Transformation Matrices; Geometric and Otherwise As examples, consider the transformation matrices of the C 3v

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1 Transformation Matrices; Geometric and Otherwise As examples, consider the transformation matrices of the v group. The form of these matrices depends on the basis we choose. Examples: Cartesian vectors: ˆx x ˆ, y ˆ, ˆ z ŷ ẑ p orbitals on the N atom of NH the three s orbitals on the hydrogen atoms of NH ˆx ŷ ẑ σ v E φ φ N H φ Cartesian basis: σ v y x y σ v φ φ x C σ v φ σ v σ v z : up

2 Example y x z : up φ φ N H φ x y φ σ v σ v φ φ σ C v Three s orbitals on the hydrogen atoms of NH Example, Answers φ φ N H φ φ σ v σ v φ φ σ C v E σ v σ v σ v

3 Transformation of d orbitals d (l,m l ) d ± (l,m l ±) ( ) d ± (l,m ±) d xz [ d + d ] d yz i [d + d ] d x y [d + d ] 4 d xy i [d d ] d z d Y 4 5 π (cos θ ) 5 sinθ cosθe±iϕ π 5 π sin θ e ±iϕ 5 π sinθ cosθ [eiϕ + e iϕ ] 5 π sinθ cosθ i [eiϕ e iϕ ] 5 π sin θ [eiϕ + e iϕ ] 4 5 π sin θ i [eiϕ e iϕ ] 5 π ( cos θ ) 4 see, e.g., Atkins & de Paula, Physical Chemistry 5 π sinθ cosθ cosϕ 4 5 π sinθ cosθ sinϕ 4 5 π sin θ cosϕ 4 5 π sin θ sinϕ 4 ( ) 5 π z r d [d xz + id yz ] ; d [d xz id yz ] d [d x y + id xy ] ; d [d x y id xy ] 5 π xz 5 π yz ( ) 5 π x y 5 π xy Group epresentations epresentation: A set of matrices that represent the group. That is, they behave in the same way as group elements when products are taken. A representation is in correspondence with the group multiplication table. Many representations are in general possible. The order (rank) of the matrices of a representation can vary.

4 Example - show that the matrices found earlier are a representation eg., E (σ v ) σ v educible and Irreducible eps. If we have a set of matrices, {A, B, C,...}, that form a representation of a group and we can find a transformation matrix, say Q, that serves to block factor all the matrices of this representation in the same block form by similarity transformations, then {A, B, C,...} is a reducible representation. If no such similarity transformation is possible then {A, B, C,...} is an irreducible representation.

5 Similarity Transformation maintains a epresentation Suppose the group multiplication rules are such that AB D, BC F, etc... Now perform similarity transforms using the transformation matrix Q: A Q - AQ, B Q BQ, C Q CQ, etc. Multiplication rules preserved: A B (Q AQ)(Q BQ) (Q DQ) D B C (Q BQ)(Q CQ) (Q FQ) F, etc. educing a epresentation by Similarity Transformations A A Q AQ B B Q BQ A B C C Q CQ

6 Blocks of a educed ep. are also epresentations This must be true because any group multiplication property is obeyed by the subblocks. If, for example, AB C, then A B C, A B C and A B. Example: Show that the matrix at left, Q, can reduce the matrices we found for the representation given earlier. Q A Block Factoring Example 6 Q Q Q Q Q σ v Q

7 Significance of Transformations Irreducible epresentations are of pivotal importance Chosen properly, similarity transformations can reduce a reducible representation into its irreducible representations With the proper first choice of basis, the transformation would not be necessary Important Future goal: finding the basis functions for irreducible representations Great Orthogonality Theorem () mn ][Γ j () m n ] h δ ij δ m m δ n n l i l j Proofs: Eyring, H.; Walter, J.; Kimball, G.E. Quantum Chemistry; Wiley, Γ i () matrix that represents the operation in the i th representation. Its form can depend on the basis for the representation. () mn ] matrix element in m th row and n th column of Γ i () l i the dimension of the i th representation h the order of the group (the number of operations) δ ij if ij, otherwise

8 Great Orthogonality Theorem - again Vectors formed from matrix elements from the m th rows and n th columns of different irreducible representations are orthogonal: () mn ][Γ j () mn ] if i j Such vectors formed from different row-column sets of the same irreducible representation are orthogonal and have magnitude h/l i : () mn ] () m n ] (h li )δ m m δ n n The First Sum ule The sum of the squares of the dimensions of the irreducible representations of a group is equal to the order of the group, that is, l i l + l + l + h i this is equivalent to: i [ χ i (E)] h

9 Second Sum ule The sum of the squares of the characters in any irreducible representation equals h, the Proof From the GOT: let mm nn : () mn ] () m order of the group [χ i ()] n ] (h li )δ m m δ n n () mm ] () mm ] (h li ) h Characters of Different Irreducible epresentations are Orthogonal The vectors whose components are the characters of two different irreducible representations are orthogonal, that is, χ i ()χ j () when i j

10 Proof Setting m n in first GOT statement: Γ i () mm Γ j () mm if i j compare this to the statement (i j): χ i ()χ j () Γ i () mm m m Γ j () mm Γ i () mm Γ j () mm m Matrices in the Same Class have Equal Characters This statement is true whether the representation is reducible or irreducible This follows from the fact that all elements in the same class are conjugate and conjugate matrices have equal characters.

11 # of Classes # of Irred. eps. The number of irreducible representations of a group is equal to the number of classes in the group. χ i ()χ j () hδ ij if the number of elements in the m th class is g m and there are k classes, k χ i ( p )χ j ( p )g p hδ ij p