10: Representation of point group part-1 matrix algebra CHEMISTRY. PAPER No.13 Applications of group Theory

Transcription

1 1 Subject Chemistry Paper No and Title Module No and Title Module Tag Paper No 13: Applications of Group Theory CHE_P13_M10

2 2 TABLE OF CONTENTS 1. Learning outcomes 2. Introduction 3. Definition of a matrix 3.1 Some special types of matrices for common use 3.2 Matrix algebra: Addition subtraction multiplication 3.3 Some characteristics/rules of binary matrix operations 4. Direct Product of matrices. 5. Summary

3 3 1. Learning Outcomes After studying this module, you shall be able to o Need for linking symmetry properties of molecules with group theory o Definition of a matrix o Special types of matrices o Know how to add the matrices? o How to subtract the matrices? o How to multiply the matrices? o Know the special properties of the binary operations of matrices o Know how find the Direct Product i.e. Kronecker product of matrices 2. Introduction In earlier modules on symmetry elements and symmetry operations, presence of symmetry elements and symmetry operations generated by them has been explained in detail. Group and its characteristics were discussed and scheme of classification of molecules into point group explained in detail with the help of flow chart and various examples. Now question arises how to make use of these symmetry properties of the molecules and the group theory in solving complex spectroscopic chemical problems. This can be achieved only when we find a procedure to link the symmetrical properties of a molecule and mathematical property ie group theory This can be achieved through matrix representation of a point group ie each and every symmetry operation of a point group can be represented by a matrix where a matrix forms the basis for representation. These matrices behave very similar to symmetry operations. Just as various symmetry operations constitute a point group, similarly matries corresponding to each symmetry operation of the point group constitute a group. All properties of symmetry operations thus can be reflected into their matrix representations. Now question arises how to represent the effects of symmetry operation on the molecules in mathematical forms. Here matrix representation of symmetry operation comes to our rescue. There are number of theories in mathematics about matrix representation. These theories can be used for solving chemical problems. Before finding matrix representations for a point group it is very necessary to

4 4 know the basics of matrix algebra. In this module on representation matrix algebra will be discussed in very very brief so the reader can have the idea how to use these matrices. 3. Definition of a matrix It is a rectangular array of elements (symbols or numbers) in rows and columns enclosed in square bracket [ ] different from determinant. Matrix A is represented as : a 11 a a 1n A a 21 a a 2n a m1 a m a mn Horizontal lines are called rows and vertical lines are called columns. Dimension (or size) of a matrix means how many rows and how many columns are there in the matrix. In matrix A, there are m rows and n columns. So the dimension of matrix A is m x n. Represented generally as [ A] m x n. first subscript is for row and second subscript is for column. Where row 1 elements are a 11, a 12, ---a 1n and column 1 elements are a 11, a 21, -----a m1. The elements in rows and columns are also known as the entries of the matrix. In general each entry of the matrix can be written as a ij ie the element at the intersection of i th row and j th column. 3.1 Some special types of matrices for common use: Only very few of these special matrices will be given here. These are: (I) Vector A vector is a matrix that has only one row or one column. These are of two types. (i) Row vector: If matrix has only one row it is called row vector. [a 1,a 2,-----a n ]. (ii) Column vector: If matrix has only one column it is called column vector. c 1 c2 c m (II) Square matrix: If the number of rows (m) of a matrix is equal to the number of columns (n) ie m n it is called square matrix. Matrix [B] below is square matrix

5 B It is 3x3 matrix In a square matrix, the elements on the diagonal are called diagonal elements.here it is shown in between dotted lines and these are 1, 5,and 8. (III) Diagonal matrix A square matrix is one in which all entries off the diagonal are zero ie only the diagonal entries are non zero such as [C] ie a ij 0, i j C diagonal elements One or more or all the diagonal elements/entries can be zero (IV) Identity matrix: A diagonal matrix in which all diagonal elements are equal to one is called identity matrix. It can be of any size, a ij 0, i j and a ii 1 for all values of i D Identity matrix can be denoted by symbol E or I (V) Symmetric and Antisymmetric matrices: Square matrices for which a ij a ji for all values of i and j are called symmetric about the main diagonal or simply symmetric matrices. Square matrices for which a ij - a ji for all values of i and j are called anti symmetric or skew symmetric matrices. E F Symmetric about the diagonal Antisymmetric about the diagonal Off diagonal elements ie above and below are same but sign is reversed (VI) Transpose: The transpose of a matrix [G] mxn is denoted by A

6 6 Transpose of matrix [G] can be obtained by changing its rows into columns to give[ G ] ie rows of [ G ] are the columns of [G] and the rows of [G] are the columns of [G ]. Example below shows the [G] and its [ G ] G , 2x3 [ G ] x2 (VII) Orthogonal matrix A square matrix is said to be orthogonal if it is when multiplied by its transpose, gives identity matrix I or E. 3.2 Matrix algebra: To understand the applications of matrices one must understand how these are combined (addition, substraction, multiplication and a special way of division). Let us look at these. (i)matrix addition: Two matrices A and B can be added only if they are of the same size/dimension Addition is shown as: A B C A +B C add the corresponding elements of both A and B to give entries of new matrix C C (5+6) (7+2) 3-2 (1+3) (2+5) (19+7) (ii) Matrix subtraction: Two matrices A and B can be subtracted only if they are of the same size and subtraction is given by [A]- [B] [D]

7 7 A B C A -B C subtract the corresponding elements of both B from that of A to give entries of new matrix C C (5-6) (2-7) 3-(-2) (1-3) (2-5) (7-19) (iii) Matrix Multiplication: Two matrices A and B can be multiplied only if the number of columns of A is equal to the number of rows of B ie these should be conformable [A] mxp x [B] pxn [C] nxm ie if A is of mxp dimension matrix and B is pxn dimension matrix, the resulting matrix C will be of mxn dimension

8 8 A now AB C and 2x3 These are conformable x x3+2x5+3x9 5x-2+ 2x-8+3x-10 c 11 c 12 1x3+2x5+7x9-2x1+-8x2-10x7 c 21 c 22 A x B C Now take 1 st row of A multiply it with corresponding elements of 1st column of B and added together to get c 11. To get c 12, multiply first row of A with 2 nd column of B. Similarly take 2 nd row of A multiply with corresponding elements of 1 st column of B to get c 21. To get c 22 multiply 2 nd row of A with 2 nd column of B. Therefore, C Division of matrices will not be discussed as it is not straight forward. 3.3 Some characteristics/rules of binary matrix operations: (i) Commutative law of addition: If A and B matrices are of m x n dimension then [A]+[B] [B]+[A] is true. (ii) Associative law of addition: If [A], [B], [C] matrices are of m x n dimensions, then [A]+[B]+[C] [A]+{[B]+[C]} {[A]+[B]}+[C] (iii) Associative law of multiplication: If [A], [B], [C] matrices are of m x n, n x p and p x q dimensions respectively, then [A][B][C] [A][BC] [AB][C] is valid provided order of multiplication is not changed (iv) Distributive Law: If A and B matrices are of m x n dimension and C and D are of n x p dimensions, then A(C+D) AC+AD and (A+B)C AC+BC are valid and resulting matrices will be of m x p dimensions

9 9 A B C BC ABC Now let us take AB (AB)C C (v) Is AB BA? If product AB exists, then number of columns of A must be equal to the number of rows of B. If BA exists then the number of columns of B must be equal to the number of rows of A. Now for AB BA, the resulting matrices from AB and BA have to be of the same size. This is possible only when A and B are square matrices and are of the same size. Even then in general, AB BA. Product of matrices in general is non commutative Find AB and BA given 6 3 A 2 5 B then AB and BA ie AB BA (vi) If A 0, B 0 AB may be zero, BA may not be zero A and B AB and BA Vectors and operators will not be discussed in this module.

10 10 4. Direct Product of matrices: Direct product or Kronecker product of matrices finds applications in quantum chemistry and group theory. If a matrix A is of nxn dimension and B is of mxm dimension, then direct product is defined as nm x nm matrix ie A B C. Symbol is a standard notation for direct product and it is not a sign of simple multiplication. Let us take two matrices A a 11 a 12 a 21 a 22 and B then direct producta B a 11 a 12 b 11 b 12 a 21 a 22 b 21 b 22 b 11 b 12 b 21 b 22 a 11 a 12 a 21 a 22 B a 11 B a 12 B a 21 B a 22 B then A B a 11 b 11 a 11 b 12 a 12 b 11 a 12 b 12 a 11 b 21 a 11 b 22 a 12 b 21 a 12 b 22 a 21 b 11 a 21 b 12 a 22 b 11 a 22 b 12 a 21 b 21 a 21 b 22 a22 b 21 a 22 b 22 a 4x4 matrix is formed let A Therefore, A B and B x3 4x1 3x3 3x1 4x2 4x1 3x2 3x1 2x3 2x1 1x3 1x1 2x2 2x1 1x2 1x

11 11 5.Summary Need for matrix representation stressed Matrix defined and explained Only some special types of matrices were discussed Matrix algebra with special reference to addition, subtraction and multiplication has been discussed Some special characteristics of binary operations of matrices have been mentioned with out proof. How to find Kronecker product i.e. direct product of matrices has been explained by taking suitable example.