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Symmetry and Point Groups, Study notes of Molecular Chemistry

The complete set of symmetry operations possessed by an object defines its point group. For example, the point group of staggered ethane is ...

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Symmetry and Point Groups
Chapter 4
Monday, September 28, 2015
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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