Download - Topic 6 (character table)
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.