CHEM%20515%20Spectroscopy

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CHEM 515Spectroscopy

• Lecture 9
• Matrix Representation of Symmetry Groups

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Matrix Representations

• Symmetry operators and symmetry operations can be
  expressed in forms of matrices. Matrices can be
  used as representations of symmetry operators.
• A vector which is used to define a point in a
  space can be represented with a one-dimensional
  matrix.

row vector
column vector
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Matrices

• A matrix is any rectangular array of numbers
  setting between two brackets.
• The general form of a matrix is
• or in a more compact form A aij
• The above matrix have a dimension of mn .

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Matrix Representations

• We are concerned here with square matrices that
  have equal dimensions (n n),
• and with column matrices for vector
  representations

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Matrix Addition and Subtraction

• If and only if the dimensions of the two matrices
  A and B are the same, they can be added or
  subtracted. (Both matrices have the identical
  amount of rows and columns)
• Addition can be performed by adding the
  corresponding elements aijbij

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Matrix Addition and Subtraction
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Matrix Multiplication

• When the number of columns of the first matrix is
  the same as the number of rows in the second
  matrix, then matrix multiplication can be
  performed.

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Matrix Multiplication
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Matrix Multiplication

• Example

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The Determinant of a Matrix

• Determinant of a 22 Matrix
• For the matrix
• Its determinant A is given by

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The Determinant of a Matrix

• Determinant of a 33 Matrix
• For the matrix
• Its determinant A is given by

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The Determinant of a Matrix

• Determinant of a 33 Matrix
• The above matrix is said to be a singular matrix.

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Transpose of Matrices

• The transpose of a matrix is found by exchanging
  rows for columns.
• For the matrix A (aij) , its transpose is
  given byAT(aji)

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Transpose of Matrices

• In the case of a square matrix (nn), the
  transpose can be used to check if a matrix is
  symmetric.
• For a symmetric matrix A AT

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Inverse of Matrices

• Assuming we have a square matrix A, which is
  non-singular, then there exists an nn matrix A-1
  which is called the inverse of A, such that this
  property holds
• AA-1 A-1A E where E is the identity matrix.

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Application of Matrices on Symmetry Operations

• Matrices can be used as representations of
  symmetry operations. The effect of symmetry
  operations is going to be considered on a point
  defined by a column matrix
• where x, y and z represent the location of that
  vector in space with respect to the point of
  origin.

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The C2h Point Group as an Example

• For the C2h symmetry, we have the elements E, i,
  C2 and sh.
• Operator E (Identity) does nothing to the vector.

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The C2h Point Group as an Example

• The operator i can be represented by the
  following matrix that exchanges each coordinate
  into minus itself.

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The C2h Point Group as an Example

• The operator sh leaves x and y coordinates
  unchanged but inverts the sign of z.

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The C2h Point Group as an Example

• The operator C2 (to be taken to set along the
  z-axis) changes x and y coordinates but leaves z
  unchanged.
• It is better to derive a general matrix for an
  n-fold rotation that is applicable for a rotation
  through any angle ?.

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The C2h Point Group as an Example

• Before a rotation takes a place, the coordinates
  for the vector of length l is

The rotated vector has coordinates
From trigonometry
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The C2h Point Group as an Example
Which can be expressed using matrices
For the C2 rotation, ? p
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The C2h Point Group as an Example

• The final matrix representation for C2 rotation
  is
• The general form of matrix representing the C2
  rotation is

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The Representations of the Four Operators for the
C2h Point Group

• These four matrices form a mathematical group
  that obeys the same mathematical table for the
  C2h point group as the operations there.

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The Representations of the Four Operators for the
C2h Point Group
C2h E C2 sh i
E E C2 sh i
C2 C2 E i sh
sh sh i E C2
i i sh C2 E