Symmetry and Molecular Structures

Some Readings Chemical Application of Group Theory F. A. Cotton Symmetry through the Eyes of a Chemist I. Hargittai and M. Hargittai The Most Beautiful Molecule - an Adventure in Chemistry H. Aldersey-Williams Perfect Symmetry J. Baggott

The Point Group Tutorial Some Links http://www.emory.edu/chemistry/pointgrp/index.html Tables for Group Theory http://www.oup.com/uk/orc/bin/9780199264636/01student/tables/ Symmetry Groups http://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html For Fun Wallpaper Groups (Plane Symmetry Groups) http://www.clarku.edu/~djoyce/wallpaper/ The Geometry Junk Yard http://www.ics.uci.edu/~eppstein/junkyard/

Shapes, Geometry and Symmetry

Types of Symmetry and Group Point, Plane, Space Molecular Symmetry Point Symmetry Symmetry elements Symmetry operations

Symmetry Operations Defined: A well-defined, non-translational movement of an object that produces a new orientation that is indistinguishable from the original object. Orientation A is indistinguishable from Orientation B, but not necessarily identical. These are Equivalent Configurations.

Symmetry Element Defined: A point, line or plane about which the symmetry operation is performed. Symmetry Element n-fold symmetry axis Mirror plane Center of inversion n-fold axis of improper rotation Symmetry Operation Identity Rotation by 2 /n Reflection Inversion Rotation by 2 /n Followed by reflection perpendicular to rotation axis Symbol E C n i S n

Identity (E) All the points in a molecule can be described with Cartesian coordinates x, y and z. These points can be transformed by the following matrix equation: E operation

* C 2 Operation Rotation by 180 o Symmetry Element: 2 fold rotation axis C 2 *

C 3 Operation Rotation by 120 o Symmetry Element: 3 fold rotation axis C 3 * * *

Proper Rotation (C n ) The proper rotation (clockwise direction) about the z axis can be described by the following matrix equation. C n operation is the angle of rotation

A C n axis generates n operations The four operations generated are: C 41, C 2 (C 42 ), C 43 and E (C 44 ).

* * v Operation Reflection Symmetry Element: Mirror plane v Parallel to rotation axis * *

Mirror Planes ( σ ) If the mirror plane coincides with the xy, xz, or yz Cartesian planes, they can be described by the following matrix equations: operation

σ x σ = E

Some Mirror Planes of Benzene Mirror plane v Parallel to rotation axis Mirror plane d Parallel to rotation axis Bisects the angles of C-C Mirror plane h Perpendicular to rotation axis

Inversion Operation Symmetry Element: Center of inversion i i i = E

Center of Inversion (i) The inversion operation changes the sign of all the x, y and z coordinates: i operation

i Operation C 2 Operation

S 4 Operation Rotation by 90 o - followed by reflection perpendicular to rotation axis Symmetry Element: S 4

Improper Rotation (S n ) The improper rotation about the z axis can be described as a proper rotation followed by changing the sign of the z coordinate. S n operation is the angle of rotation When n is even, S n generates n operations. When n is odd, S n generates 2n operations. S nn is equivalent to σ, and S n 2n is equivalent to E.

S 1 = h S 2 = i

When n is even, S n generates n operations. S 6 has generated six operations: S 61, C 31 (S 62 ), i (S 63 ), C 32 (S 64 ), S 65 and E (S 66 ).

When n is odd, S n generates 2n operations. The 10 operations generated by S 5 : S 51, C 52 (S 52 ), S 53, C 54 (S 54 ), σ, C 51 (S 56 ), S 57, C 53 (S 58 ), S 59 and E (S 5 10 ).

Symmetry Elements of C2v and C3v

What is Group Theory? Groups A fairly recent branch of mathematics. Fedorov pioneered application of group theory in crystallography. Group Theory is the closest many chemists get to truly modern mathematics. Can be used to simplify complicated geometric system (structures) and derive physical properties Used in spectroscopy and molecular orbital theory

Properties of Group The product of any two elements in the group and the square of each element must be an element in the group. One element in the group must commute with all others and leave them unchanged. The associative law of multiplication must hold. Every element must have a reciprocal, which is also an element of the group. The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order. (AB) -1 = B -1 A -1 (ABC) -1 = C -1 B -1 A -1

Types of Point Groups I Groups with a single symmetry element: C 1 (E only) C s (E and σ) C i (E and i) C n (E and C n ) S n (n is even and generates n operations)

Types of Point Groups II Groups with more than one symmetry element: D n C n axis, and n C 2 axes perpendicular to C n D nh C n axis, and n C 2 axes perpendicular to C n, and a σ perpendicular to C n D nd C n axis, and n C 2 axes perpendicular to C n, and n dihedral σ parallel to C n and bisect the angles between the n perpendicular C 2 axes

Types of Point Groups II Groups with more than one symmetry element: C nh C n axis, and one σ perpendicular to C n C nv C n axis, and two or more σ that contain C n

Types of Point Groups III Special groups: C v linear molecules lacking a center of symmetry D h linear molecules with a center of symmetry

Types of Point Groups III Special groups: T d tetrahedral groups also includes T h and T O h octahedral groups includes O I h icosahedral groups includes I

Decision Tree for Molecular Point Group

Polar Molecules A molecule cannot be polar if it has 1. a center of inversion any group with i 2. an electric dipole moment perpendicular to any mirror planes any of the groups D and their derivatives 3. an electric dipole moment perpendicular to any axis of rotation the cubic groups T, O, the icosahedral I, and their modifications

Chiral Molecules A molecule is not chiral if 1. it posses an improper rotation axis S n 2. it belongs to the group D nh or D nd 3. it belongs to T d or O h

Some Molecules

Total Representation for C 2v Individually block diagonalized matrices Reduced to 1D matrices x [ 1] [-1] [ 1] [-1] y [ 1] [-1] [-1] [ 1] z [ 1] [ 1] [ 1] [ 1] z irreducible representation Γ x = 1-1 1-1 Γ y = 1-1 -1 1 Γ z = 1 1 1 1 Γ Rz = 1 1-1 -1

Character Tables (yz) I: Mulliken symbol. A, B: 1D E: 2D T: 3D A: 1D symmetric about the principal axis (1) B: 1D unsymmetric about the principal axis (1) II: Irreducible representations for the group III: Transformation properties of vectors and rotations along the x, y and z axis IV: Transformation properties of squares and binary products of the coordinates

WEB Pages Point Group Theory http://newton.ex.ac.uk/people/goss/symmetry/symmetry.html The Point Group Tutorial http://www.emory.edu/chemistry/pointgrp/index.html Exercises in Point Group Symmetry http://origin.ch.ic.ac.uk/vchemlab/symmetry/index.htm Tables for Group Theory http://www.oup.com/uk/orc/bin/9780199264636/01student/tables/ Symmetry Groups http://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html