Molecular Symmetry and Group Theory

symmetry elements and operations

• symmetry element eg. C₃ - an example of a proper rotation axis - a line through the centre it was rotated about

• C = proper rotation axis

• C_n = rotation through 360º/n where n is the order of rotation

• a symmetry element can generate more than one symmetry operation

• when you do the symmetry operation enough times that the molecule returns to its starting position this is called the identity operation - any operation or repeated symmetry operation that brings the molecule back to where it started

• all objects possess the identity element E

other symmetry elements

mirror plane

• symmetry element σ generates a single operation

• σ² = E

inversion centre

• a centre of symmetry i also generates a single operation

• i² = E

improper rotation axis S

• rotation though 360º/n then reflect in plane perpendicular to the rotation axis

• a rotation-reflection axis

5 types of symmetry elements

• E

• σ

• i

• C_n

• S_n

• each can generate one or more symmetry operations

• if two symmetry operations are performed successively on a molecule, the result is always the same as the application of a single different operation

what is a point group?

• the symmetry operations of a molecule form a mathematical group

• there are only a limited number of combinations of operations that are possible

• each of these combinations is known as a point group

• if two molecules have all the same symmetry elements then that are in the same point group

chirality

a molecule may be chiral if it does not possess an improper rotation axis i.e. it possesses no symmetry other than proper rotation axes

only in point groups Cn or Dn

polarity

• a molecule may not have a dipole moment either perpendicular to an axis of rotation or perpendicular to a mirror plane or if the molecule possesses an inversion centre

• dipole moments are only seen in Cn, Cnv including low symmetry group Cs

point group determination

rules for symmetry operations of a molecule to constitute a mathematical group

1. the product of any 2 members (and the square of any one) must also be a group member

2. there must be an identity element E

3. for any three operations done successively (a, b, c) the associative law is obeyed (ab)c = a(bc)

4. every member (a) also has an inverse a^-1 in the group so that aa^-1 = E

character tables

• irreducible representations represent the symmetry operations of the point group ie. how performing them effects the orbitals

• al of the irreducible representations make up the character table

• character tables contain all the information required to apply group theory to chemical problems

using character tables

1. generate a set of numbers to represent the effect of the symmetry operations on molecular properties - this is known as a reducible representation- a linear combination (the sum) of some or all of the irreducible representations in the character table

2. find the component irreducible representations - this is known as reduction

3. interpret the results

interpreting Γ (reducible representations)

• reducible representations are the sum of a number of the irreducible representations in the character table

use the acronym MAD

M) multiply down the table eg. number of E x gamma for that operation x number of A1 etc etc

A) add the numbers calculated above for each of the operations

D) divide by the order of the group - the total number of symmetry operations in the group

degenerate representations