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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Table of Content
1 SYMMETRY AND GROUP THEORY ....................................................................................................... 3 1.1 SYMMETRY ELEMENTS AND SYMMETRY OPERATIONS ........................................................................................ 3 1.1.1 What is symmetry? ....................................................................................................................... 3 1.1.2 Symmetry elements ...................................................................................................................... 4 1.1.3 Symmetry operations .................................................................................................................... 4 2. ORDER OF AXIS AND PLANE OF SYMMETRY ....................................................................................... 9 2.1 ORDER OF AXIS ......................................................................................................................................... 9 2.1.1 Principal axis of symmetry ............................................................................................................ 9 2.2 PLANE OF SYMMETRY (Σ) .......................................................................................................................... 10 2.2.1 Vertical mirror plane (σv) ............................................................................................................ 10 2.2.2 Horizontal mirror plane (σh) ........................................................................................................ 11 3. CENTRE OF SYMMETRY, IDENTITY ELEMENT, AND IMPROPER ROTATION AXIS ................................. 13 3.1. CENTER OF SYMMETRY (I) ........................................................................................................................ 13 3.2 IDENTITY ELEMENT (E) ............................................................................................................................. 13 3.3 IMPROPER ROTATIONAL AXIS OF SYMMETRY OR ROTATION REFLECTION AXIS OF SYMMETRY (SN).............................. 13 4. REFERENCES .................................................................................................................................... 15 Page 2 of 15
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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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ents)
1.1.2
2 Symmetry
y elements
s
Of course,
c
we realize the symmetrry in the o
objects when we loo
ok at them.
However, we must
m
expres
ss them scie
entifically. T
This could be done w
with the help
p
ymmetry ele
ements. What
W
are sy
ymmetry ele
ements? T
These are nothing but
of sy
some
e physical entities such as line, plane, poin
nt etc. The next questtion is, “Are
e
they present in objects or molecules?
m
?” The answ
wer is “no”. These are imaginary..
3 Symmetry
y operation
ns
1.1.3
The
ese are som
me mechan
nical operattions, such as, rotation, reflection
n, inversion
n
etc., performe
ed about the symm
metry elem
ments so tthat indisttinguishable
e
stru
uctures are produced.
1.1.3
3.1 Axis of
o symme
etry or Prroper rottational ax
xis of sym
mmetry
(Cn).
)
It is
s an imagin
nary line pa
assing thro
ough an ob
bject or a m
molecule a
about which
h
whe
en the obje
ect or molec
cule is rota
ated by a ce
ertain angle
e, an indisttinguishable
e
stru
ucture is pro
oduced.
Exa
amples: C4 axis of ro
otation
Fig 1 1.3.1: C4 axis of rota
ation
e two structures canno
ot be disting
guished, if tthe letters a
are removed.
The
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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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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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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.
Can be viewed only on Acrobat 9.0 and above
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C6 axis of rotation
600 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.
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2. Order
O
of axis and plane of symm
metry
2.1 Order of axis
This imaginary line,
l
i.e., the axis of sy
ymmetry is represente
ed as Cn, where n is
know
wn as the orrder of the axis.
a
This te
ells how ma
any times w
we have to rotate the
objec
ct to reach the initial sttructure, i.e
e., one full rrotation.
Or, itt tells the an
ngle (3600/n)
/ by which
h we have tto rotate the
e molecule to get the
indistinguishable structure.
C4 axis
a of symm
metry, 360//4 = 900; that is, 900 ro
otation will give an in
distin
nguishable structure.
C3 axis
a of symm
metry, 360//3 = 1200; that is, 1200 rotation w
will give an
indistinguishable structure.
C2 axis
a of symm
metry, 360//2 = 1800; that is, 1800 rotation w
will give an
indistinguishable structure.
C6 axis
a of symm
metry, 360//6 = 600; that is, 600 ro
otation will give an
indistinguishable structure.
2.1.1 Principal axis of sym
mmetry
That axis for whhich the n value
v
is max
ximum is ca
alled the prrinciple axiss of
symm
metry.
Exam
mple:
Fig 2.1.1: Prrinciple axis of syymmetry
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C4 is the principal axis because n=4 is the maximum number
2.2 Plane of symmetry (σ)
It is an imaginary plane cutting the molecule or object into two halves
which are mirror images.
2.2.1 Vertical mirror plane (σv)
This is the mirror plane parallel to the principal axis of symmetry.
Fig 2.2.1: σv plane of symmetry
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2.2.2 Horizontal mirror plane (σh)
When the mirror plane is perpendicular to the principal axis, it is called horizontal
plane of symmetry.
C2
No
h
C2
Fig 2.2.2: σh 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, σh 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 σh plane.
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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 followed by reflection in a plane
perpendicular to C4 axis gives an indistinguishable structure.
Example: SiF4
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Fig 3.2 S4 axis of symmetry
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4. References
1. “Inorganic Chemistry: Principles of Structure and Reactivity”, James
E.Huheey, Ellen A.Keiter, Richard L.Keiter, Okhil K.Medhi, Pearson
Education, Delhi, 2006
2. ‘Chemical Applications of Group Theory”, 2/e, F.Albert Cotton, Wiley
Eastern, New Delhi, 1986
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