reducible and irreducible representations

reducible and irreducible representations

• 2.
   Representation isa set of matrices which represent the operations of a point group. It can be classified in to two types, 1. Reducible representations 2. Irreducible representations Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2s)
• 3.
   Let usconsider the C2h point group as an example. E, C2 ,sh & I are the four symmetry operations present in the group. . The matrix representation for this point group is give below. In the case of C2h symmetry, the matrices can be reduced to simpler matrices with smaller dimensions (1×1 matrices).
• 4.
  • A representationof higher dimension which can be reduced in to representation of lower dimension is called reducible representation. •Reducible representations are called block- diagonal matrices. Eg: Each matrix in the C2v matrix representation can be block diagonalized To block diagonalize, make each nonzero element into a 1x1 matrix                                                                    100 010 001 100 010 001 100 010 001 100 010 001 E C2 sv(xz) sv(yz)
• 5.
   A blockdiagonal matrix is a special type of matrices, and it has “blocks” of number through its “diagonal” and has zeros elsewhere. 5                         sincos0000 2120000 009000 00042cos 0005sin1 000436
• 6.
   The traceof a matrix (χ) is the sum of its diagonal elements.  χ = 31+2sinθ 6                         sincos0000 2120000 009000 00042cos 0005sin1 000436  i iiaA)(
• 7.
  • Because thesub-block matrices can’t be further reduced, they are called “irreducible representations”. The original matrices are called “reducible representations”. Irreducible Representations:  If it is not possible to perform a similarity transformation matrix which will reduce the matrices of representation T, then the representation is said to be irreducible representation.  In general all 1 D representations are examples of irreducible representations. • The symbol Γ is used for representations where: Γred = Γ1 Γ2 … Γn
• 10.
   1. Dimensionof the irreducible representations: If it is uni dimensional (character of E=1), term A or B is used. For a two dimensional representation, term E is used. If it is 3-D term T is used.  2. Symmetry with respect to principle axis: If the 1-D irrep is symmetrical with respect to the principle axis () [i.e., the character of the operation is +1], the term A is used. However if the 1-D represent is unsymmetrical with respect to the principal axis (i.e., the character of is -1) the term B is used.  3. Symmetry with respect to subsidiary axis or plane: If the irrep is symmetrical with respect to the subsidiary axis, or in it absence to plane, subscripts, , ,are used if it is unsymmetrical subscripts ,,, are used.
• 11.
   4. Primeand double prime marks are used over the symbol of the irrep to indicate its symmetry or anti symmetry with respect to the horizontal plane ().  5. If there is a centre of symmetry in the molecule, subscripts g and u are used to indicate the symmetry or anti symmetry of irreps res. Suppose the point group has no centre of symmetry, g or u subscripts are not used. Term ‘g’ stands for gerade (centrosymmetric) & ‘u’ for ungerade ( non- centrosymmetric).  On the basis of the above, symbols can be assigned to the irreducible representations of point group.