Lecture 34: Symmetry Elements The material in this lecture covers the following in Atkins. 15 Molecular Symmetry The symmetry elements of objects 15.1 Operations and symmetry elements 15.2 Symmetry classification of molecules (a) The groups C1,Ci, and Cs (b) The groups Cn,Cnv, and Cnh

(c) The groups Dn,Dnh, and Dnd Lecture on-line Symmetry Elements (PowerPoint) Symmetry Elements (PDF)Handout for this lecture

Some of the symmetry elements of a cube.The twofold, threefold, and fourfold axesare labelled with the conventional symbols.

Symmetry elements The systematic discussion of symmetry is calledSome objects are more symmetrical than others

Any rotation of sphere around axis throughcenter brings sphereover into itself Only some rotations of sphere

brings cube into itself

(b) an H2Omolecule has a twofold (C2) axis. Both have other symmetry elements too.

Symmetry elements

(a) An NH3 molecule has a threefold (C3) axis

NH3 has more rotation symmetry than H 2O

Symmetry elements An action that leaves an object the same after it has been carried out is called a :

Symmetry operations are :

For each symmetry operation there is a : symmetry element Which is

with respect to which the operation is performed

Operators and Symmetry elements The classification of objects according tosymmetry elements corrsponding to operations that leave at least one common point unchanged gives rise to the :

There are five kinds of symmetry operations and corresponding symmetry elements

I

F

ClBr

1. The identity E Consists of doing nothing

Only element for CFClBrI

Operators and Symmetry elements

(b) an H2Omolecule has a twofold (C2) axis. C3 and C3

-1

C6

C6 ; C3 ; C2 and C6

-1; C3-1

C5The principle rotation axisis the axis of the higest fold

Operators and Symmetry elements

The H2O molecule has two mirrorplanes. They are both vertical(that is containthe vertical axis)and as so are denotedσv and σv'

Benzene has one mirror plane perpendicularto the principleC6 -axis (σh)

Operators and Symmetry elements

C2C2

C2

C6

Dihedral mirror planes (σd) bisect the C2 axis perpendicular to theprinciple axis

A regular octahedron has a centre of inversion (i).

Operators and Symmetry elements

(a). A CH4 molecule has a fourfold improper rotation axis (S4): the molecule is indistinguishable after a 90° rotation followed by a reflection across the horizontal plane, but neither operation alone is a symmetry operation.

Operators and Symmetry elements

C4

σ

(b) The staggered form of ethane has an S6 axis composed of a 60° rotation followed bya reflection.

Operators and Symmetry elements

The Symmetry Classification of Molecules

The Schoenflies system

Hermann−Mauguin

The Symmetry Classification of Molecules

The groups C1 , Ci , Cs

A molecule belongs to C1 if ithas only the identitity E I

F

ClBr

A molecule belongs to Ci if it has only the identitity E and i

HO HCOOH

OHHOOC

H

Meso-tartaricacid

N

Quinoline

A molecule belongs to Cs if ithas only the identity and a mirror plane

C1 , Ci , Cs

The Symmetry Classification of Molecules

The groups Cn, Cnv, and C nhA molecule belongs to Cn if ithas a Cn axis and the identitity E only

O O

H H

C2

H2O2

C2 C3C4 C5 C6

Cn

The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh

A molecule belongs to Cnv if ithas a Cn axis and n vertical mirror planes

C2v C3vC4v C5v C6v C∞v

cone

CO Heteronuclear diatomics areC∞v

Cnv

H2O C2vC3v

NH3

The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh

A molecule belongs to Cnh if ithas a Cn axis and a horizontal mirror plane

C2h C3h C4h C5h C6h

C

C

Cl H

H Cl

Trans CHCl=CHCl

B

O

OO

H

H

H

B(OH)3

Cnh

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

The Symmetry Classification of Molecules The groups Cn, Cnv, and C nh

Cnh

C2

σ

i

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh

A molecule belongs to D n if ithas a Cn axis and n two fold axes (C2) perpendicular to Cn.

A molecule with n twofold rotation axes perpendicular to an n-fold rotation axis belongs to the group Dn.

Dn

Pt

NN

N

NNN

D3

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh

D2 D3D4 D5 D6

Dn

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nhA molecule belongs to Dnh if ithas a Cn axis and n two fold axes (C2) perpendicular to Cn as well asa horizontal mirror plane.

A molecule with a Mirror plane perpendicular to a Cn axis, and with n twofold axes in the plane, belongsto the group Dnh.

Dnh

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nh

A molecule belongs to D nh if ithas a Cn axis and n two fold axes (C2) perpendicular to Cn as well asa horizontal mirror plane.

Dnh

D2h D3h

D4h D5h D6h

C

C

H H

H H

C4

C2

C2

σh

D2h

Cl

Cl ClAu

Cl

C4

C2

C2

C'2

C2

σh

D4h

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nhA molecule belongs to Dnh if ithas a Cn axis and n two fold axes (C2) perpendicular to Cn as well asa horizontal mirror plane.

E , C6, 3C2 ,3C'2, σh

B

F

FFE , C3, 3C2 , σh

Dnh

The Symmetry Classification of Molecules Dnh

F PF

F

F

F

C3

C2

C2

C2

σh

The groups Dn, Dnv, and D nh

D3h

The Symmetry Classification of Molecules The groups Dn, Dnv, and D nhA molecule belongs to Dnd if in addition to the elements of Dn it posesses n dihedral mirror planes

Dnd

D2d D3d D4d D5d D6d

C C C C2

C2

C2

D2d

What you must learn from this lecture

1. Given a molecule, you should be able to identify the differentsymmetry elements (Cn,σv,σh,σd,i,Sn, etc. )

2. Having identified the symmetry elements of a molecule,you should be able to establish the point group to whichthe molecule belong