Symmetry and Group Theory

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1 4 Smmetr and Group Theor

2 4 Smmetr and Group Theor

3 4 Smmetr and Group Theor

4 4 Smmetr and Group Theor

5 Smmetr Operation and Smmetr Elements Smmetr Operation: A well-defined, non-translational moement of an object that produces a new orientation that is indistinguishable from the original object. Smmetr Element: A point, line or plane about which the smmetr operation is performed. line: smmetr element 8 o rotation: smmetr operation

6 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Proper rotation ais C n Mirror plane s Center of smmetr or center of inersion i Improper rotation ais S n

7 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Proper rotation ais C n : Rotation about C n ais b p/n n-fold smmetr ais. A C n ais generates n operations. * Principal rotational ais: highestfold rotational ais. If more than one principal aes eist, an one can be the principal ais. *: E : Identit *: Right-handed rotation

8 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Proper rotation ais C n : Rotation about C n ais b p/n n-fold smmetr ais. A C n ais generates n operations.

9 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Mirror plane s: Reflection about the s plane How man mirror planes?

10 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Mirror plane s: Reflection about the s plane *s h : mirror planes perpendicular to the principal ais. *s : mirror planes containing the principal ais Unless it is s d. *s d : mirror planes bisecting,, or ais or bisecting C aes perpendicular to the principal ais. How to define molecular aes,,?. The principal ais is the ais.. If there are more than one possible principal ais, then the one that connects the most atoms is the ais.. If the molecule is planar, then the ais is the principal ais in that plane. The ais is perpendicular to that plane. in Miessler 4. If a molecule is planar and the ais is perpendicular to that plane, then the ais is the one that connects the most number of atoms. Actuall,, 4 are arbitrar.

11 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr Center of smmetr or center of inersion i : Inersion of all objects through the center. i is Pt atom. i is a point in space.

12 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr 4 Improper rotation ais S n : Rotation about an ais b p/n followed b a reflection through a plane perpendicular to that ais or ice ersa. S 6

13 Four kinds of Smmetr Elements and Smmetr Operations Required in Specifing Molecular Smmetr 4 Improper rotation ais S n : Rotation about an ais b p/n followed b a reflection through a plane perpendicular to that ais or ice ersa. S n generates n operations for een n and n operations for odd n.

14

15 Point Groups

16 Point Groups

17 Point Groups

18 Point Groups

19 Point Groups

20 Point Groups

21 Point Groups

22 Point Groups

23 Properties and Representations of Groups Properties of a Group

24 Properties and Representations of Groups Matri Representations of Smmetr Operations : E E E : C C C : s s s : s s s

25 Properties and Representations of Groups Character Tables Schoenfliers Smbol of the Point Group Smmetr Operations R h = 4 Order of the Group = S# of Operations Smmetr Tpes of the Irreducible representations Irreducible Representations G Characters c Classes # of classes = # of smmetr tpes C p p [ ] p Character : Numerical representation of the operation Etent of originalit after the operation C [], s R R [ ] R, s [] C [], C, s d d [] [ ] d,, s C d d [] d [], s s s [] s, s d d [] d

26 Properties and Representations of Groups Character Tables h = 4 Characters c Character : Numerical representation of the operation Etent of originalit after the operation : E Tr E E E E c : C Tr C C c Tr s s c s : Tr s s c s : E Tr E E c G reducible representation = A +B +B

27 Properties and Representations of Groups Character Tables = A +B +B * The biggest possible alues of c is.

28 Properties and Representations of Groups Character Tables Great Orthogonalit Theorem h = 4

29 Co H N H N NH NH Br Cl E = C 4 = C = s = s d = A E = C 4 = C =- s = s d = E = C 4 =- C =- s =- s d =, are interconertable. degenerate - E 4 4 d d C C C C E s c s s c s c c c Properties and Representations of Groups Character Tables

30 s : E E E : cos sin sin cos cos sin sin cos cos sin sin cos C C C p p p p p p p p : s s s Properties and Representations of Groups Character Tables C

31 h = 6 E : : C block diagonalied, are degenerated. Properties and Representations of Groups Character Tables C s :

32 Properties and Representations of Groups Smmetr Labels Character Tables. Degeneracies Smmetr Labels A : smmetric about C n cc n = B : antismmetric about C n cc n = - E T. Subscript labels Meanings smmetric about C C n, cc = antismmetric about C C n, cc = -. Subscript labels when no C C n Meanings smmetric about s, cs = antismmetric about s, cs = - 4. Subscript labels Meanings g smmetric about i, ci = u antismmetric about i, ci = - 5. Superscript labels Meanings smmetric about s h, cs h = " antismmetric about s h, cs h = -

33 Applications of Smmetr Polarit Polar molecule: a molecule with a permanent electric dipole moment. A molecule with a center of inersion i cannot hae a permanent dipole. A molecule cannot hae a permanent dipole perpendicular to an mirror plane. A molecule cannot hae a permanent dipole perpendicular to an ais of smmetr. Therefore, molecules haing both a C n ais and a perpendicular C ais or σ h cannot hae a dipole in an direction. Molecules belonging to an C nh, D, T, O or I groups cannot hae permanent dipole moment.

34 Applications of Smmetr Chiralit A chiral molecule is a molecule that is distinguished from its mirror image in the same wa that left and right hands are distinguishable. A molecule that has no ais of improper rotation S n is chiral. S n : including S = σ and S = i

35 Applications of Smmetr Molecular Spectroscopies S Intersstem Crossing T Virtual state S UV/VIS Fluorescence IR Raman Stokes anti- Stokes Phosphorescence

36 Applications of Smmetr Molecular Vibrations S Virtual state S IR Raman IR-actie: when there is a change in dipole moment in a molecule as it ibrates Raman-actie: when there is a change in polariabilit a molecule as it ibrates Stokes anti- Stokes

37 Applications of Smmetr Molecular Vibrations H O G 9 Little Orthogonalit Theorem G R i i R R R g h n c c ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 ] 9 [ 4 B B A A n n n n B B A A G B B A B B A rot trans G G B A ib G Water

38 Applications of Smmetr Molecular Vibrations Water H O Both IR and Raman actie G 9 IR actie Raman actie G ib A B

39 Applications of Smmetr Molecular Vibrations Selected Vibrational Modes G nco G nco = A + B A : B : G nco G nco = A g + B u A g : B u : IR, Raman actie Raman actie IR actie

40 Applications of Smmetr Molecular Vibrations Selected Vibrational Modes Eclusion Rule : In a molecule with i smmetr element, IR-actie and Raman -actie ibrational modes are not coincident.