Chapter 3. Molecular symmetry and symmetry point group

Transcription

1 hapter Molecular symmetry and symmetry point group

2 Why do we study the symmetry concept? The molecular configuration can be expressed more simply and distinctly. The determination of molecular configuration is greatly simplified. It assists giving a better understanding of the properties of molecules. To direct chemical syntheses; the compatibility in symmetry is a factor to be considered in the formation and reconstruction of chemical bonds.

3 Symmetry elements and symmetry operations Symmetry exists all around us and many people see it as being a thing of beauty. A symmetrical object contains within itself some parts which are equivalent to one another. The systematic discussion of symmetry is called : Some objects are more symmetrical than others.

4 . Symmetry elements and symmetry operations symmetry operation A action that leaves an object the same after it has been carried out is called symmetry operation. Example: Any rotation of sphere around axis through center brings sphere over into itself

5 Example: (a) An NH molecule has a threefold ( ) axis (b) an H O molecule has a twofold ( ) axis.

6 symmetry elements Symmetry operations are carried out with respect to points, lines, or planes called symmetry elements. Example: (a) An NH molecule has a threefold ( ) axis (b) an H O molecule has a twofold ( ) axis. NH has higher rotation symmetry than H O

7 Symmetry elements Some of the symmetry elements of a cube, the twofold, threefold, and fourfold axes.

8 Symmetry operations are: The corresponding symmetry elements are:

9 ) The identity (E) Operation by the identity operator leaves the molecule unchanged. All objects can be operated upon by the identity operation. F l I Br

10 ) Inversion and the inversion center (i) An object has a center of inversion, i, if it can be reflected through a center to produce an indistinguishable configuration. A regular octahedron has a centre of inversion (i).

11 For example These have a center of inversion i. These do not have a center of inversion.

12 z y x z y x z y x i Its matrix representation Inverts all atoms through the centre of the object

13 ) Rotation and the n-fold rotation axis ( n ) Rotation about an n-fold axis (rotation through 6 o /n) is denoted by the symbol n. Example: Rotation of trigonal planer BF. One three-fold ( ) rotation axes. (=/) The principle rotation axis is the axis of the highest fold.

14 The matrix representations: + (x,y) (x,y ) x y x =-rsin(+)= -rsincos -rcossin = (-/)x + (-/)y y =rcos(+)= rcoscos - rsinsin = (/)x + (-/)y z y x y x z y x z y x cos sin sin cos z y x z y x z y x y x z y x z y x cos sin sin cos z y x z y x

15 The matrix representations: onditions: The centre of mass of the molecule is located at the origin of the artesian oordinate System Principle axis is aligned with the z-axis (-x,-y) x x cos sin x x x y y sin cos y y y z z z z z y (x,y) n x k n cos sin sin cos k n

16 For example The principle rotation axis is the axis of the highest fold. 6 5

17 If reflection of an object through a plane produces an indistinguishable configuration then that plane is a plane of symmetry (mirror plane) denoted. 4) Reflection and the Mirror plane () xy z y x z y x z y x xy (x, y,-z) (x,y,z) z y x

18 There are three types of mirror planes: If the plane is perpendicular to the vertical principle axis then it labeled h. If the plane contains the principle axis then it is labeled v. If a plane contains the principle axis and bisects the angle between two adjacent -fold axes then it is labeled d.

19 If the plane is perpendicular to the vertical principle axis then it labeled h. Example: BF also has a h plane of symmetry.

20 If the plane contains the principle axis then it is labeled v. Example: Water Has a principle axis. Has two planes that contain the principle axis, v and v. H O H v v

21 If a plane contains the principle axis and bisects the angle between two adjacent -fold axes then it is labeled d.(dihedral mirror planes ) Example: BF Has a principle axis Has three- axes. Has three d planes (?). v

22 Example: 6H6 6 Benzene has one mirror plane perpendicular to the principle 6 axis ( h) Dihedral mirror planes ( d) bisect the axis perpendicular to the principle axis. d v

23 Example: H ==H d d

24 5) The improper rotation axis a. n-fold rotation + reflection, Rotary-reflection axis (S n ) Rotate 6 /n followed by reflection in mirror plane perpendicular to axis of rotation S 4

25 Example: H -H S 6 h S 6 The staggered form of ethane has an S 6 axis composed of a 6 rotation followed by a reflection.

26 Special ases: S and S S h h S h i

27 Stereographic Projections o x x o o We will use stereographic projections to plot the perpendicular to a general face and its symmetry equivalents, to display crystal morphology o for upper hemisphere; x for lower

28 E S S S S S S ; ; ; ; ; S h h S

29 E S S S S ; ; ; E S S S S S S S S S S ; ; ; ; ; ; ; ; ; x S 4 x S4 S S h h h S S h

30 I i I E I i I I i I I i I i I i I i i I i I h n n, x x b. n-fold rotation + inversion, Rotary-inversion axis(i n ) Rotation of n followed by inversion through the center of the axis

31 Summary Element Name Operation n n-fold rotation Rotate by 6 /n Mirror plane Reflection through a plane i S n enter of inversion Improper rotation axis Inversion through the center E identity Do nothing Rotation as n followed by reflection in perpendicular mirror plane

32 . ombination rules of symmetry elements A. ombination of two axes of symmetry The combination of two axes intersecting at angle of /n, will create a n axis at the point of intersection which is perpendicular to both the axes and there are n axes in the plane perpendicular to the axis. n + ( ) n ( )

33 B. ombination of two planes of symmetry. If two mirrors planes intersect at an angle of /n, there will be a n axis of order n on the line of intersection. Similarly, the combination of an axis n with a mirror plane parallel to and passing through the axis will produce n mirror planes intersecting at angles of /n. n + v n v v v v v Ex. H O, NH v ' v v

34 . ombination of an even-order rotation axis with a mirror plane perpendicular to it. ombination of an even-order rotation axis with a mirror plane perpendicular to it will generate a centre of symmetry at the point intersection. Each of the three operations xy, n and i is the product of the other two operations xy xy (x,y,z) (-x,-y,-z) (-x,-y,z) y x i h m m h

35 Groups and group multiplications. Definition: A mathematical group, G = {G,}, consists of a set of elements G = {E, A,B,,D,...} (a) losure. The product of any two elements A and B in the group is another element in the group. (b) Identity operation. The set includes the identity operation E such that AE=EA=A for all the operations in the set. (c) Associative rule. If A, B, are any three elements in the group then (AB) = A(B). (d) Inversion. For every element A in G, there is a unique element X in G, such that XA = AX = E. The element X is referred as the inverse of A and is denoted A -.

36 '' ',,,, E, E ) ( ) ( E symmetry elements: losure. Identity operation. Associative rule. Inversion. E Example: NH Therefore, these symmetry elements constitute a group, V

37 Example: G = {E,,, v (), v (), v () } NH : V v () v () v () v () + v () v () + v () v () v () + v () v () v v v v

38 . Group Multiplication Example: H O z x y v xz yz Its total symmetry elements: E,, xz yz

39 . Group Multiplication Example: H O Multiplication table of v v E xz yz E E xz yz E yz xz xz xz yz E yz yz E xz yz xz

40 Multiplication table of v v E xz yz E E xz yz E yz xz xz xz yz E yz yz xz E (). In each row and each column, each operation appears once and only once. () We can identify smaller groups within the larger one. For example, {E, } is a group. () The group order is the total number of the group

41 Example: NH v Its total symmetry elements:e,,, v, v, v Multiplication table of v v E v v v E v v v

42 Group Multiplication v E v v v E E v v v E E v v v v v v - = E

43 Group Multiplication v E v v v E E v v v E v v v E v v v v v v v v v v v v v v v v = v

44 Group Multiplication v E v v v E E v v v E v v v E v v v v v v v E v v v v E v v v v E v B A v v =

45 Multiplication table of v v E v v v E E v v v E v v v E v v v v v v v E v v v v E v v v v E

46 Point groups, the symmetry classification of molecules Point group: All symmetry elements corresponding to operations have at least one common point unchanged.

47 . The groups, i, and s The group A molecule belongs to the group if it has no element of symmetry other than the identity. Example: BrlF F l I Br

48 The group i It belongs to i if it has the identity and inversion alone. Example: meso-tartaric acid, HlBr-HlBr

49 The group s It belongs to s if it has the identity and a mirror plane alone. N S l 4 O

50 A molecule belongs to if it has only the identity E. l I F Br OOH HO H A molecule belongs to i if it has only the identity E and i. A molecule belongs to s if it has only the identity E and a mirror plane. H HOO OH Meso-tartaric acid N Quinoline HlBr-HlBr H =lbr

51 . The groups n, nv, nh and S n The group n A molecule belongs to the group n if it possess an only n-fold axes. Example: H O

52 H O O H H l H O 6 H (H ) H l

53 The group nv If in addition to a n axis it also has n vertical mirror planes v, then it it belongs to the nv group. n, n v =O v v v

54 v v 6 H N H 4 O P 4 S v 4 H

55 The group nh Objects having a n axis and a horizontal mirror plane belong to nh. n, h trans-hl=hl

56 h H l I 7 - H 6 l

57 i The presence of a twofold axis and a horizontal mirror plane jointly imply the presence of a centre of inversion in the molecule.

58 h h 4h 5h 6h l H O H H l Trans Hl=Hl h h H O B O H B(OH)

59 The group S n Objects having a S n improper rotation axis belong to S n. Group S = i Group S = s S 4

60 S same as i OOH HO H S 4 S 6 S 8 S 4 : S 4,,S 4,,i,S6 5,S6 S ; 4 ;S 8, S 8 5, 4,S8 7 H HOO OH Meso-tartaric acid S 4 S 4 Single-axis group

61 . The group D n, D nh, D nd The group D n A molecule that has an n-fold principle axis and n twofold axes perpendicular to n belongs to D n. n, n

62 D D D 4 D5 D6 o(dien) H l (mediate state) H 6

63 The groups D nh A molecule with a Mirror plane perpendicular to a n axis, and with n two fold axes in the plane, belongs to the group D nh. D n, h H H H 4 H h D h l 4 l h Au D 4h l l '

64 D nh F F P F F h F D h

65 D h Dnh H 4 H SiF 4 ( 5 H 5 N) D h BF Pl 5 Tc 6 l 6

66 D 4h D nh [Ni(N) 4 ] - [M (OOR) 4 X ] Re l 8 6 H 6 D r( 6h 6 H 5 ) D h O==O

67 The group D nd A molecule that has an n-fold principle axis and n twofold axes perpendicular to n belongs to D nd if it posses n dihedral mirror planes. D n, n d The order of group=4n D d

68 D d D nd N 4 S 4 Pt 4 (OOR) 8

69 D d Til 6 - TaF 8 - D 4d S 8 D 5 d

70 4. High order point groups Molecules having three or more high symmetry elements may belong to one of the following: T: 4, (T h : + h ) (T d : +S 4 ) O: 4, 4 (O h : + h ) I: 6 5, (I h : +i)

71 T d Species with tetrahedral symmetry O h Species with octahedral symmetry (many metal complexes) tetrahedral symmetry group octahedral symmetry group Icosahedral symmetry group I h Icosahedral symmetry (Buckminsterful lerene, 6 )

72 ubic groups 4, T: 4, (T h : + h ) (T d : +S 4 ) Shapes corresponding to the point groups (a) T. The presence of the windmill-like structures reduces the symmetry of the object from Td.

73 ubic groups T h {E,4,4,,I,4S 6,4S 65,σ h }

74 ubic groups S 4 d T d {E,,8,6S 4,6σ d }

75 T T h T d (H ) 4 Ti 8 + H 4 T d o 4 (O) P 4 O 6

76 ubic groups O 4 O: 4, 4 (O h : + h ) Shapes corresponding to the point groups (b) O. The presence of the windmill-like structures reduces the symmetry of the object from O h.

77 ubic groups d S 6 O h S 4 4 F F F S F F F

78 ubic groups SF 6 8 H 8 OsF 8 O h Rh

79 I group 5 B H (with hydrogen omitted) I: 6 5, (I h : +i) H

80 I h {E, 5, 5,,5,i,S,S,S 6,5σ} 6, the bird-view from the 5-fold axis and 6-fold axis

81 Y molecule Linear? N Y D h i? N v I h? Two or more n? (n>) Y Y Y 5? Y 4? O h? Y N Y N h? N T Y d? D h? Y Y n? Is there n perpendicular to n? N Y Y d? N N Y,i? Y N? i? N h? N Y v? nh N s i I h O h T h T d D nh D nd D n nv n

82 4 Application of symmetry H H. hirality Br F l F l Br A chiral molecule is a molecule that can not be superimposed on its mirror image These molecules are: cannot be superimposed on its mirror image. a pair of enantiomers (left- and right-handed isomers) does not possess an axis of improper rotation, S n Ability to rotate the plane of polarized light (Optical activity ) S n (i=s ; )

83 Optical activity is the ability of a chiral molecule to rotate the plane of plane-polarized light. polarizer sample planepolarized dextrorotatory (d) or (+) levorotatory (l) or ( ) optically inactive optically active

84 Optical activity Optically inactive: achiral molecule or racemic mixture - 5/5 mixture of two enantiomers Optically pure: % of one enantiomer Optical purity (enantiomeric excess) = percent of one enantiomer percent of the other e.g., 8% one enantiomer and % of the other = 6% e.e. or optical purity

85 A chiral molecule does not possess S n (i, ) n and D n may be chiral (no S n improper axis)

86 . Polarity, Dipole Moments and molecular symmetry A polar molecules is one with a permanent electric dipole moment. Dipole Moments are due to differences in electronegativity depend on the amount of charge and distance of separation in debyes (D), = 4.8 (electron charge) d (angstroms) For one proton and one electron separated by pm, the dipole moment would be: d 9 D (.6 )( m) 4. 8D.4 m

87 Bond Dipole Moments

88 Molecular Dipole Moments Depend on bond polarity and bond angles Vector sum of the bond dipole moments Symmetric molecules may have zero net dipole --- O : O==O Lone pairs of electrons contribute to the dipole moment 8/*/8/*/

89 Molecular Dipole Moments 8/*/8/*/

90 Molecular Dipole Moments

91 Permanent Dipole Moments HO OOH H (a) A permanent dipole moment can not exist if inversion center is present. Only molecules belonging to the groups n, nv and s may have an electric dipole moment H HOO OH Meso-tartaric acid inversion (b) Dipole moment cannot be perpendicular to any mirror plane or n. ( h ) H x OH O H = x OH

92 Molecular Dipole Moments and molecular symmetry O H H l O I F O H Br N Quinoline in plane H O Trans Hl=Hl along along along inversion l H H O H B l H O B(OH) h symmetry