Symmetry and MolecularStructures

Some Readings

Chemical Application of Group TheoryF. A. Cotton

Symmetry through the Eyes of a ChemistI. Hargittai and M. Hargittai

The Most Beautiful Molecule - an Adventure in ChemistryH. Aldersey-Williams

Perfect SymmetryJ. Baggott

Some LinksThe Point Group Tutorialhttp://www.emory.edu/CHEMISTRY/pointgrp/index.htmlTables for Group Theoryhttp://www.oup.com/uk/orc/bin/9780199264636/01student/tables/

Symmetry Groupshttp://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html

For FunWallpaper Groups (Plane Symmetry Groups)http://www.clarku.edu/~djoyce/wallpaper/The Geometry Junk Yardhttp://www.ics.uci.edu/~eppstein/junkyard/

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Shapes, Geometry and Symmetry

Types of Symmetry and Group

Point, Plane, SpaceMolecular 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 isindistinguishable 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.

SymmetryElement

n-fold symmetry axisMirror planeCenter of inversionn-fold axis ofimproper rotation

SymmetryOperation

IdentityRotation by 2/nReflectionInversionRotation by 2/nFollowed by reflectionperpendicular to rotation axis

Symbol

ECn

iSn

Identity (E)All the points in a molecule can be described withCartesian coordinates x, y and z.These points can be transformed by the followingmatrix equation:

E operation

C2Operation –Rotation by 180o

Symmetry Element:

2 fold rotation axis C2

*

*

C3 Operation –Rotation by 120o

Symmetry Element:3 fold rotation axis C3

*

*

*

Proper Rotation (Cn)

The proper rotation (clockwise direction) aboutthe z axis can be described by the followingmatrix equation.

is the angle of rotation

Cn operation

A Cn axis generates n operations

The four operations generated are:C4

1, C2 (C42), C4

3 and E (C44).

v Operation –Reflection

Symmetry Element:Mirror plane v

Parallel to rotation axis

*

*

* *

Mirror Planes ( σ )

If the mirror plane coincides with the xy, xz, oryz Cartesian planes, they can be described bythe following matrix equations:

operation

σ x σ = E

Some Mirror Planes of Benzene

Mirror plane vParallel to rotation axis

Mirror plane dParallel to rotation axisBisects the angles of C-C

Mirror plane hPerpendicular to rotation axis

Inversion Operation

Symmetry Element:Center of inversion ii × i = E

Center of Inversion (i)

The inversion operation changes the sign of allthe x, y and z coordinates:

i operation

i Operation

C2 Operation

S4 OperationRotation by 90o

- followed by reflectionperpendicular to rotationaxisSymmetry Element: S4

Improper Rotation (Sn)

The improper rotation about the z axis can bedescribed as a proper rotation followed bychanging the sign of the z coordinate.

is the angle of rotationSn operation

When n is even, Sn generates n operations.When n is odd,

Sn generates 2n operations.Sn

n is equivalent to σ , and Sn2n is equivalent to E.

S1 = h S2 = i

When n is even, Sn generates n operations.

S6 has generated six operations:S6

1, C31 (S6

2), i (S63), C3

2 (S64), S6

5 and E (S66).

When n is odd, Sn generates 2n operations.

The 10 operations generated by S5:S5

1, C52 (S5

2), S53, C5

4 (S54), σ,

C51 (S5

6), S57, C5

3 (S58), S5

9 and E (S510).

Symmetry Elements ofC2v and C3v

Groups

What is Group Theory?•A fairly “recent”branch of mathematics. Fedorov

pioneered application of group theory incrystallography.

•Group Theory is the closest many chemists get totruly “modern”mathematics.

•Can be used to simplify complicated geometricsystem (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 thegroup.

•One element in the group must commute with allothers 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 reverseorder.

(AB)-1 = B-1A-1 (ABC)-1 = C-1B-1A-1

Types of Point Groups IGroups with a single symmetry element:

C1 (E only) Cs (E and σ) Ci (E and i)

Cn (E and Cn) Sn (n is even and generates n operations)

Types of Point Groups IIGroups with more than one symmetry element:Dn Cn axis, and n C2 axes

perpendicular to Cn

Dnd Cn axis, and n C2 axesperpendicular to Cn, and n dihedralσ parallel to Cn and bisect theangles between the n perpendicularC2 axes

Dnh Cn axis, and n C2 axes perpendicularto Cn, and a σ perpendicular to Cn

Types of Point Groups IIGroups with more than one symmetry element:

Cnh Cn axis, and one σ perpendicular to Cn

Cnv Cn axis, and two or more σ that contain Cn

Types of Point Groups IIISpecial groups:C∞v linear molecules lacking a center of symmetry

D∞h linear molecules with a center of symmetry

Types of Point Groups IIISpecial groups:

Td tetrahedral groups –also includes Th and TOh octahedral groups –includes O

Ih icosahedral groups –includes I

Decision Tree for Molecular Point Group

Polar Molecules

A molecule cannot be polar if it has1. a center of inversion

–any group with i

2. an electric dipole moment perpendicular toany mirror planes–any of the groups D and their derivatives

3. an electric dipole moment perpendicular toany axis of rotation–the cubic groups T, O, the icosahedral I, and

their modifications

Chiral Molecules

A molecule is not chiral if1. it posses an improper rotation axis Sn

2. it belongs to the group Dnh or Dnd

3. it belongs to Td or Oh

Some Molecules

Some Molecules

Total Representation for C2vIndividually block diagonalized matrices

Reduced to 1D matricesx [ 1] [-1] [ 1] [-1]y [ 1] [-1] [-1] [ 1]

z [ 1] [ 1] [ 1] [ 1]

irreducible representationΓx = 1 -1 1 -1Γy = 1 -1 -1 1Γz = 1 1 1 1ΓRz = 1 1 -1 -1z

Character Tables

I: Mulliken symbol.A, B: 1D E: 2D T: 3DA: 1D symmetric about the principal axis (1)B: 1D unsymmetric about the principal axis (1)

II: Irreducible representations for the groupIII: Transformation properties of vectors and

rotations along the x, y and z axisIV: Transformation properties of squares and binary

products of the coordinates

(yz)

WEB Pages

Point Group Theoryhttp://newton.ex.ac.uk/people/goss/symmetry/Symmetry.html

The Point Group Tutorialhttp://www.emory.edu/CHEMISTRY/pointgrp/index.html

Exercises in Point Group Symmetryhttp://origin.ch.ic.ac.uk/vchemlab/symmetry/index.htm

Tables for Group Theoryhttp://www.oup.com/uk/orc/bin/9780199264636/01student/tables/

Symmetry Groupshttp://www.chemistry.nmsu.edu/studntres/chem639/symmetry/group.html