1. Lecture # 9 Matrix Representation of Symmetry Groups
CHEM 515 Spectroscopy
Lecture # 9
Matrix Representation of Symmetry Groups

2. 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

3. 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 m×n .

4. Matrix Representations
We are concerned here with square matrices that have equal dimensions (n × n),
and with column matrices for vector representations

5. 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: aij+bij

6. Matrix Addition and Subtraction

7. 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.

8. Matrix Multiplication

9. Matrix Multiplication
Example:

10. The Determinant of a Matrix
For the matrix:
Its determinant |A| is given by:

11. The Determinant of a Matrix
For the matrix:
Its determinant |A| is given by:

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

13. Transpose of Matrices
The transpose of a matrix is found by exchanging rows for columns.
For the matrix A = (aij) , its transpose is given by AT=(aji)

14. Transpose of Matrices
In the case of a square matrix (n×n), the transpose can be used to check if a matrix is symmetric.
For a symmetric matrix A = AT

15. Inverse of Matrices
Assuming we have a square matrix A, which is non-singular, then there exists an n×n 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.

16. 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.

17. The C2h Point Group as an Example
For the C2h symmetry, we have the elements: E, i, C2 and σh.
Operator E (Identity) does nothing to the vector.

18. The C2h Point Group as an Example
The operator i can be represented by the following matrix that exchanges each coordinate into minus itself.

19. The C2h Point Group as an Example
The operator σh leaves x and y coordinates unchanged but inverts the sign of z.

20. 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 θ.

21. 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:

22. The C2h Point Group as an Example
Which can be expressed using matrices:
For the C2 rotation, θ = π

23. 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:

24. 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.

25. The Representations of the Four Operators for the C2h Point Group