Ch 4 Lecture 1 Symmetry and Point GroupsI. Introduction

A. Symmetry is present in nature and in human culture

B. Using Symmetry in Chemistry

1. Understand what orbitals are used in bonding

2. Predict IR spectra or Interpret UV-Vis spectra

3. Predict optical activity of a molecule

II. Symmetry Elements and OperationsA. Definitions

1. Symmetry Element = geometrical entity such as a line, a plane, or a point, with respect to which one or more symmetry operations can be carried out

2. Symmetry Operation = a movement of a body such that the appearance after the operation is indistinguishable from the original appearance (if you can tell the difference, it wasn’t a symmetry operation)

B. The Symmetry Operations

1. E (Identity Operation) = no change in the object

a. Needed for mathematical completeness

b. Every molecule has at least this symmetry operation

2. Cn (Rotation Operation) = rotation of the object 360/n degrees about an axis

a. The symmetry element is a line

b. Counterclockwise rotation is taken as positive

c. Principle axis = axis with the largest possible n value

d. Examples:

C23 = two C3’s

C33 = E

C17 axis

3. (Reflection Operation) = exchange of points through a plane to an opposite and equidistant point

a. Symmetry element is a plane

b. Human Body has an approximate operation

c. Linear objects have infinite ‘s

d. h = plane perpendicular to principle axis

e. v = plane includes the principle axis

f. d = plane includes the principle axis, but not the outer atoms

O C O OH H

h v

d

4. i (Inversion Operation) = each point moves through a common central point to a position opposite and equidistant

a. Symmetry element is a point

b. Sometimes difficult to see, sometimes not present when you think you see it

c. Ethane has i, methane does not

d. Tetrahedra, triangles, pentagons do not have i

e. Squares, parallelograms, rectangular solids, octahedra do

5. Sn (Improper Rotation Operation) = rotation about 360/n axis followed by reflection through a plane perpendicular to axis of rotation

a. Methane has 3 S4 operations (90 degree rotation, then reflection)

b. 2 Sn operations = Cn/2 (S24 = C2)

c. nSn = E, S2 = i, S1 =

d. Snowflake has S2, S3, S6 axes

C. Examples:

1. H2O: E, C2, 2

2. p-dichlorobenzene: E, 3, 3C2, i

3. Ethane (staggered): E, 3, C3, 3C2, i, S6

4. Try Ex. 4-1, 4-2

OH H

v

dC2

Cl Cl

C CH

H

H

HH

H

III. Point GroupsA. Definitions:

1. Point Group = the set of symmetry operations for a molecule

2. Group Theory = mathematical treatment of the properties of the group which can be used to find properties of the molecule

B. Assigning the Point Group of a Molecule

1. Determine if the molecule is of high or low symmetry by inspection

a. Low Symmetry Groups

b. High Symmetry Groups

2. If not, find the principle axis

3. If there are C2 axes perpendicular

to Cn the molecule is in D

If not, the molecule will be in C or S

a. If h perpendicular to Cn then Dnh or Cnh

If not, go to the next step

b. If contains Cn then Cnv or Dnd

If not, Dn or Cn or S2n

c. If S2n along Cn then S2n

If not Cn

C. Examples: Assign point groups of molecules in Fig 4.8

C∞v D∞h Td C1 Cs Ci Oh Ih

Rotation axes of “normal” symmetry molecules

Perpendicular C2 axes

Horizontal Mirror Planes

Vertical or Dihedral Mirror Planes and S2n Axes

Examples: XeF4, SF4, IOF3, Table 4-4, Exercise 4-3

D. Properties of Point Groups1. Symmetry operation of NH3

a. Ammonia has E, 2C3

(C3 and C23) and 3v

b. Point group = C3v

2. Properties of C3v (any group)a. Must contain E

b. Each operation musthave an inverse; doing bothgives E (right to left)

c. Any product equals another group member

d. Associative property