1 / 10

CHEM 515 Spectroscopy

CHEM 515 Spectroscopy. Lecture # 10 Matrix Representation of Symmetry Groups. 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. The Trace of a Matrix.

kinsey
Download Presentation

CHEM 515 Spectroscopy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  4. Reducible and Irreducible Representations • In the case of C2h 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: Γred = Γ1Γ2 … Γn

  5. Determination of the Characters of a Point Group (C3v) • On the side are the matrix representations for the six symmetry operations for the C3v group.

  6. Characters for C3v 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. 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. Condensed Character Tables • For the C3v point group, “2C3” and “3σh” mean that there are two operations in the class of the C3 type and three operations of the σh type. • This type of matrices is called square matrices (no. of classes equals no. of irreducible representations)

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

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

More Related