Symmetry - intro. Matti Hotokka Department of Physical Chemistry Åbo Akademi University

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1 Symmetry - intro Matti Hotokka Department of Physical Chemistry Åbo Akademi University Jyväskylä 008

2 Symmetrical or not The image looks symmetrical. Why? Jyväskylä 008

3 Symmetrical or not The right hand side is similar to the left hand side. Jyväskylä 008

4 Symmetrical or not No! They are mirror images. If one reflects the left hand side to the right and vice versa, then one will obtain an image exactly similar to the original. Jyväskylä 008

5 Symmetry operation Such a change of an object that the image cannot be distinguished from the original. Reflection: The mirror image is similar to the original but we know that changes have been made. Jyväskylä 008

6 Symmetry operations Identity Rotation Reflection Rotation-reflection Inversion (Translation) Jyväskylä 008

7 Notations Shönfliess (for chemists) and International (for physicists) Jyväskylä 008

8 Notations Identity E Rotation C n n Reflection h, v, d m Rotation-reflection S n n Inversion i Jyväskylä 008

9 Equilateral triangle The symmetry operations are E C C h - S, S C, C ', C " v, v, v Jyväskylä 008

10 Equilateral triangle E S C " C S - V C C V h C ' V Jyväskylä 008

11 Multiplication Product of the symmetryoperators A and B, A B, means that you operate with B on the original and then with A on the intermediate result. ( A B) = A( B ) Jyväskylä 008

12 Multiplication table E C E E C C C C Jyväskylä 008

13 Multiplication table E C E E C C C C E C E C C C C Jyväskylä 008

14 Jyväskylä 008 Multiplication table

15 Mathematical group The set of symmetry operations and the multiplication operator form a group if (P) A (B (P) E A (P) A C)=(A B) C (Associative law) E=E A=A A (Identity element) A - A A - =A - A=E (Inverse element) Jyväskylä 008

16 Group theory A huge mathematical literature is based on the three postulates. A few examples (A B) - = B - A - A B B A (generally; they may be equal) Order of group, #, is number of symmetry operations in that group For the equilateral triangle, #D h = Isomorphic groups contain different sets of objects but the multiplication table is similar Generators is a subset of the group that will generate all the remaining symmetry operations Jyväskylä 008

17 Generators D h has elements: E, C, C, The generators are C, h, C h, S, S -, C, C ', C ", v, v, Another set of generators can be chosen C, h, v v Jyväskylä 008

18 Group theory Similarity transformation C=A - Jyväskylä 008 B A Conjugated elements B and C are conjugated if there is such an A in the group that C=A - B A (Notation C B) Class All elements for which one can find an A G such that the elements are conjugated form a class For the equilateral triangle C, C 'and C " form a class Each element can only belong to one class All elements in a class behave similarly

19 Group theory The classes of the equilateral triangle are {E} {C, C } {C, C ', C "} { h} {S, S - } { v, v, v } Jyväskylä 008

20 The point groups Schönfliess notation C, C s, C i, C n (n=,,4,...) C nh, C nv (n=,,4,...) D n, D nd, D nh (n=,,4,...) S n (n=,,4,...) C v, D h T d, T h, O h, I h, K Jyväskylä 008

21 The point groups Schönfliess notation is used by most chemists The international notation lists the generators Jyväskylä 008

22 Jyväskylä 008 The point groups

23 The point groups Classification is done by using a flow diagram Jyväskylä 008

24 Special point group? Finish C v, D h, T d, T h, O h, I h YES NO C n (n>)? NO? YES C s YES C i YES NO i?? NO Collinear S n? YES S n NO C YES NO YES C axes? NO D nh YES h? h? YES C nh NO NO D nd YES v? v? YES C nv Jyväskylä 008 D n C n

25 Character table C v E C v (xz) v (yx) A A B B C x y Jyväskylä 008

26 Matrix representation A rotation in the xy plane can be given as x' = x cosφ y sinφ y' = x sinφ + y cosφ This can be written as a matrix equation x' y' z'!!"!!!"! # % $ %"%= cosφ sinφ 0 sinφ cosφ x y z # %! %"%!"! $ # % %"% $ Jyväskylä 008

27 Matrix representation Rotation through 0 C = & ( ( ( ', i.e., C ) * Rotations of the corners of an equilateral triangle ',../. - 0,../. - /= ' ' /, - 0. /./. C Jyväskylä 008

28 @ What s common The common feature between the matrices is trace C = : < </< ; Sum = 0 Sum = 0 =??/? > Trace is called character ( theory ) in symmetry Jyväskylä 008

29 M N Isomorphy Point groups: C C =C Matrix groups: Rotation through 40 C = G I I I H J L L L K Indeed, Jyväskylä 008 A C C C B A D FC FC B FC E A C C C B D F F E F= D F F F E

30 Similarity transformation Two set of matrices both represent the point group Therefore the matrices must be related Yes, they are, through a similarity transformation Jyväskylä 008

31 S O V Y Q X T P Z W ` ` ` ` b^ b^ ^ a a a a 6 [ [R[ 6 b b b Example XRX S = 0 Choose S = UURU therefore QRQ 0 \ ^ ^ ^ ] b bcb= 0 \ ] bcb ^c^ \ b_b ^c^ ] 6 6 \ ^_^ ] S C S = Jyväskylä 008

32 Goal The simplest possible matrices: diagonal Jyväskylä 008

33 Homomorphy Like isomorphy but one matrix for several symmetry operations Still a similar multiplication table Makes smaller matrices possible Goal: still the simplest possible matrices which are x, in some cases x or x Still, several representations can be constructed of a set of matrices Jyväskylä 008

34 d d Example C v E C v (xz) v (yx) Jyväskylä 008 A A B B The simplest matrices are (+) and (-). Still, four different combinations can be formed so that the multiplication tables are similar. Thus there are four representations.

35 e e e Irreducible representation The simplest possible matrix representation Cannot be reduced to a simpler form: irreducible Any matrix representation Can be formed out of these simplest building blocks (and a similarity transformation) Conversely, any matrix can be reduced to its building blocks. This is called reducing a representation. Jyväskylä 008

36 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Example E = 0 0 l n n/n m o q q/q p 0 0 C = 0 0 f h h/h g Water, C v 0 0 i k k/k j σ v = 0 0 r t t/t s u w w/w v σ ' v = 0 0 x z z/z y { } }/} C v E C v v E E C v v C C E v v v v v E C v v v C E Please verify the multiplication table! Jyväskylä 008

37 Œ Œ Exmple Water ƒ " 0 0 = 0 0 " S = 0 S ƒ 0 0 Let 0 Then Š Œ Œ Œ ˆ Š Œ Œ Œ= Š ˆ Œ Œ Š ˆ Œ"Œ 0 0 = 0 0 " ˆ S C S Jyväskylä 008

38 Ÿ Example Summarising Water E = 0 0 / / 0 0 C = 0 0 Ž 0 0 σ v = 0 0 / š œ ž ž/ž σ ' v = 0 0 / / The matrices consist of B A B Γ = A + B C v E C v (xz) v (yx) Jyväskylä 008 A A B B

39 ª ««Reducing a representation Notations The general (reducible) representation is The irreducible representations of the point group are In C v, A, A, B, B Number of symmetry operations in the point group is g = # The classes of symmetry operations in the point group are i The number of symmetry operations in class i is g i The character of class i in irreducible representation is i The character of class i in is i The integer weight of irreducible representation in is a Thus Γ = a µ µ µ a µ = g i χ χ µ g i i i Jyväskylä 008

40 Example C v ()E ()C () v (xz) () v (yx) A A B B = = 0 4 a [ ( ) ( )] A a [ ( ) ( ) ( ) ( )] A a [ ( ) ( ) ( ) ( )] B a [ ( ) ( ) ( )] B = = 4 = = = = Jyväskylä 008 a µ = g i χ χ µ g i i i

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

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: