Molecular symmetry and group theory

Dr. Md. Monirul Islam Department of Chemistry

University of Rajshahi

• The representation of symmetry operation requires strong background about linear algebra.

Vectors

(x, y, z)

Reference system

A (x, y, z) or, xyz

Norm or magnitude of vectors

|𝑨|=‖x , y , z‖=√ x2+ y2+z2

Angle between two vectors () and is

Presentation of symmetry operation

Linear transformations and matrices

Ai(xi, yi, zi)

Af (xf, yf, zf)

Reference

system( x fy fzf )=(a11 a12 a13

a21 a22 a23a31 a32 a33

)( x iy izi )Final

vectorInitial vector

Transformation matrix

is the element of matrix.

Convention is that the principal axis of rotation (rotation axis with highest n) positioned to be coincident with the z axis

• Symmetry operation can be represented by a matrix.

1) Identity :x1

y1

z1

E ?x1

y1

z1

= =x1

y1

z1

matrix satisfying this condition is:

1 0 00 1 00 0 1

1 0 00 1 00 0 1

Therefore, E = E is always the unit matrix

2) Reflection :x1

y1

z1

(xy)x1

y1

-z1

= (xy) =1 0 00 1 00 0 -1

Similarly, (xz) =1 0 00 -1 00 0 1

and (yz) =-1 0 00 1 00 0 1

3) Inversion :x1

y1

z1

i-x1

-y1

-z1

= =-1 0 00 -1 00 0 -1

4) Proper axis of rotation:

o Because of convention, φ, and hence zi, is not transformed under Cn(θ) projection into xy plane need only be ∴considered… i.e., rotation of vector v(xi,yi) through θ

x1 = v cosy1 = v sin

Cn() x2 = v cos [-] = v cos y2 = v sin [- (] = - v sin (

Using identity relations

x2 = v cos (= v cos cos + v sin sin = x1 cos + y1 sin y2 = - v sin (= -v sin cos + v cos sin =- x1 sin + y1 cos

Reformulating in terms of matrix representation:

x1

y1

z1

Cn() = x1 cos + y1 sin - x1 sin + y1 cos

z1

= cos sin 0 - sin cos 0 0 0 1

Where,

The above matrix representation is completely general for any rotation θ…

For example, C3,

3 =

cos sin 0 - sin cos 0 0 0 1

0 0 0 0 1=

5) Improper axis of rotation:

Like operators themselves, matrix operations may be manipulated with simple matrix algebra…above direct product yields matrix representation for Sn..

h Cn() = Sn()

1 0 00 1 00 0 -1

cos sin 0 - sin cos 0 0 0 1

cos sin 0 - sin cos 0 0 0 -1

. =