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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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Chapter 4
Monday, September 28, 2015
So far we can say staggered ethane has three operations:
3
, and
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!
σ
d
σ
d
σ
d
σ
d
σ
d
σ
d
of the C-C bond (the center of the molecule).
Finally, staggered ethane also has an improper rotation axis.
It is an
6
2n
) axis that is coincident with the
3
axis.
It turns out that there are several redundancies when counting up the unique improper rotations:
So the improper rotations add only two unique operations.
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.,
6
2
3
); in these
cases, the convention is to list the simpler operation.
These point groups only contain one or two symmetry operations
1 {
s
σ
h
i
E, i
h
example:
In addition to
d
h
, and
I^ h
, there are corresponding point groups that
lack the mirror planes (
, and
Adding an inversion center to the
point group gives the
h
point group.
These point groups have a
∞
axis as the principal rotation axis
∞
h
∞
φ
…
2
, i,
∞
φ
σ
v
∞
v
∞
φ
…
σ
v
These point groups have a principal axis (
n
) but no
2
axes
nv
n
n
n
σ
v
nh
depends on n,
with h = 2n
n
n
n
2
2
3
v
3
σ
v
2
h
2
, i,
σ
h
If an object has a principal axis (
n
) and an
2
n
axis but no
2
axes
and no mirror planes, it falls into an
2n
group
2
n
depends on n, with h = 2n
4
4
2
4
3
cyclopentadienyl (Cp)
ring =
Co
4
Cp
4