Chapter 1

Symmetry operationsSymmetry elements

Multiplication of symmetry elements & operations

- commutative & non-commutativePoint groups (Schoenflies symbols)

TKT2023 – noorshida mohd ali

SYMMETRY OPERATIONS

An operation perform on an object which leaves it in a configuration

that is indistinguishable from and superimposable on the original configuration

SYMMETRY ELEMENTS

A symmetry operation is carried out with respect to LINES, POINTS or PLANES


rotate by 180°

H1

OH2 H2

OH1

Axis of symmetry - Cn

about axis C1,C2,…Cn..C

Cn 360/n Cn

n n(360/n) = E = C1

Clockwise rotation

A molecule having a Cn axis can be rotated by 360°/n about the axis and the configuration will remain unchanged

e.g: H2S, SO2 , H2O2

Axis lying in the plane of the paper

C2


F3

BF2

F1

e.g: BCl3, SO3, AlCl3, [NO3]-, NH3, PBr3, [SO4]2-

Axis perpendicular to the plane of the paper (passing through the central atom)

FB

F

F

C3

F2

BF1

F3

C3 rotate by 120°


Axis passing through the central atom in the molecule [ICl4]-,

e.g: XeF4, [PtCl4]2-

C4

rotate by 90°

C4

ClI

Cl

Cl

Cl

12

34

ClI

Cl

Cl

Cl

41

23

ClI

Cl

Cl

Cl

12

34


C

CC

CC

H1

H5

H4

H3

H2

C

CC

C

CC

H6

H5

H4

H3

H2

H1

Axis is perpendicular to the plane of the paper & passing through the centre of the

ring of carbon atoms

C5H5- C6H6 C7H7

1

C

C

CC

C

CC

H1

H2

H3

H4

H5

H6

H7

C8H82-

C

C

C C

C

C

CC

H1 H2

H3

H4

H5H6

H7

H8 C5


e.g : linear molecule

(Symmetrical)

C

N NH H C H C N C

(Asymmetrical)


FB

F

F

C3

C32 = C3

-1

Cnm m(360/n)

C31 = C3

-2

C2

+m = clockwise rotation-m = anti clockwise rotation

Cnm


ClI

Cl

Cl

Cl C2

C42 = C4

-2 = C2

C41 = C4

-3

C43 = C4

-1

C4


F

Xe

FF

Fv

v

h

d

d

C4

C2

C2

C2

C2

Vertical reflection plane

•Passing through the origin & the axis with the highest

symmetry

Horizontal reflection plane

•Passing through the origin

•Perpendicular to the axis with the highest symmetry

Dihedral / diagonal reflection plane

•Passing through the origin & the axis with the highest symmetry

•Bisecting the angle between the two-fold axes perpendicular to the symmetry axis

A molecule having a plane symmetry is unchanged in configuration if all the atoms are reflected across the plane

Plane of symmetry - TKT2023 – noorshida mohd ali

O

HH

C2

v

v’

H1

OH2

C2v

v’

H1

OH2

C2

v’

O

HH

C2

v’

The plane in which the molecule lies

The plane bisects the molecule


Centre of symmetry - iIf the molecule has a centre of symmetry i, reflection of the position of the all atoms through this centre that

changing all their positions from (x, y, z) to (-x, -y, -z) will leave the configuration of the molecule unchanged

i = S2e.g : 1)

centrosymmetric moleculei

2) Staggered conformation

CHClBr-CHClBr

3) Br

H

Cl

Br

H

Cl

i

C C

F

H

H

Fi

C6


XeF

F F

F

SF

F F

F

F

F

C C

F

F

F

Fi

4) 5) i

6)

i


Fe

Fe

C

CH3

NN

C

CH3

N

NO

O

H

O

O

H

O

O

O

O

H

H

centrosymmetric molecule


Rotation-reflection axis of symmetry - Sn

C

C

H

H

C2

C

C

H

H

C

C

H

H

clockwise rotation by 360°/n about the axis reflection by across a plane perpendicular to axis and passing through the centre of the molecule

Two-fold rotation-reflection axis S2

i


FB

F

F

C3

FB

F

F

FB

F

F

Three-fold rotation-reflection axis S3


HC

H

H HHC

H

HH

C4

HC

H

HH

Four-fold rotation-reflection axis S4


ferosena Fe(C5H5)2 – staggered conformation

FeFe

C10

i.e

Ten-fold rotation-reflection axis S10


Identity element of symmetry – E

All molecules possess the identity element of symmetry, E that is rotation of the molecule

through 0° leaves the configuration unchanged

E = C1


SYMMETRY ELEMENTS SYMMETRY OPERATIONSProper rotation :Rotation axis (Cn)

Clockwise rotation about axis C1,C2,…Cn

Cn 360/n

Cnm m(360/n)

Cnn n(360/n)

Reflection plane () Reflection in plane (v, h & d)

Centre of symmetry (i) Inversion at centre (i)Improper rotation :Rotation & reflection axis (Sn)

Clockwise rotation about axis & reflection in a plane perpendicular to the axis S1,S2,…Sn

Identity (E) or (I)


MULTIPLICATION OF SYMMETRY ELEMENTS AND OPERATIONS

Symmetry operation A and BThis multiple operation is written B x A

(Carry out operation A first and then operation B)

there are four symmetry operations which are : A, B, C and DA x B = C x D

then the right and left operations are said to commute or to be commutative

But if A x B C x D then the right and left operations are said todo not commute or are non-commutative


e.g: NH3 commutative and non-commutative ?

v

NHH

H2

NHH

H1

2 3C3

NHH

H1

2

3

C3

2

1 3

h

S3

NHH

H1

2

3

C3

C3 x v(H(2)-N) S3 x h

non-commutative

3

1

C3 x v(H(2)-N) = S3 x h

NHH

H


e.g: PF5 C2(F(3)-P) x v(F(2)-P) = C3 x hcommutative and non-commutative ?

F PF

F

FF

12

3

4

5F PF

F

FF1

23

4

5

v h

F PF

F

FF

2

4

5

31

4

F PF

F

FF

1

23

5

C2

C3

C2

F PF

F

FF

1

2

3

5

4

C3

C2(F(3)-P) x v(F(2)-P) = C3 x h

commutative


The complete set of symmetry

elements in any particular molecule

Schoenflies symbols C1, Cs, Ci,

Cn, Cnv, Cnh, Cv,Sn,

Dn, Dnd, Dnh, Dh, Td & T, Oh & O, Ih

POINT GROUPS

Groups with low symmetry (C1, Cs, Ci)

Groups with an n-fold rotational axis (Cn, Cnv, Cnh, Cv)

Dihedral groups (Dn, Dnd, Dnh, Dh)

Groups with very high symmetry (Td, T, Oh, O, Ih)


Point groups Examples

1. One symmetry element : E = I = C1

all molecules possess C1

Groups with low symmetry (C1, Cs, Ci)

C1 = E = I


2. Contain a plane of symmetryCs

Cs=C1v=C1h

O

FBr

O

bromofluoromethanediol



3. Contain a centre of inversion iCi

i i.e

staggered conformation CHClBr-CHClBr

BrCl

Br

H

Cl H

Br

Cl HBr

ClH

S2 = i


Groups with an n-fold rotational axis (Cn, Cnv, Cnh, Cv)


1.Molecules containing only one axis e.g: molecule gauche H2O2 ( axis C2 lies in

the plane of paper)

CnC2

O O

HH

C2

hydrogen peroxide

111°



- axis perpendicular to the paper (not

planar)C3

P

triphenylphosphine

C3


2.

A plane perpendicular to the Cn axis

Cnh C1h = C1v = Cs

rarer systemh


Point groups

Examples

Contains one C2 axis of symmetry A plane perpendicular to the C2 axis If n is even the point group necessarily contains a centre of symmetry (i)

C2h

h

C C

Cl

H

H

ClF

NN

Fi i

trans-1,2-dichloroethenetrans-N2F2

h

hC2

C2


Point groups Examples C3h

Orthoboric acid : B(OH)3

BO

O

O

H

H

H

C3

h


Point groups

Examples

3. Contains one C2 axis of symmetryn plane that containing the C2 axis

Cnvv

C2v

v’

O

HH

v

C2

dihydrogen oxide

Cl Cl

H

HH

H

dichlorobenzene

Cl

Cl

H

H

H

H

v’

v’v

v

C2

C2



C3v

ammoniaN

HHH

C3

v

vv

e.g : all the trigonal pyramidal in shape

- PCl3, phosphine (PH3), arsine (AsH3)

C4v

IF

FF

F

F

C4

v

vd

d

IF5

e.g : all the trigonal bipyramidal in shape

- BrF5, XeOF4

C5v & C6v

rarer system



4. ∞ (infinite) plane of symmetry Contain one axis All linear molecule (non symmetry / asymmetry) – without a centre of symmetry (i) e.g: HCl, O=C=S

Cvv

C∞

H C N C I Cl C

vv


Dihedral groups (Dn, Dnd, Dnh, Dh)

Point groups Examples 1.

2.

No mirror plane

Molecules possessing nC2 axes perpendicular to the principal axes Cn

Cn

1 other n if n = even number : (i)

DnD3

3+

C3

Dnh

n C2 Cnh

tris(ethylenediamine)cobalt(III) cationCo

N N

N

N

N N



-

- C2

- 1

- 2 - if n = 2 : (i)

D2h

v

h

C2

C2

C2

i

i

C2

C2

C2

naphtalene

ethylene

2 C2 C2

C CHH

HH

3



- C3

- 1

- 3

D3h

v

h

3 C2 C3

C3

h

carbonate ion CO32-

C2v

C2v

C2

v

CO

O

O



- C3

- 1

- 3

D3h 3 C2 C3

v

h

F PF

F

FF

C3

h

v

C2

C2

vC2

v

phosphorus pentafluoride



- if n = 4 : (i) - C4

- 1

- 2 and 2

D4h 4 C2 C4

v

h

C4

h

v

C2

C2

v

C2

d

C2dtrans-tetraaminedichlorocobalt(III) cation

+

Co

Cl

Cl

NH3

H3N NH3

H3N

Co

Cl

Cl

NH3

H3N NH3

H3N i

d



3. Molecules possessing nC2 axes perpendicular to the principal axes Cn

Cn

S2n

n d plane if n = odd number : (i)

S4

C2

2 d

Dnd

n C2 Cn

alleneD2d 2 C2 C2

C C C

H

H

HH

C2

d

d

C2C2



C3

S6

3 d if n = 3 : (i)

D3d 3 C2 C3

H

C H

H

H

H

H

C

C3

d

C2

C2

C2

d

d

i

staggered ethane



linear molecule with centre of symmetry (i) (symmetrical) 1 e.g: O2, H2, Cl2

Dh

C2 C

Be FF

i

i C

h

v

h

C2

C C HH C

C2h v

v


Groups with very high symmetry (Td, T, Oh, O, Ih)(Cubic point group)

Tetrahedral Td : 4C3, 3C2, 6d, 3S4

T : 4C3, 3C2 (only containing rotation of Td)

e.g: - CH4

- CCl4

Icosahedral Ih : 6C5, 10C3, 15C2, 15, 6S10, 10S6, i =S2

e.g: - [B12H12]2-

Octahedral Oh : 4C3, 3C4, 6C2, 3h, 6d, 4S6, i =S2, 3S4

O : 4C3, 3C4, 6C2

(only containing rotation of Oh)

e.g: - SF6

- [Fe(CN)6]3-

- [Co(CN)6]3-

hexacyanocobaltate(III) anion


Tetrahedral Td : 4C3, 3C2, 6d, 3S4

T : 4C3, 3C2 (only containing rotation of Td)

C

C

C

C

C3

C3

CHH

H

H

e.g: - CH4

Top view:Side view:


C2

H

H

H

H

C

C2

HC

H

HH

C2

C2

C2

H

H

H

H

C

C2H

H

H

H

C

H

C2

H

H H

C

H H

H H

C C2

H H

H H

C

C2

Top view:Side view:

Position of H atoms in the cubic

above

bottom

centre


H

H

H

H

C

d

d d

d

H

H

H

H

C

dd

dH

H

H

H

C

H

H

H

H

C

C2

H

H

H

H

CH

H

H

H

C C2

C2

Side view:TKT2023 – noorshida mohd ali

HC

H

HH

S4

HC

H

HH

C4

HC

H

H H

Rotate 90º

HC

H

HH

Reflection h

Side view:


Octahedral Oh : 4C3, 3C4, 6C2, 3h, 6d, 4S6, i=S2, 3S4

O : 4C3, 3C4, 6C2 (only containing rotation of Oh)

e.g: - SF6

S

F

FF

F

F

F

SF

F F

F

F

F

FF

F

F

F

F

S

i = S2

Side view:


S

F

F F

F F

F

F

F FS

F

FF

C3 , S6

Top view:

FF

F

F

F

F

S

C4

C4

C4

FF

F

F

F

F

S

h

h

h


FF

F

F

F

F

S

d

d

C2

C2h

FF

F

F

F

F

S

d

d

C2

C2h

C2

FF

F

F

F

F

S

d

h

d

C2


Icosahedral Ih : 6C5, 10C3, 15C2, 15, 6S10, 10S6, i=S2

e.g: - [B12H12]2-

C5

C2C3


Chapter 1

Documents

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h wv c3 h h n h1c3 h3

w tkt2023 noorshida

axis reflection

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