Molecular symmetry and group theory

Dr. Md. Monirul Islam Department of Chemistry

University of Rajshahi

Character table

The triangular basis set does not uncover all Γirr of

the group defined by {E, C3, C32 , σv, σv, σv″ }. A

triangle represents Cartesian coordinate space (x, y,

z) for which the Γis were determined. May choose

other basis functions in an attempt to uncover other

Γis. For instance, consider a rotation about the z-

axis,

Character table

Character table

Character table

Character table

Character table

Character table

Thus the same result is obtained

Note, the derivation of the character table in this section is based solely on the properties of characters; the table was derived algebraically.

Character table

Area II Area I Area III Area IV

Area I : The characters of the irreducible representation i of the group.

Area II : Mulliken symbols for the irreducible representation i of the group.

• All one-dimensional representations: A or B

• Two-dimensional representations : E

• Three-dimensional representations : T or sometimes F

• One-dimensional representations that are symmetric with respect to

rotation by 2/n, i.e. (Cn) = 1 are designated A

• One-dimensional representations that are antisymmetric with respect

to rotation by 2/n, i.e. (Cn) = -1 are designated B

• Subscript 1 is attached to A when symmetric with respect to C2.

• Subscript 2 is attached to B when antisymmetric with respect to C2.

• Prime is attached to A or B when symmetric with respect to

h.

• Double prime is attached to A or B when antisymmetric with

respect to h.

• Subscript g is attached to A or B when symmetric with

respect to i.

• Subscript u is attached to A or B when antisymmetric with

respect to i.

• Subscripts are used to E or F for two dimensional

representations.

Area III

• x, y and z represent coordinates.

• Rx, Ry and Rz stand for rotations about the specified axes.

Area IV

• The squares and binary products of coordinates according to

their transformation properties

• For example, the pair of functions xz and yz must have the

same transformtion properties as the pair x, y, since z goes

into itself under all symmetry operation in the group.

• Accordingly, (xz, yz) are found opposite the E representation.