SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS

Department of Chemistry

Central university of rajasthan

Presentation on

SYMMETRY ELEMENTS AND SYMMETRY OPERATIONSSubmitted to :-

DR. Malli bhanuchandraAstt. ProfessorDepartment of chemistry

Submitted by :-Roopendra singh madhukarInt. M.sc. B.ed. Chemistry2015imsbch023

GROUP THEORY :-

Fundamentals of group theory are developed by Evariste Galois.

It is the study of symmetry. It is purely mathematics concept which has wide applications

in physical sciences.

When applied to Chemistry, it can be used, for example, to……. to predict whether or not a molecule has a dipole moment to predict if a molecule will show optical activity To derive selection rules for spectroscopic transitions to determine which AOs to be used to construct hybrid

orbitals. to predict which molecular vibrations lead to IR spectra. to label and designate MOs etc.

What is symmetry ?

Symmetry is when a shape looks identical to its original shape after being flipped or turned.

Nature loves symmetry Most objects found in nature have symmetry Symmetry is associated with beauty

e.g. Flowers, diamonds, butterflies, snail shells,leaves, etc are all beautiful, highly symmetrical because of harmony and attractiveness of their forms and proportions.

Symmetry in nature :-

Symmetry in the human body :-

A flower, crystal or a molecule, is said to have symmetry if it has two ormore orientations in the space that are indistinguishable. The criteria forJudging these are based on symmetry elements and symmetry operations.

What is symmetry element and symmetry operation ?A symmetry element is a geometrical entity

such as a line, a plane, or a point about which one can perform an operation of rotation, reflection, or inversion.

A symmetry operation is movement of a molecule/object about an symmetry element such that resulting configuration is indistinguishable from the original.

A symmetry operation will transform a molecule into an equivalent or identical configuration.

for example:- H2O molecule is rotated about an axis through oxygen atom and bisecting H-O-H bond angle, through 180.

0

HO

H

water

C2 a b HO

H

waterb a H

OH

watera b

I II III

The configurations I, II and III are indistinguishable, therefore this operation is a symmetry operation.

The symmetry element is the imaginary line (axis).

The symmetry operation is the rotation of a molecule about this axis through 180.

I and II are equivalent. II and III are equivalent.But I and III are identical.

0

Symmetry elements and symmetry operations :-

Symmetry Elements Symmetry Operations

1. Identity [E] Doing nothing

2. Proper Rotation axis or Axis of Symmetry [Cn]

Rotation about the axis through some angle

3. Mirror Plane or Plane of Symmetry []

Reflection about the plane

4. Inversion Centre or Centre of Symmetry [ i ]

Inversion { inversion is a reflection about a point}

5. Improper Rotation axis or Rotation- Reflection axis

[Sn]

Rotation about an axis through some angle followed by a reflection in a plane perpendicular to the rotation axis

1. Identity [E] :-This is an operation which brings molecule

back to its original orientation.This operation does nothing. It is simplest

of all the symmetry elements. It is the only element/operation possessed

by all molecules. It is denoted by E. for example:- CHBrFCl

F Br

Cl

1-chloro-1-bromo-1-fluoromethane

2. Axis of symmetry [Cn] :- It is called n-fold rotational axis. If the rotation of a molecule about an axis

through some angle results in a configuration which is indistinguishable from the original, then the molecule is said to possess a proper rotation axis.

It is denoted as Cn.n is order of rotation axis.Rotation about an axis by an angle of

360/n.For example:- water molecule

HO

H

watera b

HO

H

waterb a1800

Order of rotation axis = 2Symmetry element = C2 axisOperations = C2

1, C22 = E

Operation 2: Cn, Proper Rotation:Rotation about an axis by an angle of 2/n = 360/n

How about: NFO2?

H2ONH3

C2 C3

NH3 N

H’’’ H’ H’’

N

H’ H’’ H’’’

N

H’’ H’’’ H’

C3

v

1200 1200

1200

C3 C3

C31 C3

2

C33

v v

BF3

F

B F F

Ni

C3 C4 C6

F

B F F

Niv

v

v

v

v

v

v

v

vv

v

vv

3 C2 4 C2 6 C2

Ni[CN]4 C6H6

v v v

Principal and Subsidiary Axes : In molecules with more than one axis of symmetry, the axis with the highest foldsymmetry (highest n in Cn) is called the Principal Axis. The other axes are called Subsidiary Axes.

In case there are more than one axes of same order, the axis passing through maximum number of atoms is the Principal Axis.

The axis of symmetry can be C∞ .

H Cl H H v v C∞

HCl H2

Symbol of the proper rotation axis

Order of rotation axis

3600 /n

1. C2 (= C63) 2 180

2. C3 (= C62) 3 120

3. C4 4 904. C5 5 725. C6 6 60

Symmetry operations associated with axis of symmetry :-

In general a Cn axis can generated n operationCn , Cn

2, Cn3, Cn

4......... Cnn

Cnn = E

Cnn+1 = Cn

Cnn+2 = Cn

2 and so on

C2

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

C4

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

C2

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

C2

C2

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

C2

PtCl4

Proper Rotation:Cn = Rotation about an axis by an angle of 2/n

The highest order rotation axisis the principal axis

and it is chosen as the z axis

Iron pentacarbonyl, Fe(CO)5

C3 axis

What other rotational axes do we have here?

3. Plane of symmetry [] :-A mirror plane is an imaginary plane which divides a

molecules into two equal halves such that one half is the exact mirror image of the other.

It is denoted by ‘’. Atoms on the surface of plane remain unshifted during

reflection. Classification of mirror planes:- Vertical plane(v) :- The principal axis of symmetry lies in

the this plane. Horizontal plane (h):- The principal axis of symmetry is

perpendicular to the plane. Dihedral plane (d):- The plane passing through the

principal axis but passing in between two subsidiary axis, is the dihedral plane.

C2

principal axis C2 C2

σvmirror plane

σvmirror plane

σhMirror plane

Water molecule

Benzene

σv mirror plane

Allene molecule containing Dihedral plane

Benzene containing Dihedral plane

C2

4. Inversion centre of centre of symmetry :- If a line drawn through a point in a

molecule and extended in both directions encounters equivalent point in either, the point through which line is drawn is called an inversion centre.

It denoted as ‘i’.

Square planar (AB4)Ethane

1,4-dibromobenzene Trans-dibromoethene

5. Improper axis of symmetry or rotation-reflection axis or alternate axis of symmetry:-

If a molecule is rotated about an axis through some angle and the resulting configuration is reflected in a plane perpendicular to this axis, if new configuration is indistinguishable from the original, then the axis is called an improper axis.

It denoted as ‘Sn’

The symmetry element is denoted as S2.

Cl

H

H

Cl

a

b

1800

H

Cl

Cl

H

b

a

Cl

H

H

Cla

b

Methane molecule showing S4 symmetry element

Operations generated by Sn :-The no. Of operations generated by Sn

depends on whether n is odd or even. If ‘n’ is even then generated operations are

‘n’. If ‘n’ is odd then generated operations are

‘2n’.

References :-

Molecular symmetry and group theory by Robert L. Carter.

Chemical Applications of Group Theory by F. Albert Cotton.

http://symmetry.otterbein.edu/tutorial/methane.html

Google

Thank you