Symmetry and Point Groups, Study notes of Molecular Chemistry

Download Symmetry and Point Groups and more Study notes Molecular Chemistry in PDF only on Docsity!

Symmetry and Point Groups

Chapter 4

Monday, September 28, 2015

Symmetry in Molecules: Staggered Ethane

So far we can say staggered ethane has three operations:

E

,^

C

3

, and

C

3

2

Now we’ve added three reflections:

σ

d

σ

d

′, and

σ

d

Note that there is no

σ

h

for staggered ethane!

Symmetry in Molecules: Staggered Ethane

σ

d

σ

d

σ

d

σ

d

σ

d

σ

d

Symmetry in Molecules: Staggered Ethane^ Ethane also has an inversion center that lies at the midpoint

of the C-C bond (the center of the molecule).

Symmetry in Molecules: Staggered Ethane

Finally, staggered ethane also has an improper rotation axis.

It is an

S

6

S

2n

) axis that is coincident with the

C

3

axis.

Symmetry in Molecules: Staggered Ethane

It turns out that there are several redundancies when counting up the unique improper rotations:

So the improper rotations add only two unique operations.

Summary

Symmetry Elements and Operations

-^

elements are imaginary points, lines, or planes within the object.

-^

operations are movements that take an object between equivalentconfigurations – indistinguishable from the original configuration,although not necessarily identical to it.

-^

for molecules we use “point” symmetry operations, which includerotations, reflections, inversion, improper rotations, and theidentity. At least one point remains stationary in a point operation.

-^

some symmetry operations are redundant (e.g.,

S

6

2

C

3

); in these

cases, the convention is to list the simpler operation.

Low-Symmetry Point Groups

These point groups only contain one or two symmetry operations

C

1 {

E

C

s

E,

σ

h

C

i

E, i

High-Symmetry Point Groups

T

h

example:

In addition to

T

d

,^

O

h

, and

I^ h

, there are corresponding point groups that

lack the mirror planes (

T

,^

O

, and

I

Adding an inversion center to the

T

point group gives the

T

h

point group.

Linear Point Groups

These point groups have a

C

∞

axis as the principal rotation axis

D

∞

h

E,

^2

C

∞

φ

,^

…

C

2

, i,

S

∞

φ

σ

v

C

∞

v

E,

^2

C

∞

φ

,^

…

σ

v

C

Point Groups

These point groups have a principal axis (

C

n

) but no

C

2

axes

C

nv

E,

n

C

n

,^

n

σ

v

C

nh

depends on n,

with h = 2n

C

n

E,

n

C

n

C

2

E, C

2

C

3

v

E, 2C

3

^3

σ

v

C

2

h

E, C

2

, i,

σ

h

S

Point Groups

If an object has a principal axis (

C

n

) and an

S

2

n

axis but no

C

2

axes

and no mirror planes, it falls into an

S

2n

group

S

2

n

depends on n, with h = 2n

S

4

E, S

4

, C

2

, S

4

3

cyclopentadienyl (Cp)

ring =

Co

4

Cp

4