CHEM 515 Spectroscopy

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CHEM 515Spectroscopy

• Lecture 10
• Matrix Representation of Symmetry Groups

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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.

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The Trace of a Matrix

• The trace of a matrix (?) is the sum of its
  diagonal elements.
• ? 312sin?

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Reducible and Irreducible Representations

• In the case of C2h symmetry, the matrices can be
  reduced to simpler matrices with smaller
  dimensions (11 matrices).

• Because the sub-block matrices cant be further
  reduced, they are called irreducible
  representations. The original matrices are
  called reducible representations.
• The symbol G is used for representations where
• Gred G1 G2 Gn

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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.

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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.

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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
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Condensed Character Tables

• For the C3v point group, 2C3 and 3sh mean
  that there are two operations in the class of the
  C3 type and three operations of the sh type.
• This type of matrices is called square matrices
  (no. of classes equals no. of irreducible
  representations)

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Mulliken Notations

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

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Mulliken Notations

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