Objective:
 To familiarise the 3D geometry of various molecules.
 To determine the point groups.
Introduction:
The symmetry relationships in the molecular structure provide the basis for a mathematical theory, called group theory. The mathematics of group theory is predominantly algebra. Since all molecules are certain geometrical entities, the group theory dealing with such molecules is also called as the ‘algebra of geometry’.
Symmetry Element:
A symmetry element is a geometrical entity such as a point, a line or a plane about which an inversion a rotation or a reflection is carried out in order to obtain an equivalent orientation.
Symmetry Operation:
A symmetry operation is a movement such as an inversion about a point, a rotation about a line or a reflection about a plane in order to get an equivalent orientation.
The various symmetry elements and symmetry operations are listed in below table.
Symmetry element

Symmetry operation

Schoenflies symbol

HermannMauguin symbol

Centre of Symmetry or Inversion centre

inversion

I

ī

Plane of symmetry

Reflection θ

σ

m

Axis of symmetry

Rotation through

C_{n}

n

Improper axis

Rotation followed by reflection in a plane perpendicular to axis

S_{n}

ñ

Identity element

Identity Operation

E


Centre of symmetry:
A point in the molecule from which lines drawn to opposite directions will meet similar points at exactly same distance. Some of the molecules, which have a centre of symmetry, are:
N_{2}F_{2,} PtCl_{4}, C_{2}H_{6}
_{ }
1,2di chloro1,2di bromoethane(all trans and staggered)_{}
Plane of symmetry:
A plane which divides the molecule into two equal halves such that one half is the exact mirror image of the other half. The molecules, which have plane of symmetry, are:
H_{2}O, N_{2}F_{2, }C_{2}H_{4 }
The broken line in the σ plane. If we look from left side (A) into the mirror plane, H_{A }appears to have gone on the other side and its image appears exactly at H_{B}. Similarly viewing the structure of H_{2}O molecule from the right side (B), the reflection of H_{B }appears at H_{A }configuration II is the result of this reflection operation and is equivalent to I. Another round of this operation on the molecule (configuration II) yields configuration III which is identical to configuration I.
Proper Axis of symmetry:
An axis passing through the molecule about which when the molecule is rotated 360/n an equivalent orientation is produced. This is an axis of nfold symmetry or an axis of order as shown below.
H_{2}O, N_{2}H_{3, }BF_{3}
Initially, H_{2}O molecule is in configuration I, lying flat on the plane of the paper , and after rotating it through an angle θ = 180^{0} about an axis passing through O atom (Zaxis) and having HOH angle, the configuration II will be obtained. The configuration II is equivalent to configuration I, but not identical. By another similar rotation about Zaxis on configuration II, the molecule goes into configuration III. Here configuration III is identical to the initial or original configuration I.
Principle Axis:
If there is more than one axis of symmetry, in many cases one of the axes is identified as principal axis, which will be selected in the following order:
 The only axis
 The highest order axis
 The axis passing through maximum number of atoms
 The axis perpendicular to the plane of the molecule
The principal axis is taken as the vertical axis that is in the zdirection. The subsidiary axis is perpendicular to the principal axis and will, hence, be in the horizontal direction.
Molecule

Principal axis

Subsidiary axis

Water

C_{2}

Nil

Ammonia

C_{3}

Nil

BF_{3}

C_{3}

3C_{2}

XeF_{4}

C_{4}

4C_{2}

Cyclopentadienyl anion

C_{5}

5C_{2}

Benzene

C_{6}

6C_{2}

H_{2}O_{2}

C_{2}

Nil

XeOF_{4}

C_{4}

Nill

(a C_{n } axis can combine with only n C_{2} axis perpendicular to it or with no subsidiary axis.)
The plane of symmetry is also classified on the basis of the principal axis. The planes including or involving the principal axis are called vertical planes (σ_{v}) and the planes perpendicular to the principal axis is called horizontal plane (σ_{h})_{.}
Molecule

Principal axis

Vertical planes

Horizontal planes

Water

C_{2}

Nil

Nil

Ammonia

C_{3}

Nil

One

BF_{3}

C_{3}

2

Nil

XeF_{4}

C_{4}

3

Nil

Cyclopentadienyl anion

C_{5}

3

One

Benzene

C_{6}

4

One

H_{2}O_{2}

C_{2}

6

One

XeOF_{4}

C_{4}

4

Nil

Improper Axis of symmetry:
An axis passing through the object about which when the object is rotated through 360/n followed by reflection in a plane perpendicular to the axis produces an equivalent orientation.
For example ethane molecule (staggered form).
Configuration I and II are not equivalent i.e., θ = 60^{0 }and the consequence C_{6} – rotational operation is not a valid symmetry operation by itself. Similarly, II and III are not equivalent, thus showing that σ operation perpendicular to the so called C_{6} rotational axis is also not a genuine symmetry operation. But the configurations I and III are equivalent, so that C_{6 }followed by σ perpendicular to C_{6 }is a genuine, through the combined operation this product operation results in an element called S_{6 }axis.
Identity Element:
This element is obtained by an operation called identity operation. After this operation, the molecule remains as such. This situation can be visualized by two ways. Either
 We do not do anything on the molecule or
 We rotate the molecule by 360^{0}.
Every molecule has this element of symmetry and it coexists with the identity of the molecule, hence the name identity element.
Point group:
The symmetry elements can combine only in a limited number of ways and these combinations are called the point groups.
Nomenclature of the point group:
 There are certain conventions developed by two schools of thought for naming these point groups.
 The Schoenflies nomenclature is popularly used molecular point groups than that of HermannMauguin.
 Crystal and space groups are named after HermannMauguin symbolism.
 H_{2}O and pyridine are assigned the point group symbolC_{2v }which means the molecules contain a C_{2 }axis and 2 σ_{v} planes.
Identification of molecular point groups:
The whole molecules are divided into three broad categories.
 Molecules of low symmetry (MLS).
 Molecules of high symmetry (MHS).
 Molecules of special symmetry (MSS).
Molecules of Low Symmetry (MLS):
The starting point could be the molecules containing no symmetry elements other than E, such molecules are unsymmetrically substituted and these molecules are said to be belongs to C_{1 }point group.
The TeCl_{2}Br_{2 }molecules with its structure in gaseous phase belongs to C_{1 }point group, and tetrahedral carbon and silicon compounds of the formula AHFClBr (A=C,Si).
Molecules of High Symmetry (MHS):
In this category all the molecules containing C_{n }axis (invariably in the absence or presence of several other types of symmetry of elements) are considered. There are three main types of point groups C_{n,} D_{n,} and S_{n}.
C_{n }type point group:
The molecules which contain only one C_{n},_{ }proper axis are considered. The presence of C_{n} implies the presence of (n1) distinct symmetry elements whether n is even or odd. Since C_{n }generates a set of n elements including E, the order of this group is n, (h=n) the molecules belonging this group are designated as C_{n }point groups.
This group contains a C_{n }axis and n σ_{v }planes of symmetry. When n is odd, all the planes are σ_{v }type only, and if n is even, there are n/2 planes of σ_{v }type and another n/2 planes of σ_{v}^{’ }type
This set of point group can by adding a horizontal plane (σ_{h}) to a proper rotational axis, C_{n}. This group has a total of 2n elements –n elements from C_{n }and other n elements can be generated by a combination of C_{n }and σ_{h}, leading to the corresponding S_{n} axes. When n is even, C_{nh }point group molecules necessarily contains a centre of inversion, i.
Strans1,3Butadiene  C_{2h }Boric acid  C_{3h}
D_{n} type point groups:
These are purely rotational groups that are they contain only rotational axis of symmetry. When the molecule containing only one type C_{n }axis, it was classified as C_{n }point group. if in addition to one the C_{n } axis, a set of n C_{n } axes perpendicular to C_{n } are added, it belongs to another point group called D_{n }point group. The order, h, of this rotational group is 2n, since C_{n }generates (n1)+E elements and the number of C_{2}_{s} are n more.
For example gauche or skew form of ethane contains D_{3} point group.
Biphenyl (skew)  D_{2}
This point group can be obtained by adding a horizontal (σ_{h}) plane to a set of D_{n }group elements. The order of this D_{nh }group is 4n. In addition to the n elements of C_{n }when n is even, the elements generated are quite distinct and different from what has already been obtained. However when n is odd, we get set of n elements based on S_{n }axis.
Example is  B_{2}H_{6 } D_{2h}
This point group can be obtained by adding a set of dihedral planes (nσ_{d}) to a set of D_{n }group elements. This would thus require that there is a C_{n }proper axis along with nC_{2 }s perpendicular to C_{n }axis and nσ_{d }planes, constituting a total of 3n elements thus far.
Example is  Cyclohexane (chair form)  D_{3d}
S_{n} type point groups:
S_{n }axis is the only group generator for the S_{n }(n= even) point group of molecules. The point groups C_{nh}, D_{nh}, and D_{nd} .when n is odd, the presence of S_{n }axis implies the presence of 2n elements, in which a plane of symmetry (σ) makes an independent appearance. Thus the presence of a plane perpendicular to C_{n }or S_{n }axis and other additional elements would lead to the other point group such as C_{nh }, D_{nh },or D_{nd }when n is even and there is no plane perpendicular (σ_{h }) to this axis, the presence of other elements in addition to S_{n }axis leads to only D_{nd }point group.
Example is  SiO_{4}(CH_{3})_{4 } S_{4}
Point Groups and their Detailed List of Symmetry Elements are Included in the Below Table.
Point group

Order of group, h

Type of symmetry elements

C_{1}

1

E (=C_{1})

C_{1}

2

E, I (=S_{2 })

C_{1}

2

E, σ

C_{n }– groups: ( h = n )

C_{2}

2

E, C_{2}

C_{3}

3

E C_{3}^{1}, C_{3}^{2}

C_{4}

4

E, C_{4}^{1}, C_{4}^{2 }(=C_{2}), C_{4}^{3}

C_{5}

5

E, C_{4}^{1}, C_{4}^{2 },C_{4}^{3}, C_{4}^{4}

C_{nv }– groups: ( h = 2n )

C_{2v}

4

E, C_{2 }, σ

C_{3v}

6

E,C_{3}^{1}, C_{3}^{2 }, 3σ_{v}

C_{4v}

8

E,C_{4}^{1},C_{4}^{2}(=C_{2}), C_{4}^{3}, 2σ_{v}, 2σ_{v}^{’}

C_{nh }– groups: ( h = 2n )

C_{2h}

4

E, C_{2 },i=( S_{2 }), σ_{h}

C_{3h}

6

E, C_{3}^{1}, C_{3}^{2 }, S_{3}^{1}, S_{3}^{5}, σ_{h}

C_{4h}

8

E,C_{4}^{1},C_{4}^{2}(=C_{2}), C_{4}^{3}, S_{4}^{1}, S_{4}^{3},σ_{h}, i=( S_{2 })

D_{n }– groups: ( h = 2n )

D_{2}

4

E, C_{2}, C_{2}^{’}

D_{3}

6

E, C_{3}^{1}, C_{3}^{2}, 3C_{2}

D_{4}

8

E,2C_{4}, C_{2}, 4C_{2}

D_{nh }– groups: ( h = 4n )

D_{2h}

8

E, C_{2 }, 2C_{2}^{’ }, i=( S_{2 }), σ_{h }, 2σ_{v}

D_{3h}

12

E, 2C_{3 },3C_{2 }, σ_{h },3σ_{v }, 2S_{3},
(S_{3}^{1}, S_{3}^{5})

D_{4h}

16

E, 2C_{4},( C_{4}^{1},C_{4}^{2 }), C_{2}=( C_{4}^{2}), 2C_{2}^{’ },2C_{2}^{”}, σ_{h },2σ_{v },3σ_{d}, i , 2S_{4 }(S_{4}^{1}, S_{4}^{3 })

D_{nd}– groups: ( h = 4n )

D_{2d}

8

E, C_{2 }, 2C_{2}^{’}, 2 σ_{d}, 2S_{4}

D_{3d}

12

E, 2C_{3 }(C_{3}^{1}, C_{3}^{2}), 3C_{2 },i, 3σ_{d}, 2S_{6}(S_{6}^{1}, S_{6}^{3 })

D_{4d}

16

E, 2C_{4, }(C_{4}^{1}, C_{4}^{3}), C_{2}=( C_{4}^{2}), _{ }4C_{2}^{’ },4σ_{d}, 4S_{8}(S_{8}^{1}, S_{8}^{3 }, S_{8}^{5}, S_{8}^{7})

S_{n }(n=even)– groups: ( h = n )

S_{4}

4

E, S_{4}^{1}, S_{4}^{3 }, C_{2}

S_{6}

6

E, S_{6}^{1}, S_{4}^{5},^{ }C_{3}^{1}, C_{3}^{2}, i

S_{8}

8

E, S_{8 },S_{8}^{1}, S_{8}^{3 }, S_{8}^{5}, S_{8}^{7}, C_{4}^{1}, C_{4}^{3}, C_{2}=( C_{4}^{2})

Infinite point group (h=∞)

C_{∞v}

∞

E, ∞, C_{∞}, ∞σ_{v}

D_{∞v}

∞

E, ∞, C_{∞}, ∞σ_{v}, σ_{h}, i

Molecules of special Symmetry:
This class has two groups of molecules:
 Linear or infinite groups and
 Groups which contain multiple higherorder axes.

Linear or infinite groups:
In addition to all the linear molecules, circleshaped and coneshaped ones also belong to this category. These can be further subdivided into two groups, C_{∞v} and D_{∞v} groups, the presence or absence of i used to distinguish between these two types of groups.
This group can be defined the same way as that of C_{nv} group, where n is infinity. The C_{∞ }axis lies along the inter nuclear molecules, and since the molecule is linear the σ_{v }planes are infinite in number. The order of this group is h = ∞. All hetero nuclear molecules, and all unsymmetrically substituted linear polyatomic molecules are belongs to this point group.
Examples are HX (X = F, Cl, Br, I), CO, NO, CN etc.
This group is an extension of D_{nh }group (∞). This group of molecules contain a C_{∞ }axis, ∞C_{2 }axes perpendicular to C_{∞ }axis and a σ_{h }plane. Then, it would also imply that the molecule possess ∞σ_{v }planes and a centre of inversion(i). So all centre of symmetric molecules are belongs to this point group.
Homo nuclear diatomic molecules such as N_{2}, O_{2 }, H_{2 }, F_{2 } and Cl_{2 }etc.
Molecules Containing Multiple HigherOrder Axes:
This is a special class of molecules which contain more than one type of rotational axes (n≥2) that are neither perpendicular to the principal C_{n }axis (nhighest), as in D_{n }and related point groups, nor bear any perpendicular relationship. These highsymmetry molecules have shapes corresponding to the five platonic solids: tetrahedral, octahedral, cube, dodecahedral and icosahedra.
Tetrahedral Point Groups:
The highestfold axis in these point groups is C_{3 }axis, which is occur in multiples. Molecules with only C_{3 }axes and additionally only C_{2 }axes belong to T, a pure rotational point group, since they contain only proper rotational axes. All other type of elements (σ_{v },i, S_{n}) are absent in three groups.
T: 8C_{3 }(4C_{3}^{1}, 4C_{3}^{2}), 3C_{ 2}, E
Si(CH_{3})_{4}
When σ_{d}, S_{4 }(collinear with C_{2 }axes) elements are added to the T group elements, we get a full group called T_{d. }The order of this group is 24.
T_{d}: 8C_{3 }(4C_{3}^{1}, 4C_{3}^{2}), 3C_{ 2}, E, 6S_{4 }(S_{4}^{1}, S_{4}^{3}), 6 σ_{d}
CCl_{4}
There is another uncommon point group, T_{h}, which can be obtained by adding three planes of symmetry (σ_{h}) to T group. The order of this is group is 24.
T_{h} :8C_{3 }(4C_{3}^{1}, 4C_{3}^{2}), 3C_{ 2}, i, 3σ_{h}, 8S_{6 }(4S_{6}^{1}, 4S_{6}^{5})
Example  Co(NO_{2})_{6}^{3}
Octahedral Point Groups:
This is another class of cubic groups. Additionally, octahedral point groups have multiple C_{4 }axes when compared to that of tetrahedral groups.
When the group contains only rotational axes, it is labelled as O group, h, of this group are 24.
O: E, 6C_{4 }(3C_{4}^{1}, 3C_{4}^{2}), 8C_{3 }(4C_{3}^{1}, 4C_{3}^{2}), 6C_{ 2}, 3C_{ 2 }‘=3C_{4}^{2}
To the O group elements, if 3σ_{h }and 6 σ_{d }planes are added, a group of higher symmetry can be generated. The order of this group is 48.
O_{h }E, 6C_{4 }(3C_{4}^{1}, 3C_{4}^{2}), 3C_{ 2 }‘=3C_{4}^{2}, 6C_{ 2}, 8C_{3 }(4C_{3}^{1}, 4C_{3}^{2}), i, 3σ_{h}, 6σ_{d}, 6S_{4 }(S_{4}^{1}, S_{4}^{3}), 8S_{6 }(4S_{6}^{1}, 4S_{6}^{5})
Cubane
Icosahedral Groups:
This group contains molecules with either icosahedral or pentagonal dodecahedral shapes and belongs to I_{h }point groups. The molecules containing only the rotational elements are said to be belongs to I point group. The order of this point group is 60, whereas that full group is 120.
I E, 24C_{5 }(6C_{5}^{1}, 6C_{5}^{2}, 6C_{5}^{3}, 6C_{5}^{4}), 20C_{3 }(10C_{3}^{1}, 10C_{3}^{2}), 15C_{ 2}
I_{h } E, 24C_{5}, 20C_{3}, 15C_{ 2},24S_{10 }(6S_{10}^{1}, 6S_{10}^{3}6S_{10}^{7},6S_{10}^{9}), 20S_{6 }(10S_{6}^{1}, 10S_{6}^{5}), i, 15σ
Fullerene
Great Orthogonality Theorem:
The matrices of the different Irreducible Representations (IR) possess certain well defined interrelationships and properties. Orthogonality theorem is concerned with the elements of the matrices which constitute the IR of a group.
The mathematical statement of this theorem is,
Where,
i, j – Irreducible Representations
l_{i}, l_{j} – Its dimensions
h – Order of a group
Γ_{i}(R)_{mn} – Element of m^{th} row, n^{th} column of an i^{th} representation
Γ_{j}(R)'m'n'  Element of m' ^{th} row, n' ^{th} column of j' ^{th} representation
δ_{ij} δ_{mm'} δ_{nn'} – Kronecker delta
Kronecker delta can have values 0 and 1. Depending on that the main theorem can be made into three similar equations.
i.e.,
1. When, Γ_{i} ≠ Γ_{j} and j ≠ i, then δ_{ij} = 0
Therefore, Σ_{R} [ Γ_{i}(R)_{mn} ] [ Γ_{j}(R)'m'n' ]^{*} = 0
2. When, Γ_{i} = Γ_{j} and j = i, then δ_{ij} = 1
Therefore, Σ_{R} [ Γ_{i}(R)_{mn} ] [ Γ_{i}(R)'m'n' ]^{*} = 0
From these two equations we can say the Orthogonality theorem as, “the sum of the product of the irreducible representation is equal to zero”.
3. When i = j, m = m', n = n'
Then, Σ_{R} [ Γ_{i}(R)_{mn} ] [ Γ_{i}(R) mn]^{*} =
From the above equations some important rules of the irreducible representations of a group and there character were obtained.
Five Rules Obtained:
1. The sum of the squares of the dimensions of the representation = the order (h) of the group.
i.e., Σ_{li}^{2} = l_{1}^{2} + l_{2}^{2} + l_{3}^{2} + …… l_{n}^{2} = h
Γ_{i}(E) –the character of the representation of E in the ith IR which is equal to the dimension of the representation.
i.e., Σ_{i} [ Γ_{i}(E)]^{2} = h
2. The sum of the squares of the characters in any IR is equal to ‘h’.
i.e., Σ_{R} [ Γ_{i}(R)]^{2} = h
3. The vectors whose components are the characters of two different IR are orthogonal.
i.e., Σ_{R} Γ_{i}(R) Γ_{j}(R) = 0 when i ≠ j.
4. In a given representation (reducible/irreducible) the characters of all matrices belonging to operations in the same class are identical.
Eg: in C_{3v} point group there are, E, 2C_{3}, 3 σ_{v}. there characters are same for a particular IR.
5. No: of irreducible representation in a group = No: of classes in a group.
Applications:
Applying these 5 rules we can develop the character table for various point groups. For most chemical applications, it is sufficient to know only the characters of the each of the symmetry classes of a group.
Steps for The Construction of A Character Table::
 Write down all the symmetry operations of the point group and group them into classes.
 Note that the no: of the IR is found out using the theorem.
 Interrelationships of various group operations are to be carefully followed.
 Use the orthogonality and the normality theorem in fixing the characters.
 Generate a representation using certain basic vectors. Try out with X, Y, Z, R_{σ}, R_{y}, R_{z} etc. as the bases and check.
Character Table for C_{2v} Point Group:
1. For C_{2v} point group, there are 4 symmetry operations, Γ_{1}, Γ_{2}, Γ_{3}, Γ_{4 }therefore, it contains 4 classes. i.e., E, C_{2z}, σ_{xz}, σ_{yz}. And character of E is denoted as l_{1}, l_{2}, l_{3}, l_{4}.
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
l_{1} 



Γ_{2} 
l_{2} 



Γ_{3} 
l_{3} 



Γ_{4} 
l_{4} 



2. The sum of the squares of the dimensions of the symmetry operations = 4.
i.e., l_{1}^{2} + l_{2}^{2} + l_{3}^{2} + l_{4}^{2} = h = 4.
This can only be satisfied by four one dimensional representations.
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 



Γ_{2} 
1 



Γ_{3} 
1 



Γ_{4} 
1 



The unknowns for Γ_{1} is a_{1}, b_{1}, c_{1} , for Γ_{2} is a_{2}, b_{2}, c_{2}.
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 
a_{1} 
b_{1} 
c_{1} 
Γ_{2} 
1 
a_{2} 
b_{2} 
c_{2} 
Γ_{3} 
1 
a_{3} 
b_{3 } 
c_{3} 
Γ_{4} 
1 
a_{4} 
b_{4} 
c_{4} 
3. Sum of the squares of the characters of any IR is equal to the order of the group.
i.e., 1^{2} + a_{1}^{2} + b_{1}^{2} + c_{1}^{2} = 4.
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 
1 
1 
1 
Γ_{2} 
1 
a_{2} 
b_{2} 
c_{2} 
Γ_{3} 
1 
a_{3} 
b_{3 } 
c_{3} 
Γ_{4} 
1 
a_{4} 
b_{4} 
c_{4} 
4. The orthogonality theorem must be satisfied by all the symmetry operations.
i.e., Σ_{R} Γ_{i}(R) Γ_{j}(R) = 0
i.e., for Γ_{1} . Γ_{2}
i.e., 1.1 + a_{1} .1 + b_{2} . 1 + c_{2} .1 = 0
Let a_{2} = 1, b_{2} = 1 and c_{2} = 1
Then Γ_{1} . Γ_{2} = 0
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 
1 
1 
1 
Γ_{2} 
1 
1 
1 
1 
Γ_{3} 
1 
a_{3} 
b_{3 } 
c_{3} 
Γ_{4} 
1 
a_{4} 
b_{4} 
c_{4} 
For Γ_{3} . Γ_{1}
i.e., 1.1 + a_{3} .1 + b_{3} . 1 + c_{3} .1 = 0
Let a_{3} = 1, b_{3} = 1 and c_{3} = 1
Then Γ_{1} . Γ_{2} = 0
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 
1 
1 
1 
Γ_{2} 
1 
1 
1 
1 
Γ_{3} 
1 
1 
1_{ } 
1 
Γ_{4} 
1 
a_{4} 
b_{4} 
c_{4} 
For Γ_{4} . Γ_{1}
i.e., 1.1 + a_{4} .1 + b_{4} . 1 + c_{4} .1 = 0
Let a_{4} = 1, b_{4} = 1 and c_{4} = 1
Then Γ_{1} . Γ_{2} = 0
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Γ_{1} 
1 
1 
1 
1 
Γ_{2} 
1 
1 
1 
1 
Γ_{3} 
1 
1 
1_{ } 
1 
Γ_{4} 
1 
1 
1 
1 
Rules For Assigning Mullicon Symbols:
1. If the IR is unidimensional term A or B is used.
If it is two dimensional E is used.
If it is three dimensional T is used.
2. If one dimensional IR is symmetric with respect to the principle axis C_{n}, i.e., character of C_{n} is +1, the term A is used. If it is 1, the term B is used.
3. If IR is symmetric with respect to subsidiary axes then subscript 1 is given and is antisymmetric then subscript 2 is given.
4. Prime and double prime marks are used for indicating symmetric or antisymmetric with respect to horizontal plane.
5. ‘g’ and ‘u’ subscripts are given for those which are symmetric and antisymmetric respectively with respect to centre of symmetry then,
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
A_{1} 
1 
1 
1 
1 
A_{2} 
1 
1 
1 
1 
B_{3} 
1 
1 
1_{ } 
1 
B_{4} 
1 
1 
1 
1 
In any character table there are 4 different areas.
Area I – Characters of symmetry operations
Area II – Mullicon Symbols
Area III – Cartesion coordinates of rotation axes.
Area IV – Binary Products
Area III:
In order to assign the cartesion coordinates, different operations are performed on each of the axes. Here we find the symbols X, Y, Z represents coordinates and rotations R_{x}, R_{y} and R_{z}.
Consider a vector along with Z axes, the identity doesn’t change the direction of the head of the vector. On doing C_{2}, σ_{xz}, σ_{yz} operations no change will occur. Hence its characters are 1 1 1 1. Therefore the vector ‘Z’ transforms under A_{1}.
Similarly,
The characters are 1 1 1 1 corresponding to B_{1}. And with respect to vector Y, 1 1 1 1 and therefore corresponds to B_{2}. Similar arrangement could be made to rotation axes Rx, Ry, Rz representing rotation about XZ axes. In order to see how they transformed, a curved arrow should be considered around the axes. If the direction of the head of the curved arrow doesn’t change due to operation, the character is +1, otherways it is 1.
The characters are 1 1 1 1. Therefore it will be A_{2} and it becomes Rz.
The characters are 1 1 1 1. Therefore it will be B_{1} and it becomes R_{x}. Similarly B_{2} become R_{y}.
Therefore,
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Linear Functions, Rotations 
A_{1} 
1 
1 
1 
1 
Z 
A_{2} 
1 
1 
1 
1 
R_{z} 
B_{1} 
1 
1 
1 
1 
X, R_{y} 
B_{2} 
1 
1 
1 
1 
Y, R_{x} 
Area IV:
Which represents the squares and binary products.
A_{1} = Z = 1 1 1 1
A_{1}^{2} = Z^{2} = 1 1 1 1 = A_{1}
B_{1} = X = 1 1 1 1
B_{1}^{2} = X^{2} = 1 1 1 1 = A_{1}
B_{2} = Y = 1 1 1 1
B_{2}^{2} = Y^{2} = 1 1 1 1 = A_{1}
XY = B_{1} . B_{2} = 1 1 1 1 = A_{2}
XZ = B_{1} . A_{1} = 1 1 1 1 = B_{1}
YZ = B_{2} . A_{1} = 1 1 1 1 = B_{2}
Therefore the actual character table for C_{2v} point group will be,
C_{2v} 
E 
C_{2z} 
σ_{xz} 
σ_{yz} 
Linear Functions, Rotations 
Quadratic 
A_{1} 
1 
1 
1 
1 
Z 
X^{2}, Y^{2}, Z^{2} 
A_{2} 
1 
1 
1 
1 
R_{z} 
XY 
B_{1} 
1 
1 
1 
1 
X, R_{y} 
XZ 
B_{2} 
1 
1 
1 
1 
Y, R_{x} 
YZ 
Character Table for C_{3v} Point Group:
1. For C_{3v} point group, there are 6 symmetry operations and 3 classes, i.e., Γ_{1}, Γ_{2}, Γ_{3}.
2. The sum of the squares of the dimensions of the symmetry operations = 6.
i.e., l_{1}^{2} + l_{2}^{2} + l_{3}^{2} = h = 6.
This can only be satisfied by, 2 one dimensional and 1 two dimensional representations.
C_{3v} 
E 
2C_{3} 
3σ_{v} 
Γ_{1} 
1 
a_{1} 
b_{1} 
Γ_{2} 
1 
a_{2} 
b_{2} 
Γ_{3} 
2 
a_{3} 
b_{3} 
3. The sum of the dimensions of Γ_{1} also 6.
Therefore, its characters are (1 1 1).
C_{3v} 
E 
2C_{3} 
3σ_{v} 
Γ_{1} 
1 
1 
1 
Γ_{2} 
1 
a_{2} 
b_{2} 
Γ_{3} 
2 
a_{3} 
b_{3} 
4. All operations must satisfy the orthogonality condition, Σ_{R} Γ_{i} (R) Γ_{j} (R) = 0
i.e., For Γ_{1} . Γ_{2}
i.e., 1.1 + 2 . a_{2} .1 + 3 . b_{2} . 1 = 0
Let a_{2} = 1 and b_{2} = 1
Then Γ_{1} . Γ_{2} = 0
C_{3v} 
E 
2C_{3} 
3σ_{v} 
Γ_{1} 
1 
1 
1 
Γ_{2} 
1 
1 
1 
Γ_{3} 
2 
a_{3} 
b_{3} 
i.e., For Γ_{3} . Γ_{2}
i.e., 2.1 + 2 . a_{3} .1  3 . b_{3} . 1 = 0
Let a_{3} = 1 and b_{3} = 0
Then Γ_{3} . Γ_{2} = 0
C_{3v} 
E 
2C_{3} 
3σ_{v} 
Γ_{1} 
1 
1 
1 
Γ_{2} 
1 
1 
1 
Γ_{3} 
2 
1 
0 
For any character table there are 4 areas.
For Area I:
Assign the Mullicon symbols.
C_{3v} 
E 
2C_{3} 
3σ_{v} 
A_{1} 
1 
1 
1 
A_{2} 
1 
1 
1 
B 
2 
1 
0 
For Area III:
In order to assign the Cartesian coordinates different operations are performed on each of the axes. Here we were finding the symbols X, Y, Z represents coordinates and rotations R_{x}, R_{y} and R_{z}.
Consider,
The characters are 1 1 1 corresponding to A_{1}.
The characters are 1 1 1, the character corresponding to C_{3} will be 1. Therefore it will be E. Similarly for vector Y, we get 1 1 1 and this also E.
Similar arrangement could be made to rotation axes R_{x}, R_{y}, R_{z}.
The characters are 1 1 1. Therefore it corresponds to E and it will become R_{x}.
The characters are 1 1 1. Therefore it corresponds to A_{2} and it will become R_{z}.
Similarly for E the characters are 2 1 0 and it will become R_{y}.
C_{3v } 
E 
2C_{3} 
3σ_{v} 
Linear Functions, Rotations 
A_{1} 
1 
1 
1 
Z 
A_{2} 
1 
1 
1 
R_{z} 
E 
2 
1 
0 
(X, Y) (Rx, Ry) 
Area IV:
Which represents the squares and binary products.
A_{1} = Z = 1 1 1
A_{1}^{2} = Z^{2} = 1 1 1 = A_{1}
XY = E = 2 1 0 = E
XZ = E . A_{1} = 2 1 0 = E
YZ = E . A_{1} = 2 1 0 = E
Therefore the actual character table for C_{3v} point group will be,
C_{3v } 
E 
2C_{3} 
3σ_{v} 
Linear Functions, Rotations 
Quadratic 
A_{1} 
1 
1 
1 
Z 
Z^{2} 
A_{2} 
1 
1 
1 
R_{z} 

E 
2 
1 
0 
(X, Y) (Rx, Ry) 
(XY), (XZ), (YZ) 
Some Important Character Tables for Molecular Point Groups:
 Character Table for Non Axial Point Groups:
 Character Table for C_{n} Point Groups:
 Character Table for C_{nv} Point Groups:
 Character Table for C_{nh} Point Groups:
 Character Table for D_{n} Point Groups:
 Character Table for D_{nh} Point Groups:
 Character Table for D_{nd} Point Groups:
 Character Table for S_{n} Point Groups:
 Character Tables for Higher Point Groups:
 Character Tables for Linear Point Groups:
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