Irreducible representation of symmetry elements.

1. BITS Pilani
Pilani Campus
08/04/2016
Irreducible representation and character
table

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Translation through z-axis

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Translation through x-axis

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Rotation through px orbital
If a px orbital on the central atom
of a molecule with C2v symmetry
is rotated about the C2 axis, the
orbital is reversed, so the
character will be -1.
is reflected in the yz plane, the
orbital is also reversed, and the
character will be -1.

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Translation through y-axis

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Rotation through pz orbital
Rotation about z-axis, Rz, can be treated by the following way
1. The z-axis is pointing out of the screen
2. If the rotation is still in the same direction then the
result is symmetric
3. If the rotation in opposite direction then anti-symmetric

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Character table (C2v)
1. Characters of +1 indicate that the basis function is unchanged by
the symmetry operation.
2. Characters of -1 indicate that the basis function is reversed by the
symmetry operation.

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Character table (C2v)
The functions to the right are called basis
functions. They represent mathematical functions
such as orbitals, rotations, etc.

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Character table representation
1. An A representation indicates that the
functions are symmetric with respect to
rotation about the principal axis of rotation.
2. B representations are asymmetric with
respect to rotation about the principal axis.
3. E representations are doubly degenerate.
4. T representations are triply degenerate.

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Character table representation
5. 1 and 2 indicate symmetric and asymmetric to
reflection in a vertical plane (σv)
6. ‘ and “ indicate symmetric and asymmetric to
reflection in a horizontal plane (σh)
7. Subscrips u and g indicate asymmetric
(ungerade) or symmetric (gerade) with respect
to a center of inversion.

11. BITSPilani, Pilani Campus
Character table (C2v)
Last column: this gives the transformation properties of the binary
products of Cartesian coordinates xy, yz, zx etc.
and the square of the coordinates x2, y2, z2, x2+y2, x2+y2+z2 etc.
These Cartesian coordinates are referred to as “basis function” on
which the symmetry operation operate

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Five important rules
Rule 1: The sum of the squares of the dimensions of the irreducible
representations of a group is equal to the order of the group, that is,
Σli
2 = l1
2 + l2
2 + l3
2 + l4
2
Rule 2: The sum of the squares of the characters in any irreducible
representation equals h, that is,
Σ [χi(R)]2 = h
R
Rule 3: The vectors whose components are the characters of two
different irreducible representations are orthogonal, that is,
Σ χi(R) χj(R) = 0, when, i ≠ j
R

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Five important rules
Rule 4: In a given representation (reducible or irreducible) the
characters of all matrices belonging to operations in the same
class are identical
Rule 5: The number of irreducible representations of a group is
equal to the number of classes in the group