group theory (theory) _ inorganic chemistry virtual lab _ chemical sciences

8/12/2019 Group Theory (Theory) _ Inorganic Chemistry Virtual Lab _ Chemical Sciences

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Group Theory

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

Hermann-Mauguin symbol

Centre of Symmetry or

Inversion centre

inversion I ī

Plane of symmetry Reflection θ σ m

Axis of symmetry Rotation through Cn

n

Improper axis Rotation followed by reflection in a

plane perpendicular to axis

Sn

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:

N2F

2, PtCl

4, C

2H

6

1,2-di chloro-1,2-di 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

2O, N

2F

2,C

2H

4

The broken line in the σ- plane. If we look from left side (A) into the mirror plane, HA

appears to have gone on the other side and its image

p Theory (Theory) : Inorganic Chemistry Virtual Lab : Chemical Sc... http://amrita.vlab.co.in/?sub=2&brch=193&sim=10

6 05-Aug-14



For example ethane molecule (staggered form).

Configuration I and II are not equivalent i.e., θ = 600

and the consequence C6 – 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 C6 rotational axis is also not a

genuine symmetry operation. But the configurations I and III are equivalent, so that C6

followed by σ perpendicular to C6

is a genuine,

through the combined operation this product operation results in an element called S6

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 3600.

Every molecule has this element of symmetry and it co-exists 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 Hermann-Mauguin.Crystal and space groups are named after Hermann-Mauguin symbolism.

H2O and pyridine are assigned the point group symbol-C

2vwhich means the molecules contain a C

2axis 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 C1

point group.

The TeCl

2Br

2molecules with its structure in gaseous phase belongs to C

1point 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 Cn

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 Cn,

Dn,

and Sn.

Cn

type point group:

Cn

point groups:

The molecules which contain only one Cn, proper axis are considered. The presence of C

n implies the presence of (n-1) distinct symmetry


6 05-Aug-14



elements whether n is even or odd. Since Cn

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 Cn

point groups.

Cnv

point groups:

This group contains a Cn

axis and n σv

planes of symmetry. When n is odd, all the planes are σv

type only, and if n is even, there are n/2

planes of σv

type and another n/2 planes of σv

’type

Cnh

point groups:

This set of point group can by adding a horizontal plane (σh) to a proper rotational axis, C

n. This group has a total of 2n elements –n

elements from Cn

and other n e lements can be generated by a combination of Cn

and σh, leading to the corresponding S

n axes. When n is

even, Cnh

point group molecules necessarily contains a centre of inversion, i.

S-trans-1,3-Butadiene - C2h

Boric acid - C3h

Dn

type point groups:

Dn

point groups:

These are purely rotational groups that are they contain only rotational axis of symmetry. When the molecule containing only one type Cn

axis, it was classified as Cn

point group. if in addition to one the Cn

axis, a set of n Cn

axes perpendicular to Cn

are added, it belongs to

another point group called Dn

point group. The order, h, of this rotational group is 2n, since Cn

generates (n-1)+E elements and the number

of C2s

are n more.

For example gauche or skew form of ethane contains D3 point group.

Biphenyl (skew) - D2


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Dnh

point groups:

This point group can be obtained by adding a horizontal (σh) plane to a set of D

ngroup elements. The order of this D

nhgroup is 4n. In

addition to the n elements of Cn

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 Sn

axis.

Example is - B2H

6- D

2h

Dnd

point group:

This point group can be obtained by adding a set of dihedral planes (nσd) to a set of D

ngroup elements. This would thus require that there is

a Cn

proper axis along with nC2

s perpendicular to Cn

axis and nσd

planes, constituting a total of 3n elements thus far.

Example is - Cyclohexane (chair form) - D3d

Sn

type point groups:

Sn

axis is the only group generator for the Sn

(n= even) point group of molecules. The point groups Cnh

, Dnh

, and Dnd

.when n is odd, the

presence of Sn

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 Cn

or Sn

axis and other additional elements would lead to the other point group such as Cnh

, Dnh

,or Dnd

when n is even and there is no plane perpendicular (σh

) to this axis, the presence of other elements in addition to Sn

axis leads to only Dnd

point group.

Example is - SiO4(CH

3)4

- S4

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

C1

1 E (=C1)

C1

2 E, I (=S2

)

C1

2 E, σ

Cn

– groups: ( h = n )

C2

2 E, C2

C3

3E C3

1

, C3

2

C4

4 E, C4

1, C

4

2(=C

2), C

4

3

C5

5 E, C4

1, C

4

2,C

4

3, C

4

4

Cnv

– groups: ( h = 2n )

C2v

4 E, C2

, σ

C3v

6 E,C3

1, C

3

2, 3σ

v

C4v

8 E,C4

1,C

4

2(=C

2), C

4

3, 2σ

v, 2σ

v

’

Cnh


C2h

4 E, C2

,i=( S2

), σh

C3h

6 E, C3

1, C

3

2, S

3

1, S

3

5, σ

h


6 05-Aug-14



Point

group

Order of

group, h

Type of symmetry elements

C4h

8 E,C4

1,C

4

2(=C

2), C

4

3, S

4

1, S

4

3,σ

h, i=(

S2

)

Dn


D2

4 E, C2, C

2

’

D3

6 E, C3

1, C

3

2, 3C

2

D4

8 E,2C4, C

2, 4C

2

Dnh


D2h 8 E, C2 , 2C2’ , i=( S2 ), σh , 2σv

D3h

12 E, 2C3

,3C2

, σh

,3σv

, 2S3,

(S3

1, S

3

5)

D4h

16 E, 2C4,( C

4

1,C

4

2), C

2=( C

4

2), 2C

2

’

,2C2

” , σ

h,2σ

v,3σ

d, i , 2S

4(S

4

1, S

4

3)

Dnd


D2d

8 E, C2

, 2C2

’ , 2 σ

d, 2S

4

D3d

12 E, 2C3

(C3

1, C

3

2), 3C

2,i, 3σ

d, 2S

6(S

6

1,

S6

3)

D4d

16 E, 2C4,

(C4

1, C

4

3), C

2=( C

4

2), 4C

2

’

,4σd, 4S

8(S

8

1, S

8

3, S

8

5, S

8

7)

Sn

(n=even)– groups: ( h = n )

S4

4 E, S4

1, S

4

3, C

2

S6

6 E, S6

1, S

4

5, C

3

1, C

3

2, i

S8

8 E, S8

,S8

1, S

8

3, S

8

5, S

8

7, C

4

1, C

4

3,

C2=( C

4

2)

Infinite- point group (h=∞)

C∞v

∞ E, ∞, C∞

, ∞σv

D∞v

∞ E, ∞, C∞

, ∞σv, σ

h, i

Molecules of special Symmetry:

This class has two groups of molecules:

Linear or infinite groups andGroups which contain multiple higher-order axes.

Linear or infinite groups:

In addition to all the linear molecules, circle-shaped and cone-shaped ones also belong to this category. These can be further sub-divided

into two groups, C∞v

and D∞v

groups, the presence or absence of i used to distinguish between these two types of groups.

C∞v

point group:

This group can be defined the same way as that of Cnv

group, where n is infinity. The C∞

axis lies along the inter nuclear molecules, and

since the molecule is linear the σv

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.

D∞v

point group:

This group is an extension of Dnh

group (∞). This group of molecules contain a C∞

axis, ∞C2

axes perpendicular to C∞

axis and a σh

plane.

Then, it would also imply that the molecule possess ∞σv

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 N2, O

2, H

2, F

2and Cl

2etc.

Molecules Containing Multiple Higher-Order 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

Cn

axis (n-highest), as in Dn

and related point groups, nor bear any perpendicular relationship. These high-symmetry molecules have shapes


6 05-Aug-14



corresponding to the five platonic solids: tetrahedral, octahedral, cube, dodecahedral and icosahedra.

Tetrahedral Point Groups:

The highest-fold axis in these point groups is C3

axis, which is occur in multiples. Molecules with only C3

axes and additionally only C2

axes

belong to T, a pure rotational point group, since they contain only proper rotational axes. All other type of elements (σv

,i, Sn) are absent in

three groups.

T: 8C3

(4C3

1, 4C

3

2), 3C

2, E

Si(CH3)4

When σd, S

4(collinear with C

2axes) elements are added to the T group elements, we get a full group called T

d.The order of this group is 24.

Td: 8C

3(4C

3

1, 4C

3

2), 3C

2, E, 6S

4(S

4

1, S

4

3), 6 σ

d

CCl4

There is another uncommon point group, Th, which can be obtained by adding three planes of symmetry (σ

h) to T group. The order of this is

group is 24.

Th :8C

3(4C

3

1, 4C

3

2), 3C

2, i, 3σ

h, 8S

6(4S

6

1, 4S

6

5)

Example - Co(NO2)6

3-

Octahedral Point Groups:

This is another class of cubic groups. Additionally, octahedral point groups have multiple C4

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, 6C4

(3C4

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 3σh

and 6 σd

planes are added, a group of higher symmetry can be generated. The order of this group is 48.

Oh

E, 6C4

(3C4

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:


6 05-Aug-14



This group contains molecules with either icosahedral or pentagonal dodecahedral shapes and belongs to Ih

point groups. The molecules

containing only the rotational elements are said to be belongs to I point group. Th order of this point group is 60, whereas that full group is

120.

I E, 24C5

(6C5

1, 6C

5

2, 6C

5

3, 6C

5

4), 20C

3(10C

3

1, 10C

3

2), 15C

2

Ih

E, 24C5, 20C

3, 15C

2,24S

10(6S

10

1, 6S

10

36S

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

li, l

j – Its dimensions

h – Order of a group

Γi(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

δij 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, Γi ≠ Γ

j and j ≠ i, then δ

ij = 0

Therefore, ΣR [ Γ

i(R)

mn ] [ Γ

j(R)'m'n' ]

* = 0

2. When, Γi = Γ

j and j = i, then δ

ij = 1

Therefore, ΣR [ 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, ΣR [ 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.


6 05-Aug-14



i.e., Σli

2 = l

1

2 + l

2

2 + l

3

2 + …… l

n

2 = h

Γi(E) –the character of the representation of E in the ith IR which is equal to the dimension of the representation.

i.e., Σi [ Γ

i(E)]

2 = h

2. The sum of the squares of the characters in any IR is equal to ‘h’.

i.e., ΣR [ Γ

i(R)]

2 = h

3. The vectors whose components are the charactors of two different IR are orthogonal.

i.e., ΣR 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 C3v

point group there are, E, 2C3, 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.1.

Note that the no: of the IR is found out using the theorem.2.

Interrelationships of various group operations are to be carefully followed.3.

Use the orthogonality and the normality theorem in fixing the characters.4.

Generate a representation using certain basic vectors. Try out with X, Y, Z, Rσ, R

y, R

z etc. as the bases and check.5.

Character Table for C2v

Point Group:

1. For C2v

point group, there are 4 symmetry operations, Γ1, Γ

2, Γ

3, Γ

4therefore, it contains 4 classes. i.e., E, C

2z, σ

xz, σ

yz. And character of

E is denoted as l1, l

2, l

3, l

4.

C2v E C

2zσ

xzσ

yz

Γ1

l1

Γ2

l2

Γ3

l3

Γ4

l4

2. The sum of the squares of the dimensions of the symmetry operations = 4.

i.e., l1

2 + l

2

2 + l

3

2 + l

4

2 = h = 4.

This can only be satisfied by four one dimensional representations.

C2v E C

2zσ

xzσ

yz

Γ1 1

Γ2 1Γ3 1

Γ4 1

The unknowns for Γ1 is a

1, b

1, c

1 , for Γ

2 is a

2, b

2, c

2.

C2v E C

2zσ

xzσ

yz

Γ1 1 a

1b

1c1

Γ2 1 a

2b

2c2

Γ3 1 a

3b

3c3

Γ4 1 a

4b

4c4


6 05-Aug-14



3. Sum of the squares of the characters of any IR is equal to the order of the group.

i.e., 12 + a

1

2 + b

1

2 + c

1

2 = 4.

C2v E C

2zσ

xzσ

yz

Γ1 1 1 1 1

Γ2 1 a

2b

2c2

Γ3 1 a

3b

3c3

Γ4 1 a

4b

4c4

4. The orthogonality theorem must be satisfied by all the symmetry operations.

i.e., ΣR Γ

i(R) Γ

j(R) = 0

i.e., for Γ1 . Γ

2

i.e., 1.1 + a1 .1 + b

2 . 1 + c

2 .1 = 0

Let a2 = 1, b

2 = -1 and c

2 = -1

Then Γ1 . Γ

2 = 0

C2v E C

2zσ

xzσ

yz

Γ1 1 1 1 1

Γ2 1 1 -1 -1

Γ3 1 a

3 b

3 c

3

Γ4 1 a4 b4 c4

For Γ3 . Γ

1

i.e., 1.1 + a3 .1 + b

3 . 1 + c

3 .1 = 0

Let a3 = -1, b

3 = 1 and c

3 = -1

Then Γ1 . Γ

2 = 0

C2v E C

2zσ

xzσ

yz

Γ1 1 1 1 1

Γ2 1 1 -1 -1

Γ3 1 -1 1 -1

Γ4 1 a

4b

4c4

For Γ4 . Γ

1

i.e., 1.1 + a4 .1 + b

4 . 1 + c

4 .1 = 0

Let a4 = -1, b

4 = -1 and c

4 = 1

Then Γ1 . Γ

2 = 0

C2v E C

2zσ

xzσ

yz

Γ1 1 1 1 1

Γ2 1 1 -1 -1

Γ3 1 -1 1 -1

Γ4 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 Cn, 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,


16 05-Aug-14



C2v E C

2zσ

xzσ

yz

A1 1 1 1 1

A2 1 1 -1 -1

B3 1 -1 1 -1

B4 1 -1 1 -1

In any character table there are 4 different areas.

Area I – Characters of symmetry operations

Area II – Mullicon SymbolsArea 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 Rx, 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 C2, σ

xz, σ

yz

operations no change will occur. Hence its characters are 1 1 1 1. Therefore the vector ‘Z’ transforms under A1.

Similarly,

The characters are 1 -1 1 -1 corresponding to B1. 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.


16 05-Aug-14



The characters are 1 1 -1 -1. Therefore it will be A2 and it becomes Rz.

The characters are 1 -1 1 -1. Therefore it will be B1 and it becomes R

x. Similarly B

2 become R

y.

Therefore,

C2v

E C2z

σxz

σyz Linear Functions, Rotations

A1 1 1 1 1 Z

A2 1 1 -1 -1 R

z

B1 1 -1 1 -1 X, R

y

B2 1 -1 -1 1 Y, R

x

Area IV:

Which represents the squares and binary products.

A1 = Z = 1 1 1 1

A1

2 = Z

2 = 1 1 1 1 = A

1

B1 = X = 1 -1 1 -1

B1

2 = X

2 = 1 1 1 1 = A

1

B2 = Y = 1 -1 -1 1

B2

2 = Y

2 = 1 1 1 1 = A

1

XY = B1 . B

2 = 1 1 -1 -1 = A

2

XZ = B1 . A

1 = 1 -1 1 -1 = B

1


16 05-Aug-14



YZ = B2 . A

1 = 1 -1 -1 1 = B

2

Therefore the actual character table for C2v

point group will be,

C2v

E C2z

σxz

σyz

Linear Functions,

RotationsQuadratic

A1 1 1 1 1 Z

X2, Y

2,

Z2

A2 1 1 -1 -1 R

z XY

B1 1 -1 1 -1 X, R

y XZ

B2 1 -1 -1 1

Y, Rx YZ

Character Table for C3v

Point Group:

1. For C3v

point group, there are 6 symmetry operations and 3 classes, i.e., Γ1, Γ

2, Γ

3.

2. The sum of the squares of the dimensions of the symmetry operations = 6.

i.e., l1

2 + l

2

2 + l

3

2 = h = 6.

This can only be satisfied by, 2 one dimensional and 1 two dimensional representations.

C3v E 2C

33σ

v

Γ1 1 a

1b

1

Γ2 1

a2

b2

Γ3 2 a

3b

3

3. The sum of the dimensions of Γ1 also 6.

Therefore, its characters are (1 1 1).

C3v E 2C

33σ

v

Γ1 1 1 1

Γ2 1 a

2b

2

Γ3 2 a

3b

3

4. All operations must satisfy the orthogonality condition, ΣR Γ

i (R) Γ

j (R) = 0

i.e., For Γ1 . Γ

2

i.e., 1.1 + 2 . a2 .1 + 3 . b

2 . 1 = 0

Let a2 = 1 and b

2 = -1

Then Γ1 . Γ

2 = 0

C3v E 2C

33σ

v

Γ1 1 1 1

Γ2 1 1 -1

Γ3 2 a

3b

3

i.e., For Γ3 . Γ

2

i.e., 2.1 + 2 . a3 .1 - 3 . b3 . 1 = 0

Let a3 = -1 and b

3 = 0

Then Γ3 . Γ

2 = 0

C3v E 2C

33σ

v

Γ1 1 1 1

Γ2 1 1 -1

Γ3 2 -1 0

For any character table there are 4 areas.

For Area I:


16 05-Aug-14



Assign the Mullicon symbols.

C3v E 2C

33σ

v

A1 1 1 1

A2 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 Rx, R

y and R

z.

Consider,

The characters are 1 1 1 corresponding to A1.

The characters are 1 -1 1, the character corresponding to C3 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 Rx, R

y, R

z.


16 05-Aug-14



The characters are 1 -1 1. Therefore it corresponds to E and it will become Rx.

The characters are 1 1 -1. Therefore it corresponds to A2 and it will become R

z.

Similarly for E the characters are 2 -1 0 and it will become Ry.

C3v E 2C

33σ

v Linear Functions, Rotations

A1 1 1 1 Z

A2 1 1 -1 R

z

E 2 -1 0 (X, Y) (Rx, Ry)

Area IV:

Which represents the squares and binary products.

A1 = Z = 1 1 1

A1

2 = Z

2 = 1 1 1 = A

1

XY = E = 2 -1 0 = E

XZ = E . A1 = 2 -1 0 = E

YZ = E . A1 = 2 -1 0 = E

Therefore the actual character table for C3v

point group will be,

C3v E 2C

33σ

v Linear Functions, Rotations Quadratic

A1 1 1 1 Z Z

2

A2 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:1.

Character Table for Cn Point Groups:2.

Character Table for Cnv

Point Groups:3.

Character Table for Cnh

Point Groups:4.

Character Table for Dn Point Groups:5.

Character Table for Dnh

Point Groups:6.

Character Table for Dnd

Point Groups:7.

Character Table for Sn Point Groups:8.

Character Tables for Higher Point Groups:9.

Character Tables for Linear Point Groups:10.

Developed under a Research grant from NMEICT, MHRD

by

Amrita CREATE (Center for Research in Advanced Technologies for Education),

VALUE (Virtual Amrita Laboratories Universalizing Education)

Amrita University, India 2009 - 2014

http://www.amrita.edu/create


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