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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
Symmetry and Group Theory
K. Sridharan
Dean
School of Chemical & Biotechnology
SASTRA University
Thanjavur 613 401
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
Table of Content
1SYMMETRYANDGROUPTHEORY.......................................................................................................3
1.1SYMMETRYELEMENTSANDSYMMETRYOPERATIONS........................................................................................31.1.1Whatissymmetry?.......................................................................................................................31.1.2Symmetryelements......................................................................................................................41.1.3Symmetryoperations....................................................................................................................4
2.ORDEROFAXISANDPLANEOFSYMMETRY.......................................................................................9
2.1ORDEROFAXIS.........................................................................................................................................9
2.2.1Verticalmirrorplane(v)............................................................................................................102.2.2Horizontalmirrorplane(h)........................................................................................................11
3.CENTREOFSYMMETRY,IDENTITYELEMENT,ANDIMPROPERROTATIONAXIS.................................13
3.1.CENTEROFSYMMETRY(I)........................................................................................................................133.2IDENTITYELEMENT(E).............................................................................................................................133.3IMPROPERROTATIONALAXISOFSYMMETRYORROTATIONREFLECTIONAXISOFSYMMETRY(SN)..............................13
4.REFERENCES....................................................................................................................................15
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2.1.1Principalaxisofsymmetry............................................................................................................92.2PLANEOFSYMMETRY()..........................................................................................................................10
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
1 Symmetry and group theory It is very important to understand
the symmetry and point group of orbitals and
molecules so that their behaviors under different circumstances
are clearly
understood. The point groups are based on the shapes of orbitals
and structures
of molecules. For example, s orbital is spherical and has a
particular symmetry,
while p orbital has dumbbell shape and has different symmetry.
Similarly, d
orbitals have different shapes and hence different symmetries.
Methane has a
tetrahedral shape and its symmetry is Td, while benzene is
hexagonal planar
and its symmetry is D6h. Water is V shaped and its point group
is C2v. In order
to understand the splitting of orbitals in different
environments and the spectral
characteristics of complexes, their symmetries and point groups
must be
understood.
1.1 Symmetry elements and symmetry operations
1.1.1 What is symmetry?
In simple language we can say that an object has symmetry, if it
has some
special characteristics, such as pleasing designs, while we look
at it. As an
example, when we see the telephone posts or electric lamp posts,
we say
that there is symmetry because they are arranged in a straight
line at
equal distance. Similarly, when we look at the gates of houses,
they will
appear symmetric because of their designs. Naturally, our eyes
will compare
the design on one half of the gate with that of the other half
and if they find
some characteristic feature such as mirror image or other, then
we feel there
is symmetry. A suspension bridge, a butterfly, the rose petal
etc. are some
examples to show the pleasing designs and hence, they are
symmetric.
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements) Note: It can be viewed only on
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C3 axis of rotation
Fig 1 1.3.2: C3 axis of rotation
The angle between any two spheres is equal to 1200. Hence,
rotation by 1200
gives an indistinguishable structure. When the colors of the
spheres are
removed, the two structures cannot be distinguished.
Note :
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
Note : It can be viewed only on Acrobat 9.0 and above
C2 axis of rotation
Fig 1 1.3.3: C2 axis of rotation in water molecule
When the above V-shaped molecule is rotated by 1800 about the
axis passing through the blue sphere, red and green spheres are
interchanged. If the colors of the spheres are removed, the two
structures are indistinguishable.
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
Fig 1 1.3.4: C2 axis of rotation in a linear molecule
The angle between the blue and red spheres is 1800.Rotaion about
the vertical axis by 1800 gives an indistinguishable structure,
once the colors are removed.
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements) C6 axis of rotation
Fig 1 1.3.5: C6 axis of rotation
600 rotation about the axis perpendicular to the paper gives an
indistinguishable structure, once the colors of the spheres are
removed.
600 rotation
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15 Page 9 of
Principal axis for wh
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements) C4 is the principal axis because
n=4 is the maximum number
2.2 Plane of symmetry () It is an imaginary plane cutting the
molecule or object into two halves
which are mirror images.
2.2.1 Vertical mirror plane (v)
Fig 2.2.1: v plane of symmetry
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This is the mirror plane parallel to the principal axis of
symmetry.
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements) 2.2.2 Horizontal mirror plane
(h) When the mirror plane is perpendicular to the principal axis,
it is called horizontal plane of symmetry.
C2
C2
No h
Fig 2.2.2: h plane of symmetry
In the first case (plane triangle), the reflection could not be
distinguisdhed from the original and the mirror plane is called a
horizontal mirror plane, h plane. In the other case, (V-shaped),
the relection is inverted and we are able to distinguish this from
the original one. Hence, it is not a h plane.
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
3. Centre of symmetry, Identity Element, and Improper rotation
axis
3.1. Center of symmetry (i) If we can move in a straight line
from every atom or point in a molecule or object
through a single point at the center to an identical atom or
point on the other side
of the center, then the molecule or object is said to possess a
center of symmetry
Fig 3.1 : Centre of symmetry
3.2 Identity Element (E) This is nothing but rotating the
molecule by 3600. The original molecule is
obtained. The corresponding operation can be called as doing
nothing operation.
This is important from mathematical considerations.
3.3 Improper rotational axis of symmetry or Rotation reflection
axis of symmetry (Sn). Rotation by a particular angle followed by
reflection in a plane perpendicular to
the rotational axis leads to an indistinguishable structure.
Example: S4 axis: rotation by 360/4 = 900
4 axis gives an indistinguishable structure.
Example: SiF4
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followed by reflection in a plane
perpendicular to C
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
Fig 3.2 S4 axis of symmetry
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NPTEL Chemistry and Biochemistry Coordination Chemistry
(Chemistry of transition elements)
4. References 1. Inorganic Chemistry: Principles
2. Chemical Applications of Group Theory, 2/e, F.Albert Cotton,
Wiley
Eastern, New Delhi, 1986
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of Structure and Reactivity, James
E.Huheey, Ellen A.Keiter, Richard L.Keiter, Okhil K.Medhi,
Pearson
Education, Delhi, 2006
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