Overview - Symmetry groups of molecules

 

C1 Ci Cs

C2 C2v C2h

C3 C3v C4v C5v

D3 D2h D3h D6h D∞h

D3d D5d

S4 Td Oh

The list of characters of all possible irreducible representations is called character table.

character χ The character of an element in a representation is the trace of the matrix for that element

The Schönflies symbol appears in the top left corner of the table. The entries within the character table deliver various information on the irreducible representations of the respective point group. The operations of the group are headers of the columns. It is not necessary to indicate the character for single operations because there are classes of geometrically equivalent operations.

classes All operations of the same kind are said to form one class and each the operations within the class have always the same character.

The number of operations within a class is indicated like a factor (e.g. two in 2 C₃ in the table below). On the left edge of the table, the symmetry species of an irreducible representation appears, using the so-called MULLIKEN symbol. With the difference of employing minuscules, the MULLIKEN symbols are used as well in the classification of orbitals. Only orbitals of one class will contribute to an overlap integral. In this context, but as well to determine the transitional dipole moment, the product rules for symmetry species are important. In the two last columns, the functions with respect to their symmetry appear. It is shown, in which way carthesian coordinates x,y and z, functions of these coordinates and rotations R_i around these axis (which represent angular moments p_i) transform. For example, in group C_3v function z spans symmetry species A₁ and function (x,y) species E.
 

C3v, (3 m)	E	2 C3	3 σv	h = 6
A1	1	1	1	z, z2, x2+y2
A2	1	1	-1		Rz
E	2	-1	0	(x,y),(xy,x2-y2)(xz,yz)	(Rx,Ry)



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