d
D5d
D6d
Sn
S4
S6
S8
S10
Higher
Td
Oh
Ih
Linear
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 undersymmetry 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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