Lecture 4

Symmetry and group theory

Natural symmetry in plants

Symmetryin animals

Symmetry in the human body

Symmetry in modern artM. C. Escher

Symmetry in arab architectureLa Alhambra, Granada (Spain)

Symmetry in baroque artGianlorenzo BerniniSaint Peter’s ChurchRome

7th grade art projectSilver Star SchoolVernon, Canada

Re2(CO)10

C2F4 C60

Symmetry in chemistry

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

A molecule is said to have symmetry if some parts of it may be interchangedby others without altering the identity or the orientation of the molecule

Molecular structures

Symmetry Operation:

Transformation of an object into an equivalent or indistinguishableorientation

C3, 120º

Symmetry Elements:

A point, line or plane about which a symmetry operation is carried out

5 types of symmetry operations/elements

Identity: this operation does nothing, symbol: E

Operation 1: Identity Operation, do nothing.

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

How about: NFO2?

H2ONH3

C2 C3

180° (2/2)

C2

The Operation: Proper rotation Cn is the movement (2/n)

The Element: Proper rotation axis Cn is the line

Applying C2 twiceReturns molecule to original oreintation

C22 = E

Rotation angle Symmetry operation

60º C6

120º C3 (= C62)

180º C2 (= C63)

240º C32(= C6

4)

300º C65

360º E (= C66)

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

Operations can be performed sequentially

nnn

nn

CC

EC

1

Can perform operation several times.

mnC

...nnnmn CCCC

Means m successive rotations of 2/n each time. Total rotation is 2m/n

m times

Observe

The highest order rotation axisis the principal axis

and it is chosen as the z axis

Iron pentacarbonyl, Fe(CO)5C3 axis

What other rotational axes do we have here?

Let’s look at the effect of a rotation on an algebraic function

Consider the pz orbital and let’s rotate it CCW by 90 degrees.

px proportional to xe-r where r = sqrt(x2 + y2 + z2) using a coordinate system centered on the nucleus

x

y

x

y

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)

px C4 px

C4

oo

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.

Let’s do it analytically using C4 f(x,y,z) = f(y,-x,z)

We start with px = xe-r where r = sqrt(x2 + y2 + z2) and make the required substitution to perform C4

x

y

x

y

px C4 px

C4

oo

Thus C4 px (x,y,z) = C4 xe-r = ye-r = 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)

(reflection through a mirror plane)

NH3

Only one ?

H2O, reflection plane, perp to board

What is the exchange of atoms here?

’

H2O another, different reflection plane

What is the exchange of atoms here?

B

F F

F

If the plane containsthe principal axis it is called v

B

F F

F

If the plane is perpendicularto the principal axis

it is called h

n = E (n = even)n = (n = odd)

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 !!

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/nfollowed by reflection through perpendicular plane

S4 in methane, tetrahedral structure.

Some things to ponder: S42 = C2

Also, S44 = E; S2 = i; S1 =

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

mnC Means m successive rotations of 2/n each

time. Total rotation is 2m/n

But let’s examine some other multiplications of operators

C4

12

3

4

41

2

3

4

12

3

C4 ’

C4

We write x C4 = ’, 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 moleculeis called the point group (there are 28 point groups)

The mathematical treatment of the properties of groupsis 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 determinethe point groupof a molecule

Groups of low symmetry