Symmetry - intro Matti Hotokka Department of Physical Chemistry Åbo Akademi University Jyväskylä 008
Symmetrical or not The image looks symmetrical. Why? Jyväskylä 008
Symmetrical or not The right hand side is similar to the left hand side. Jyväskylä 008
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
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
Symmetry operations Identity Rotation Reflection Rotation-reflection Inversion (Translation) Jyväskylä 008
Notations Shönfliess (for chemists) and International (for physicists) Jyväskylä 008
Notations Identity E Rotation C n n Reflection h, v, d m Rotation-reflection S n n Inversion i Jyväskylä 008
Equilateral triangle The symmetry operations are E C C h - S, S C, C ', C " v, v, v Jyväskylä 008
Equilateral triangle E S C " C S - V C C V h C ' V Jyväskylä 008
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
Multiplication table E C E E C C C C Jyväskylä 008
Multiplication table E C E E C C C C E C E C C C C Jyväskylä 008
Jyväskylä 008 Multiplication table
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
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
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
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
Group theory The classes of the equilateral triangle are {E} {C, C } {C, C ', C "} { h} {S, S - } { v, v, v } Jyväskylä 008
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
The point groups Schönfliess notation is used by most chemists The international notation lists the generators Jyväskylä 008
Jyväskylä 008 The point groups
The point groups Classification is done by using a flow diagram Jyväskylä 008
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
Character table C v E C v (xz) v (yx) A + + + + A + + - - B + - + - B + - - + C x y Jyväskylä 008
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 0 0 0 x y z # %! %"%!"! $ # % %"% $ Jyväskylä 008
Matrix representation Rotation through 0 C = 0 0 0 0 & ( ( ( ', i.e., C ) + + + * Rotations of the corners of an equilateral triangle ',../. - 0,../. - /= ' ' 0 0 0 0 0 0 0 /, - 0. /./. C Jyväskylä 008
@ What s common The common feature between the matrices is trace C = 0 0 0 0 4 6 6 6 5 7 9 9 9 8 0 0 : < </< ; 0 0 0 0 Sum = 0 Sum = 0 =??/? > Trace is called character ( theory ) in symmetry Jyväskylä 008
M N Isomorphy Point groups: C C =C Matrix groups: Rotation through 40 C = 0 0 0 0 G I I I H J L L L K Indeed, Jyväskylä 008 A C C C B 0 0 0 0 A D FC FC B FC E 0 0 0 0 A C C C B D F F E F= 0 0 0 0 D F F F E
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
S O V Y Q X T P Z W ` ` ` ` b^ b^ ^ a a a a 6 [ [R[ 6 b b b Example 6 6 0 XRX S = 0 Choose S = UURU therefore QRQ 0 \ ^ ^ ^ ] b bcb= 0 \ ] bcb ^c^ \ b_b ^c^ ] 6 6 \ ^_^ ] 6 6 0 0 0 0 0 0 0 0 0 0 S C S = Jyväskylä 008
Goal The simplest possible matrices: diagonal Jyväskylä 008
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
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.
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
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Example E = 0 0 l n n/n m 0 0 0 0 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 0 0 0 0 u w w/w v σ ' v = 0 0 x z z/z y 0 0 0 0 { } }/} 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
Œ Œ Exmple Water ƒ " 0 0 = 0 0 " S = 0 S ƒ 0 0 Let 0 Then Š Œ Œ Œ 0 0 0 0 0 0 ˆ Š Œ Œ Œ= 0 0 0 0 Š ˆ Œ Œ 0 0 0 0 0 0 Š ˆ Œ"Œ 0 0 = 0 0 " ˆ S C S Jyväskylä 008
Ÿ Example Summarising Water E = 0 0 / 0 0 0 0 / 0 0 C = 0 0 Ž 0 0 σ v = 0 0 / š 0 0 0 0 œ ž ž/ž σ ' v = 0 0 / 0 0 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 + - - +
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
Example C v ()E ()C () v (xz) () v (yx) A + + + + A + + - - B + - + - B + - - + + - + - = + + + = 0 4 a [ ( ) ( )] A a [ ( ) ( ) ( ) ( )] A a [ ( ) ( ) ( ) ( )] B a [ ( ) ( ) ( )] B = + + + = 4 = + + 4 = = + + 4 = Jyväskylä 008 a µ = g i χ χ µ g i i i