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Lecture

CHEM 212 Lecture Notes - Symmetry Operation, Indo, Ammonia


Department
Chemistry
Course Code
CHEM 212
Professor
Richard Oakley

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Symmetry is a phenomenon of the natural world, as well as the world of human inven-
tion (Figure 4-1). In nature, many types of flowers and plants, snowflakes, insects, cer-
tain fruits and vegetables, and a wide variety of microscopic plants and animals exhibit
characteristic symmetry. Many engineering achievements have a degree of symmetry
that contributes to their esthetic appeal. Examples include cloverleaf intersections, the
pyramids of ancient Egypt, and the Eiffel Tower.
Symmetry concepts can be extremely useful in chemistry. By analyzing the sym-
metry of molecules, we can predict infrared spectra, describe the types of orbitals used
in bonding, predict optical activity, interpret electronic spectra, and study a number of
additional molecular properties. In this chapter, we first define symmetry very specifi-
cally in terms of five fundamental symmetry operations. We then describe how mole-
cules can be classified on the basis of the types of symmetry they possess. We conclude
with examples of how symmetry can be used to predict optical activity of molecules and
to determine the number and types of infrared-active stretching vibrations.
In later chapters, symmetry will be a valuable tool in the construction of molecu-
lar orbitals (Chapters
5
and
10)
and in the interpretation of electronic spectra of coordi-
nation compounds (Chapter
11)
and vibrational spectra of organometallic compounds
(Chapter
13).
A molecular model kit is a very useful study aid for this chapter, even for those who
can visualize three-dimensional objects easily. We strongly encourage the use of such a kit.
4-1
All molecules can be described in terms of their symmetry, even if it is only to say they
SYMMETRY
have none. Molecules or any other objects may contain
symmetry elements
such
as
ELEMENTS AND
mirror planes, axes of rotation, and inversion centers. The actual reflection, rotation, or
OPERATlONS
inversion is called the
symmetry operation.
To contain a given symmetry element,
a
molecule must have exactly the same appearance after the operation as before.
In
other
words, photographs of the molecule (if such photographs were possible!) taken from
the same location before and after the symmetry operation would be indistinguishable.
If a symmetry operation yields a molecule that can be distinguished from the original in

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4-1
Symmetry Elements and Operations
77
(a)
FIGURE
4-1
Symmetry in
Nature,
Art,
and
Architecture.
any way, then that operation is
not
a symmetry operation of the molecule. The examples
in Figures 4-2 through 4-6 illustrate the possible types of molecular symmetry
operations and elements.
The
identity operation
(E)
causes no change in the molecule. It is included for
mathematical completeness. An identity operation is characteristic of every molecule,
even if it has no other symmetry.
The
rotation operation
(C,)
(also called
proper rotation)
is rotation through
360°/n about a rotation axis. We use counterclockwise rotation as a positive rotation. An
example of a molecule having a threefold
(C3)
axis is CHC13. The rotation axis is coinci-
dent with the C-H bond axis, and the rotation angle is 360'13
=
120". Two
Cj
opera-
tions may be performed consecutively to give a new rotation of 240". The resulting

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78
Chapter
4
Symmetry
and
Group Theory
H
I
C1-..-
/
C,
Cl
c3:gc1
C1
C1
Top view
C3
rotations of
CHCI,
Cross section of protein disk
of tobacco mosaic virus
C2,
C,
and
C6
rotations
of
a snowflake design
FIGURE
4-2
Rotations.
The
cross section
of
ilie tobacco mosaic virus is a cover diagram from
Nature,
1976,259.
O 1976,
Macmillan Journals Ltd. Reproduced with permission of Aaron
Klug.
operation is designated
c~~
and is also a symmetry operation of the molecule. Three suc-
cessive
C3
operations are the same as the identity operation
(c~~
=
E).
The identity oper-
ation is included in all molecules. Many molecules and other objects have multiple rotation
axes. Snowflakes are a case in point, with complex shapes that are nearly always hexagonal
and nearly planar. The line through the center of the flake perpendicular to the plane of the
flake contains a twofold
(C2)
axis, a threefold
(C3)
axis, and a sixfold
(C6)
axis. Rotations
by 240"
(~3~)
and 300"
(~6~)
are also symmetry operations of the snowflake.
Rotation Angle Symmetry Operation
There are also two sets of three
C2
axes in the plane of the snowflake, one set
through opposite points and one through the cut-in regions between the points. One of
each of these axes is shown in Figure 4-2. In molecules with more than one rotation axis,
the
C,
axis having the largest value of
n
is the
highest order rotation axis
or
principal
axis.
The highest order rotation axis for a snowflake is the
C6
axis. (In assigning Carte-
sian coordinates, the highest order
C,
axis is usually chosen
as
the
z
axis.) When neces-
sary, the
C2
axes perpendicular to the principal axis are designated with primes; a single
prime
(C2')
indicates that the axis passes through several atoms of the molecule, where-
as a double prime
(C2")
indicates that it passes between the outer atoms.
Finding rotation axes for some three-dimensional figures is more difficult, but the
same in principle. Remember that nature is not always simple when it comes to
symmetry-the protein disk of the tobacco mosaic virus has a 17-fold rotation axis!
In the
reflection operation
(o)
the molecule contains a mirror plane. If details
such as hair style and location of internal organs are ignored, the human body has a left-
right minor plane, as in Figure
4-3.
Many molecules have mirror planes, although they
may not be immediately obvious. The reflection operation exchanges left and right, as if
each point had moved perpendicularly through the plane to a position cxactly as far from
the plane as when it started. Linear objects such
as
a round wood pencil or molecules
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