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Lecture 4

Symmetry and group theory

Natural symmetry in plants

Symmetry in animals

Symmetry in the human body

The platonic solids

Symmetry in modern art M. C. Escher

Symmetry in arab architecture La Alhambra,

Granada (Spain)

Symmetry in baroque art Gianlorenzo Bernini Saint

Peters Church Rome

Symmetry in Native American crafts

7th grade art project Silver Star School Vernon,

Canada

Re2(CO)10

C60

C2F4

Symmetry in chemistry

- Molecular structures
- Wave functions
- Description of orbitals and bonds
- Reaction pathways
- Optical activity
- Spectral interpretation (electronic, IR, NMR)
- ...

Molecular structures

A molecule is said to have symmetry if some parts

of it may be interchanged by others without

altering the identity or the orientation of the

molecule

Symmetry Operation Transformation of an object

into an equivalent or indistinguishable orientatio

n

Symmetry Elements A point, line or plane about

which a symmetry operation is carried out

5 types of symmetry operations/elements

Operation 1 Identity Operation, do nothing.

Identity this operation does nothing, symbol E

Operation 2 Cn, Proper Rotation Rotation

about an axis by an angle of 2?/n 360/n

NH3

H2O

How about

NFO2?

The Operation Proper rotation Cn is the movement

(2p/n) The Element Proper rotation axis Cn is

the line

180 (2p/2)

Applying C2 twice Returns molecule to original

oreintation C22 E

C2

Proper rotation axes

NH3

H2O

How about

NFO2?

Rotation angle Symmetry operation

60º C6

120º C3 ( C62)

180º C2 ( C63)

240º C32( C64)

300º C65

360º E ( C66)

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

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

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

PtCl4

Proper Rotation Cn Rotation about an axis by

an angle of 2?/n

PtCl4

Can perform operation several times.

Rotation 2?m/n

Operations can be performed sequentially

Can perform operation several times.

Means m successive rotations of 2p/n each time.

Total rotation is 2mp/n

m times

Observe

Iron pentacarbonyl, Fe(CO)5

C3 axis

The highest order rotation axis is the principal

axis and it is chosen as the z axis

What other rotational axes do we have here?

Lets look at the effect of a rotation on an

algebraic function

Consider the pz orbital and lets rotate it CCW

by 90 degrees. px proportional to xe-ar where r

sqrt(x2 y2 z2) using a coordinate system

centered on the nucleus

y

y

C4

x

o

x

o

px

C4 px

How do we express this mathematically? The

rotation moves the function as shown. The value

of the rotated function, C4 px, at point o is

the same as the value of the original function px

at the point o . The value of C4 px at the

general point (x,y,z) is the value of px at the

point (y,-x,z) Moving to a general function

f(x,y,z) we have C4 f(x,y,z) f(y,-x,z)

Thus C4 can be expressed as (x,y,z) ?(y,-x,z).

If C4 is a symmetry element for f then

f(x,y,z) f(y,-x,z)

According to the pictures we see that C4 px

yields py. Lets do it analytically using C4

f(x,y,z) f(y,-x,z) We start with px xe-ar

where r sqrt(x2 y2 z2) and make the

required substitution to perform C4

y

y

C4

x

o

x

o

px

C4 px

Thus C4 px (x,y,z) C4 xe-ar ye-ar py And

we can say that C4 around the z axis as shown is

not a symmetry element for px

Operation 3 Reflection and reflection

planes (mirrors)

s

s

? (reflection through a mirror plane)

s

NH3

Only one s?

H2O, reflection plane, perp to board

s

What is the exchange of atoms here?

H2O another, different reflection plane

s

What is the exchange of atoms here?

Classification of reflection planes

Sequential Application

Operation 4 Inversion i Center of inversion

or center of symmetry (x,y,z) ? (-x,-y,-z)

in E (n is even) in i (n is odd)

Inversion not the same as C2 rotation !!

(No Transcript)

Figures with center of inversion

Figures without center of inversion

Operation 5 Improper rotation (and improper

rotation axis) Sn Rotation about an axis by an

angle 2?/n followed by reflection through

perpendicular plane

S4 in methane, tetrahedral structure.

Some things to ponder S42 C2

Also, S44 E S2 i S1 s

Summary Symmetry operations and elements

Operation Element

proper rotation axis (Cn)

improper rotation axis (Sn)

reflection plane (s)

inversion center (i)

Identity (E)

Successive operations, Multiplication of Operators

Already talked about multiplication of rotational

Operators

Means m successive rotations of 2p/n each time.

Total rotation is 2mp/n

But lets examine some other multiplications of

operators

C4

1

2

1

4

1

C4

s

2

2

3

4

3

4

3

C4

s

s

We write s x C4 s, first done appears to

right in this relationship between operators.

Translational symmetry not point symmetry

Symmetry point groups

The set of all possible symmetry operations on a

molecule is called the point group (there are 28

point groups)

The mathematical treatment of the properties of

groups is Group Theory

In chemistry, group theory allows the assignment

of structures, the definition of orbitals,

analysis of vibrations, ...

See Chemical Applications of Group Theory by F.

A. Cotton

To determine the point group of a molecule

(No Transcript)

Groups of low symmetry

(No Transcript)

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