Based on “Inorganic Chemistry”, Miessler and Tarr,

Symmetry and Group Theory

Based on “Inorganic Chemistry”, Miessler and Tarr, 4th edition, 2011, Pearson Prentice Hall

Images from Miessler and Tarr “Inorganic Chemistry” 2011 obtained from Pearson Education, Inc.

What is symmetry?

Symmetry elements and operations� All molecules can be described in terms of their symmetry, even if it is only to say they have none symmetry

� Molecules can contain symmetry elements such as mirror planes, axes of rotation, and inversion centers

� The actual reflection, rotation or inversion is called the � The actual reflection, rotation or inversion is called the symmetry operation

� To contain a given symmetry element, a molecule must have exactly the same appearance after the operation as before� If you took a picture, you could not tell the operation was performed just by looking at the before and after

Identity operation (E)� Identity operation (E) causes no change in the molecule (included for mathematical completeness)

� It is characteristic of every molecule, even if it has no other symmetry

Rotation operation (proper rotation) - Cn� It is rotation through 360° / n about a rotation axis� We use counterclockwise rotation as positive rotation

� A molecule can have multiple rotation axes, but the highest order rotation axis will be the one with greater n and will be called principal axis

Reflection operation (σ)� The molecule contains a mirror plane

� Horizontal planes (σh) are perpendicular to principal axis of rotationaxis of rotation

� Vertical planes (σv) and dihedral planes (σd) are parallel to the principal axis of rotation

Inversion operation (i)� All points move through the center of the molecule to a position opposite the original position and as far as when it started

Inversion operation (i)

(a) figures with i (b) figures without i

Rotation-reflection (Sn) or improper rotation

Point groups

Low symmetry

High symmetry

C and D groups

Properties and representation of groups

� All mathematical groups have certain properties� Each group must contain an identity operation that commutes will other members of the group and leaves them unchanged (EA = AE = A)

� Each operation must have an inverse that, when combined with the operation, yields the identity operation (BA = E; B is inverse of A)operation, yields the identity operation (BA = E; B is inverse of A)

� The product of any two group operations must also be a member of the group ( BA = C; C is also member of the group)

� The associative property of combination must hold

A(BC) = (AB)C

Each operation must have an inverse that, when

combined with the operation, yields the identity

operation (BA = E; B is inverse of A)

The product of any two group operations must also be

a member of the group ( BA = C; C is also member of

the group)

Representations of groups� Matrixes are used to represent symmetry operations

� Transformation matrixes are used so

[new coordinates] = [transformation matrix][old coordinates]

A little bit of matrix multiplication

� The product matrix will have the same number of rows as A and the same number of columns as B

Matrixes’ character� Characters are only defined for square matrixes (equal number of rows and columns) – the sum of the numbers on the diagonal from upper left to lower right in a block diagonalized matrix

� These characters form a representation – an alternate shorthand version of the matrix representation

� Representations can be reducible or irreducible – reducible � Representations can be reducible or irreducible – reducible representations are combinations of (and can be broken down into) irreducible representations

Matrixes’ character� For example, transformation matrixes’ character for water molecule form a representation

E C2 σv (xz) σv’ (yz)3 -1 1 1

Character tables� A complete set of irreducible representations for a point group is called the character table for that group

Examples and applications of

symmetry - chirality� Chiral molecules are not superimposable on their mirror image – they are dissymetric

� Chiral molecules belong to a few point groups

Examples and applications of symmetry

– hybrid orbitals in central atom� Point vectors along with bonds

� Apply all symmetry operations to generate the reducible representation

� Reduce the representation using the reduction formula

� Select the orbitals used for hybridization� Select the orbitals used for hybridization

� Example: CH4

Examples and applications of symmetry

– IR- and Raman-active vibration bands� Point vectors along all axis in atoms

� Apply all symmetry operations to generate the reducible representation

� Reduce the representation using the reduction formula

� Substract the representations belonging to translation and � Substract the representations belonging to translation and rotation

� Select which vibrations are IR-active and/or Raman-active

� Examples: H2O, XeF4