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Symmetry Operations and Elements
•

The goal for this section of the course is to understand how symmetry arguments can be
applied to solve physical problems of chemical interest
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•

A symmetry element is an imaginary geometrical construct about which a symmetry
operation is performed
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•

A symmetry
t operation
ti carries
i every point
i t in
i the
th object
bj t into
i t an equivalent
i l t point
i t or the
th
identical point
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•

The symmetry of a molecule or ion can be described in terms of the complete collection of
symmetry operations it possesses
...
The more
symmetry operations a molecule has, the higher its symmetry is
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The Identity Operation (E)
•

The simplest of all symmetry operations is identity, given the symbol E
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If it possesses no other symmetry, the object is said to be
asymmetric
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It exists for every object, because the
object itself exists
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•

In addition,
addition identity is often the result of carrying out a particular operation successively a
certain number of times,
i
...
, if you keep doing the same operation repeatedly, eventually you may bring the object
back to the identical (not simply equivalent) orientation from which was started
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•

Thus, if a series of repeated operations carries the object back to its starting point, the result
would be identified simply as identity
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•

If a molecule has rotational symmetry Cn , rotation by
2/n = 360°/n brings the object into an equivalent
position
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•

If the molecule has one or more rotational axes, the
one with the highest value of n is the principal axis of
rotation
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•

Multiple iterations are designated by a superscript,
e
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three successive C4 rotations are identified as C43

•

The C42 and C44 operations are preferably identified as
the simpler C2 and E operations,
operations respectively
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•

The C2' and C2" axes of a planar MX4 molecule
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•

Only two notations are needed for the four axes,
axes because both C2’ axes are said to belong to
the same class, while the two C2’’ axes belong to a separate class
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e
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•

In listing the complete set of symmetry
operations for a molecule, operations of the
same class are designated by a single notation
preceded by a coefficient indicating the number
of equivalent operations comprising the class
...
g
...

)

•

Rotations are clockwise, except S4‐1 , which
is equivalent to the clockwise operation S43
...
e
...


•

Thus, there are only two operations
belonging to this class for the tetrahedral
MX4 molecule (S4 and S43 ) about this axis
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•

Consequently, each axes gives rises to two
S4 operations resulting in a class designated
as 6S4 (3S4 + 3S43 )
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Examples
•

Find all symmetry elements and operations in the following:

Defining the Coordinate System
•

Molecules are conventionally oriented relative to a right‐hand Cartesian coordinate system:

•

The following conventions of axis orientation are usually observed:
1
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2
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If there
are several highest order rotational axes, z is usually taken as the axis passing through
the greatest number of atoms
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Defining the Coordinate System (contd
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For planar molecules, if the z axis as defined above is perpendicular to the molecular
plane, the x axis lies in the plane of the molecule and passes through the greatest
number of atoms
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Defining the Coordinate System (contd
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For non‐planar molecules, once the z axis has been defined, the x axis is usually chosen
so that the xz plane contains as many atoms as possible
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Where a decision about the orientation of the x axis cannot be made on this basis, the
distinction between x and y is usually not important or is not generally fixed by
convention
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g
...


•

The order in which successive different symmetry operations are performed can affect the
result
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•

In writing multiplications of symmetry operation we use a "right‐to‐left" notation:

•



BA = X "Doing A then B has the same result as the operation X
...




It may be that either BA ≠ AB or BA = AB
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molecule affects the result
The final positions in each case are not the same, but they are related to each other by C2
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•

The symmetry elements of the CBr2Cl2 molecule are shown below
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•

As the carbon atom is unaffected by any symmetry operation all matrices need only describe
the positions of the bromine and chlorine atoms using a 1x4 column matrix
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Multiplication Tables
•

All possible binary combinations of symmetry operations can be summarized in a
multiplication table
...
g
...
For this process we can start to build a multiplication table
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operation

•

Followed by self combinations

Step 1: Combinations with identity
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Th we have
Thus
h
th following
the
f ll i relationships:
l ti hi
C2E = EC2 = C2

vE = Ev = v

v‘E = Ev‘ = v‘

Step 3: Mixed binary combinations
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•

First we perform the v operation followed by the C2 operation
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

The first column of results duplicates the list of operations in the label column
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

Every column shows every operation once and only once
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

The order of resultant operations in every column is different from any other column
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•

y
y operations
p
are the elements of a ggroup
p and define the symmetry
y
yp
point
These symmetry
group for that molecule or ion
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•

The group formed by the operations of CBr2Cl2 is named C2v where h = 4
...
e
...


Requirements of a Mathematical Group (1)
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Closure:

If A and B are in the group G, and AB = X, then X is also in G
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

For example, inspection of the multiplication table for the operations of CBr2Cl2 shows
that all binary combinations equal either E, C2 , v or v‘
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Requirements of a Mathematical Group (2)
•

Identity:

In any group G, there is an element E, such that
EX = XE = X
The symmetry operation of identity is this group element
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group



As the C2v multiplication table demonstrates, the identity operation does indeed meet
the requirements of the identity element of a group
...
If A, B, C, and X are in G, then
C(BA) = X
( )
= (CB)A
( )
But commutation is not general (e
...
, BAC ≠ CBA)
...
g
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Requirements of a Mathematical Group (4)
•

Reciprocality:

In any group G, every element A has an inverse A‐1, such that
AA–1 =  A–1A  =  E
An element may be its own inverse (e
...
, all operations of C2v)
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Subgroups
•

Within all groups there are smaller collections of elements, called subgroups, which also
obey the criteria for a group
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etc
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•

For the C2v point group (h = 4) subgroups with g = 1, 2 are possible and each exist
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•

In contrast , crystallographers prefer to use the Hermann‐Mauguin notation, which is best
suited for designating the 32 crystallographic point groups and the space groups used to
describe crystal structures
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Non‐rotational

2
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Dihedral

4
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

C1 is the point group of asymmetric molecules which only possess the identity element E
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

The Ci point group is not commonly encountered as most molecules which posses the i
element also possess other complimentary symmetry elements
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•

point ggroups
p are an example
p of the important
p
cyclic
y
groups
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G = { X, X2 , 
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•

The Cn and S2n groups are cyclic groups; e
...
,
C4 = { C4 , C42 , C43 , E }
S4 = { S4, C2 , S43 , E }

•

The multiplication tables of cyclic groups "scroll" from row to row and column to column:
e
...
,

•

To the rotations of the corresponding Cn groups the family of Cnv groups adds n vertical mirror
planes which intersect at the Cn axis
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It is the point group of all non‐centrosymmetric linear molecules
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g
...

C O

•

To generate any of the Cnh groups, we need only add a horizontal mirror plane to the series of
Cn rotations of the appropriate cyclic Cn group
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•

The S2n series are not common
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These C2
axes are called the dihedral axes
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axis
e
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the staggered conformation of ethane is of D3d symmetry and possesses 3C2 dihedral axes
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Unlike the Cn groups, the Dn groups are not cyclic
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In Dnd groups, the combination of rotational operations and vertical mirror reflections (d)
ggenerates a series of S2n operations
p
about an axis collinear with the p
principal
p axis
...

Like the Cnh groups,
groups the Dnh groups include n‐fold improper axis when n>2 and are
centrosymmetric
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•

All are characterized by the presence of multiple, intersecting, high‐order rotational axes
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•

A three‐fold axis, generating the operations C3 and C32 , emerges from each of the four
triangular faces of a tetrahedron
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•

Three S4 axes, each associated with S4 and S43 operations, are each collinear with these C2
axes
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f

•

In the cube, a fourfold axis emerges from each pair of opposite faces, whereas a threefold axis
emerges from each pair of opposite corners,
corners extending the diagonals of the cube
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•

Buckminsterfullerene C60 is an example of a high‐order polyhedron with Ih symmetry
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Icosahedron (I h)

Ih

Flow chart for systematically determining the point group of a molecule
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