1. CHEM 515 Spectroscopy Lecture # 10 Matrix Representation of Symmetry Groups

2. 2 Block Diagonal Matrices A block diagonal matrix is a special type of matrices, and it has “blocks” of number through its “diagonal” and has zeros elsewhere.

3. 3 The Trace of a Matrix The trace of a matrix ( χ ) is the sum of its diagonal elements. χ = 31+2sinθ

4. 4 Reducible and Irreducible Representations In the case of C 2h symmetry, the matrices can be reduced to simpler matrices with smaller dimensions (1×1 matrices). Because the sub-block matrices can’t be further reduced, they are called “irreducible representations”. The original matrices are called “reducible representations”. The symbol Γ is used for representations where: Γ red = Γ 1 Γ 2 … Γ n

5. 5 Determination of the Characters of a Point Group (C 3v ) On the side are the matrix representations for the six symmetry operations for the C 3v group.

6. 6 Characters for C 3v Point Group Mathematical treatments involving determining the matrix traces of the irreducible representations and applying the “great orthogonality theorem” yield character tables describing the molecular symmetry with single numbers rather than matrices.

7. 7 Character Tables “Similarity transformation” method can be also used to classify the symmetry elements within a point group. Generally, symmetry operations that produce the same effect are in the same class. Notice that the characters for any given irreducible representation must be the same for operations in the same class

8. 8 Condensed Character Tables For the C 3v point group, “2C 3 ” and “3σ h ” mean that there are two operations in the class of the C 3 type and three operations of the σ h type. This type of matrices is called square matrices (no. of classes equals no. of irreducible representations)

9. 9 Mulliken Notations Mulliken notations (A, B, E with some subscripts, such as numbers or letters) are used to name the irreducible representations. C 3v E2C32C3 3σv3σv Γ 1 = A 1 111 Γ 2 = A 2 11 Γ 3 = E20

10. 10 Mulliken Notations Mulliken notations (A, B, E with some subscripts, such as numbers or letters) are used to name the irreducible representations.