1. SYMMETRY OPERATION AND SYMMETRY ELEMENT
Symmetry
Symmetry (from Greek συμμετρία symmetria "agreement
in dimensions, due proportion, arrangement") in everyday language refers
to a sense of harmonious and beautiful proportion and balance. In
mathematics, "symmetry" has a more precise definition, and is usually
used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling.
Symmetry Operations
Some geometrical manipulation on the molecule in such
a way that an ‘equivalent’ or indistinguishable’ product configuration of the
molecule always results after the operation.
Point for remembering for
symmetry operations
·
While
doing a symmetry operation, one has to keep in mind that the ‘center of
gravity’ of the molecule must not be shifted.
·
The
new product configuration obtained after the symmetry operation need not
always be identical to that of the ‘original’, but it is sufficient if it is
equivalent to it.
Types of symmetry operation
There are mainly three kinds of symmetry operations on
molecules depending on their structure, these are
1. Rotation
2. Reflection
3. Inversion
Symmetry elements
The geometrical parts of a molecule through which the
symmetric operations can be applied in molecule are known as symmetric elements
there are five types of symmetric element using on the given three symmetry
operations and also combining some of them. The symmetric elements are
1. Rotational Axis of Symmetry (C0)
2. Plane of Symmetry (σ)
3. Improper Axis of Symmetry (Sn)
4. Inversion Center / Center of Symmetry (i)
5. Identity Elements (E)
Point to remember
All symmetry elements must either pass through or
contain the ‘center of gravity’.
Rotational axis of symmetry-
An axis with a curved arrow about it is called as the
rotational axis and should be written in such a way that it always passes
through the center of gravity of the molecule.
The curved arrow around the rotational axis indicates
the direction or choice of rotation, ‘clock-wise’ or ‘anti clock-wise’ fashion.
Example
1. In H2O molecule the rotational axis passes
through the center of gravity and lying in the plane of molecule and for water
molecule the symmetry angle is 1800. Rotational axis of symmetry is
designated by Cn, where n = 3600/ፀ.
So for water molecule n =2 and the Rotational axis is
designated as C2
2. In NH3 molecule the rotational axis passes
through the center of gravity and lying perpendicular to the plane of paper and
for molecule molecule the symmetry angle is 1200. Rotational axis of
symmetry is designated by Cn, where n = 3600/ፀ.
So for water molecule n =3 and the Rotational axis is
designated as C3.
Types of Rotational axis
A molecule can have more than one symmetry axis; the one with the
highest n is called the principal axis, and by convention
is aligned with the z-axis in a Cartesian coordinate system and the rest rotational axis of symmetry are known as the secondary rotational axis.
Point to remember
·
The secondary
axes are always perpendicular to the principal axis.
·
There are as
many secondary axes as dictated by the order n of the principal axis.
Plane of symmetry
A plane of symmetry is an imaginary plane that bisects a
molecule into halves that are mirror images of each other. It is based
on the phenomenon of reflection.
This is the conceptual mirror plane which when placed in a molecule
passes through its center of gravity and reflects one half of the molecule into
the other half and vice versa and such
Characteristics of mirror plane
1. The mirror plane is thin enough to cut through even
the smallest of the atoms such as hydrogen so that one hemi-sphere of an atom
is reflected into the other and vice versa.
2. The mirror plane possesses both the sides reflecting
capacity so that when it is placed in the molecule it reflects left portion of
the molecule into the portion and vice versa
3. The mirror plane is as large, in its dimensions, as
the size of the molecule so that the reflection of the entire molecule can be
effected.
Note- in some places
the plane of symmetry can be represented by the broken line passing through the
center of gravity of the molecule, and this can be used for showing the plane
bisecting the molecule into two halves.
Types
of planes of symmetry
There are three types of plane of symmetry
1- Vertical plane of symmetry ( σv)
2- Dihedral plane of symmetry ( σd)
3- Horizontal plane of symmetry ( σh)
Note- when the
molecule contains only one plane and in the absence any other symmetry
elements, the above classification is redundant, and the plane is called a ‘simple
plane’.
Vertical
plane of symmetry ( σv)
The plane which is parallel to the principal axis.
Examples
In the above examples both the planes, plane 1 and
plane are parallel to the principal axis C2 so the planes are
vertical plane of symmetry.
Horizontal plane of symmetry (
h)
The plane which is perpendicular to principal axis.
Examples in BF3 molecule
for more detail related to plane vertical and horizontal plane of symmetry kindly click here
Dihedral plane of symmetry
1- In the staggered molecule, the σv that additionally bisects the angle between two C2
– axes is called dihedral POS σ
example –
staggered ethane having 1C3 along C-----C so it have 3C2 or 0C2 which are represented
in its following diagram
In planer and
pyramidal molecule the σv that passes through the less no. of atom is known as
σd .
Example in PtCl4
In PtCl4 ,the structure of is planner so
the determination of the dihedral and vertical plane is on the basis of the
second deffination of dihedral plane.
Similarly we can
determine the plane of simmetry in the following compounds, which are mostly asked
in the NET CISR examination
1. BF3
2. Allene
3. H2O
4. NH3
5. H2O2
6. PCl5
7. Regular pentagon
8. Benzene
for more detail of dihedral plane of symmetry kindly click here
Improper axis of symmetry or
rotoreflection axis of symmetry
An imaginary axis
passing through the molecule symmetrically rotation about which by some angle ө
i.e. if ratation is about the
x-axis, than reflection will take place in yz-plane.
Or for ratation
about Z-axis
Sn(z)
= Cn(z) followed by σxy
Or in multiplication form
Sn(z)
= σ
where Cn(z) is first operator and σ
e.g.
1. methane (CH4 )
CH4 has S4 Improper axis of
symmetry
So by the formula
Sn(z) = Cn(z)
followed by σ
S4(z) = C4(z)
followed by σxy
For n = 4 by the formula
= 3600/n
So θ = 900
Note – in case of methane the 900 ratation
is not equivalent
So the C4 axis is absecne in methane.
2. PCl5 – this molecule has S3
improper axis of symmetry.
So
= 1200
N
E
T
examination example
The improper axis S6 is present
in
i.
B2H6
ii.
CH4
iii.
PH5
iv.
SF6
For more about improper axis of symmetry and for the solution of the given question click here
Center
of Symmetry or Inversion Center
An
imaginary point in the molecule which invert all the atoms of molecule in such
a way that resulting structure is and equivalent
e.g. 
N
E
T
examination example
The center of symmetry is present in
1.
ClF3
2.
H2O
3.
BF3
4.
Cis H2O2
5.
Trans H2O2
6.
SF6
7.
PCl5
For the
solution of the given question click here
Identity element
The operation
applied on a molecule which retain its identiy known as the identity operation
and this only possible in case of 00 and 3600 rotation.
Since for
= 3600
n = 3600 /3600
= 1
so for the
identity element the operation C 1 is applicable.
for video related to identity element and summery of the whole chapter click here
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