Group Theory Nptel
-
Upload
shailendra-agarwal -
Category
Documents
-
view
232 -
download
0
Transcript of Group Theory Nptel
-
7/23/2019 Group Theory Nptel
1/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 1 of 24Joint Initiative of IITs and IISc Funded by MHRD
Irreducible Representations and Character
Tables
K.Sridharan
Dean
School of Chemical & Biotechnology
SASTRA University
Thanjavur 613 401
-
7/23/2019 Group Theory Nptel
2/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 2 of 24Joint Initiative of IITs and IISc Funded by MHRD
Table
of
Contents
1Irreduciblerepresentationandcharactertables......................................................................... 4
1.1 Irreduciblerepresentationandgettingthesymmetriesoftranslationsalongthe
differentaxes............................................................................................................................... 4
1.1.1 Importanceofirreduciblerepresentation................................................................... 4
1.1.2Translationalongthex,yandzaxesandirreduciblerepresentation................................ 4
1.1.2.1Translationalongthexaxis............................................................................................. 4
1.2 Rotationalongthexaxis..................................................................................................... 6
1.2.1Rotationalongthexaxis.................................................................................................... 6
1.3C2vCharacterTable................................................................................................................... 7
2 Identifyingthesymmetriesoftranslationsalongtheaxesinsomeinorganicmolecules......9
2.1 D4hcharactertable........................................................................................................ 10
2.1.1Meaningof(x,y),(xz,yz)and(Rx,Ry)&C4operationonthetranslationalongthexaxis10
2.1.2OperationEonthetranslationalongthexaxis............................................................... 11
2.1.3C2operationonthetranslationalongxaxis.................................................................... 11
2.1.4EffectofC2operationonx andyvectors....................................................................... 12
2.1.5EffectofC2onx andyvectors....................................................................................... 13
2.1.6Effectofoperationionx andyvectors.......................................................................... 14
2.1.7EffectofS4operationonthex andyvectors................................................................. 14
2.1.8Effectofhoperationonx andyvectors....................................................................... 15
2.1.9Effectofvonx andyvectors......................................................................................... 15
2.1.10Effectofdonx andyvectors...................................................................................... 16
2.1.11Identifyingthesymmetryoftranslationaboutxaxis.................................................... 17
3.1Deducingsymmetriesofrotationabouttheaxesfromirreduciblerepresentations.............18
3.1.1Rotationalongthezaxis.................................................................................................. 18
3.1.1.1EffectofEonrotationalongthezaxis..................................................................... 18
3.1.1.2
Effect
ofC
2
on
rotation
about
z
axis
..........................................................................
18
3.1.1.3EffectofC2onrotationaboutzaxis......................................................................... 18
3.1.1.4EffectofC2onrotationaboutzaxis......................................................................... 19
3.1.1.5Effectofionrotationaboutzaxis............................................................................ 19
3.1.1.6EffectofS4onrotationaboutzaxis.......................................................................... 19
-
7/23/2019 Group Theory Nptel
3/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 3 of 24Joint Initiative of IITs and IISc Funded by MHRD
3.1.1.7Effectofvonrotationaboutzaxis......................................................................... 20
3.1.1.8Effectof donrotationaboutzaxis.......................................................................... 20
3.1.1.9Effectof honrotationaboutzaxis.......................................................................... 20
4ApplicationsofIrreducibleRepresentations.............................................................................. 21
5.References................................................................................................................................. 24
-
7/23/2019 Group Theory Nptel
4/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 4 of 24Joint Initiative of IITs and IISc Funded by MHRD
1 Irreducible representation and character tables
1.1 - Irreducible representation and getting the symmetries of
translations along the different axes
An irreducible representation contains characters which cannot be reducedfurther to a simpler form. In other words, this is the simplest representation.of
characters of different symmetry operations.
1.1.1 Importance of irreducible representation
The point group of a molecule consists of a number of symmetry operations.
These symmetry operations constitute a mathematical group. It means that they
exhibit interrelationship as a collection. These mathematical relationships help us
in breaking each group into its irreducible representation.These irreducible representations help us in analyzing molecular properties such
as optical activity, dipole moments and electronic properties such as IR and
Raman spectroscopy, electronic spectroscopy etc. Dynamic properties such as
translation, rotation etc can also be transformed by symmetry operations of the
point group of the molecule.
1.1.2 Translation along the x,y and z-axes and irreducible
representationLet us consider water molecule. It has point group C2v. The symmetry operations
of this point group are E, C2, v(xz), and v(yz). Now we can see how the
translation along the three axes is transformed by these symmetry operations.
1.1.2.1 Translation along the x-axis
Translation is represented by an arrow along the respective axis for the atoms in
the given molecule. Let us consider water molecule as shown in Figure 1.1.2.1.
Z
Y
X
-
7/23/2019 Group Theory Nptel
5/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 5 of 24Joint Initiative of IITs and IISc Funded by MHRD
O
HHx
y
zC2
Fig 1.1.2.1 Water molecule translation along x-axis
Identity operation, E, does not change the directions of arrows. This is called
symmetric and the character is equal to +1
The C2operationchanges the directions of arrows 1800opposite. This is called
antisymmetric and the character is equal to -1
O
H H
The v(xz) operationis not changing the direction of the arrows along the x-axis
and hence the character is equal to +1
The v(yz) operation is not changing the directions of arrows 1800 opposite.
Hence, the character is -1.
Thus the characters of the four symmetry operations can be represented asfollows:
Symmetry operations: E C2 v(xz) v(yz).
Characters: +1 -1 +1 -1From the C2vcharacter table, it is can be seen that this irreducible representation
belongs to B1symmetry.
Similarly, it can be shown that translation along the y-axis represents B2
symmetry and along z-axis represents A1symmetry.
The numbers are called characters. Since these numbers cannot be reduced to
lower values, they are called irreducible representations.
This translation operation holds good for p - orbitals also because they can be
compared to arrows: the lobe with positive sign is similar to the head and the
lobe with negative sign can be compared to the tail of an arrow. Hence, the
-
7/23/2019 Group Theory Nptel
6/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 6 of 24Joint Initiative of IITs and IISc Funded by MHRD
symmetry of a pxorbital will be the same as that of translation along the x-axis,
that of the pyorbital will be the same as that of the translation along the y-axis,
and that of the pzorbital will be the same as that of the pzorbital.
1.2 Rotation along the x-axis
A curved arrow ( ) is taken as the basis vector for rotation to understand
the effect of different operations on it.
1.2.1 Rotation along the x-axisIdentity operation, E, does not change the directions of arrows. This is called
symmetric and the character is equal to +1
The C2operationchanges the directions of arrows 1800opposite. This is called
antisymmetric and the character is equal to -1
The v(xz) operation changes the direction of the curved arrow and hence the
character is equal to -1.
The v(yz) operation does not change the direction of the curved arrow and
hence the character is equal to +1.
-
7/23/2019 Group Theory Nptel
7/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 7 of 24Joint Initiative of IITs and IISc Funded by MHRD
Thus the characters of the four symmetry operations can be represented as
follows:
Symmetry operations: E C2 v(xz) v(yz).
Characters: +1 -1 -1 +1
From the C2vcharacter table, it is can be seen that this irreducible representation
belongs to B2 symmetry. That is, the rotation about the x-axis belongs to B2
symmetry. Similarly, it can be shown that the rotation about the y-axis belongs to
B1symmetry and that about the z-axis belongs to A2symmetry.
1.3 C2v Character Table
C2v E C2 (xz) (yz)A1 1 1 1 1 z x , y , zA2 1 1 -1 -1 Rz xy
B1 1 -1 1 -1 x, Ry xzB2 1 -1 -1 1 y, Rx yz
II I III IV
On the left corner of the character table, the point group is shown. Any character
table has four main areas, I, II, III and IV.
Area Iconsists of the characters of the irreducible representations of the group.
Area II contains the Mulliken symbols. The meanings of 1.these symbols are
given below:
1. Symbols A and B are given to one dimensional representation, E to two
dimensional representation, and T to three dimensional representation.
2. When a one dimensional representation is symmetric with respect torotation by 2/n about the principal Cn axis, i.e., (Cn) = 1, symbol A is
given and B is given, if it is antisymmetric, (Cn) = -1.
-
7/23/2019 Group Theory Nptel
8/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 8 of 24Joint Initiative of IITs and IISc Funded by MHRD
3. Subscripts 1 is attached to A and B, if the operation is symmetric to vor a
C2 perpendicular to the principal axis. and subscript 2 is attached if it is
antisymmetric .
4. Prime is attached to all letters,(A, B, etc.) if the operation is symmetric
with respect to h plane. Double prime is attached (A, B, etc.) if it is
antisymmetric.
5. If a group gas centre of inversion, then subscript g is used if it is
symmetric with respect to inversion and subscript u is used if it is
antisymmetric with respect to inversion.
Area III consists of symbols x, y, z, Rx, Ry, and Rz. These represent the
Cartesian coordinates and the rotations about the three axes. If two symbolsare placed within parentheses, [ex: (x,y), (Rx, Ry)], it means that both put
together form the basis and they cannot be separated.
Area IVcontains the squares and binary products of the coordinates
-
7/23/2019 Group Theory Nptel
9/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 9 of 24Joint Initiative of IITs and IISc Funded by MHRD
2 Identifying the symmetries of translations along the
axes in some inorganic moleculesExample 1: PtCl4
2-
x
y
z
Pt Cl
Cl
Cl
Cl C'2, v
C'2, v
C"2, d
C"2, d
C4
Symmetry elements present:
1. One C4axis perpendicular to the plane of the paper (i.e. molecular plane)
2. Four C2 axes (two along the Pt-Cl bonds, shown as C2and two along the
diagonals shown as C2
3. One hplane, that is the plane of the paper (molecular plane)
4. Two vplanes containing the C2. axes
5. Two d planes containing the C2. axes
Hence, the point group is D4h
-
7/23/2019 Group Theory Nptel
10/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 10 of 24Joint Initiative of IITs and IISc Funded by MHRD
2.1 D4hcharacter table
D4h E 2C4 C2 2C2 2C2 i 2S4 h 2v 2d
A1g 1 1 1 1 1 1 1 1 1 1 x +y , z
A2g 1 1 1 -1 -1 1 1 1 -1 -1 RzB1g 1 -1 1 1 -1 1 -1 1 1 -1 x -y
B2g 1 -1 1 -1 1 1 -1 1 -1 1 xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z
B1u 1 -1 1 1 -1 -1 1 -1 -1 1
B2u 1 -1 1 -1 1 -1 1 -1 1 -1Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)
2.1.1 Meaning of (x,y), (xz,yz) and (Rx,Ry) & C4operation on thetranslation along the x-axis
The symbol (x,y) means that translation along the x- and y-axes are inseparable
in a molecule with D4h symmetry and similarly operations on the the px and py
orbitals. The same explanation holds good for the rotation about x- and y-axes,
and the operations on the dxzand dyzorbitals.
Example:
Translation along the x-axis and along the y-axis are represented by arrows in
PtCl42-as shown in Figure 2.1.1.1
-
7/23/2019 Group Theory Nptel
11/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 11 of 24Joint Initiative of IITs and IISc Funded by MHRD
Fig 2.1.1.1 C4operation &translation along the x- and y- axes in PtCl42-
Thus both the vectors have changed positions in the C4 operation and the
character of this operation is equal to zero, i.e., (C4) = 0. Also, the vectors x and
y are inseparable because when an operation is done on x-vector, y-vector is
also affected. Hence, x and y are put in parentheses and written as (x,y).
2.1.2 Operation E on the translation along the x-axis
It is a doing nothing operation and the vectors are not disturbed from their
original positions. Hence, (E) = 2.
2.1.3 C2operation on the translation along x-axis
The effect of C2 on the translation along the x-axis is shown in Figure 2.1.3.1.
The x- and y-vectors (arrows) are shifted to their negative coordinates. Hence,
(C2) = -2.
-
7/23/2019 Group Theory Nptel
12/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 12 of 24Joint Initiative of IITs and IISc Funded by MHRD
Fig 2.1.3.1 Effect of C2on the translation along the x-axis
2.1.4 Effect of C2operation on x- and y-vectors
The C2operation converts the x-vector into its negative and the y-vector remains
unchanged. Hence, (C2) = 0
Fig 2.1.4.1 Effect of C2on the translation along the x-axis
-
7/23/2019 Group Theory Nptel
13/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 13 of 24Joint Initiative of IITs and IISc Funded by MHRD
2.1.5 Effect of C2on x- and y-vectors
The C2operation interchanges the x- and y-vectors. Hence, (C2) = 0
Fig 2.1.5.1 Effect of C2on the translation along the x-axis
-
7/23/2019 Group Theory Nptel
14/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 14 of 24Joint Initiative of IITs and IISc Funded by MHRD
2.1.6 Effect of operation i on x- and y-vectors
The inversion operation, i,changes the x- and y-vectors into their negatives.
Hence, (i) = -2
Fig 2.1.6.1 Effect of ion the translation along the x-axis
2.1.7 Effect of S4operation on the x- and y-vectors
This operation rotates the molecule by 900 and reflects in the molecular plane,
that is, the plane of the paper and (S4) = 0.
-
7/23/2019 Group Theory Nptel
15/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 15 of 24Joint Initiative of IITs and IISc Funded by MHRD
Fig 2.1.7.1 Effect of S4on the translation along the x-axis
2.1.8 Effect of hoperation on x- and y-vectors
Reflection in the
hplane does not affect the x- and y-vectors. Hence, (
h) = 2
2.1.9 Effect of v on x- and y-vectors
The effect will be the same as that of C2because vcontains the C2
axis.
Hence, (v) = 0.
-
7/23/2019 Group Theory Nptel
16/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 16 of 24Joint Initiative of IITs and IISc Funded by MHRD
.
Fig 2.1.9.1 Effect of von the translation along the x-axis
2.1.10 Effect of don x- and y-vectorsDihedral plane dcontains C2
. Hence, the effect of dwill be the same as that of
C2. Thus (d) = 0
-
7/23/2019 Group Theory Nptel
17/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 17 of 24Joint Initiative of IITs and IISc Funded by MHRD
Fig 2.1.10.1 Effect of don the translation along the x-axis
2.1.11 Identifying the symmetry of translation about x-axis
Now the characters of different operations can be given as follows:
D4h E 2C4 C2 2C2 2C2
i 2S4 h 2v 2d
2 0 -2 0 0 -2 0 2 0 0
This result is compared with D4h character tableto find out the symmetry. It is
found that the symmetry is Eu. This appears in the character table as follows:
D4h E 2C4 C2 2C2 2C2 i 2S4 h 2v 2d
Eu 2 0 -2 0 0 -2 0 2 0 0 (x,y)
In the same way it can be shown that the translation about z-axis belongs to A2u
symmetry.
-
7/23/2019 Group Theory Nptel
18/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 18 of 24Joint Initiative of IITs and IISc Funded by MHRD
3.1 Deducing symmetries of rotation about the axes
from irreducible representationsA curved arrow is used as the base vector for rotation.
3.1.1 Rotation along the z-axis
3.1.1.1 Effect of Eon rotation along the z-axis
Fig 3.1.1.1.1 Effect of C4on rotation about the z-axis
3.1.1.2 Effect of C2on rotation about z-axis
This is nothing but doing C4twice. The direction or the position of the arrow will
not be changed. Hence, (C2) = +1.
3.1.1.3 Effect of C2on rotation about z-axis
The effect is shown in Figure 3.1.1.3.1. The direction of the curved arrow is
changed and hence,(C2) = -1
Pt ClCl
Cl
Cl
C4, z
Pt ClCl
Cl
Cl
C2'
-z
C2'
Fig 3.1.1.1.3 Effect of C2on rotation about the z-axis
-
7/23/2019 Group Theory Nptel
19/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 19 of 24Joint Initiative of IITs and IISc Funded by MHRD
3.1.1.4 Effect of C2on rotation about z-axis
The effect is the same as that of C2. (C2
) = -1
3.1.1.5 Effect of i on rotation about z-axis
The direction of the curved arrow is not changed and shown in Figure 3.1.1.5.1
(i) = +1.
Fig 3.1.1.5.1 Effect of i on rotation about z-axis
3.1.1.6 Effect of S4on rotation about z-axis
Pt ClCl
Cl
Cl
C4, z
C4
Pt ClCl
Cl
Cl
C4, z
h
Pt ClCl
Cl
Cl
Fig 3.1.1.6.1 Effect of S4on rotation about z-axis
It is C4 operation followed by reflection in sh plane. The direction of the curved
arrow has not changed. Hence, (S4) = +1.
-
7/23/2019 Group Theory Nptel
20/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 20 of 24Joint Initiative of IITs and IISc Funded by MHRD
3.1.1.7 Effect of von rotation about z-axis
The vplane contains the C2axis. The direction of the arrow changes as shown
in Figure 3.1.1.7.1. Hence, (v) = -1.
Fig 3.1.1.7.1 Effect of von rotation about z-axis
3.1.1.8 Effect of don rotation about z-axis
The dplane contains C2
axis. The effect is the same as C2
operation.
(d) = -1.
3.1.1.9 Effect of hon rotation about z-axis
The hplane is the molecular plane and the direction of the arrow is not
changed. Hence, (h) = +1.
Now the characters of the different operation are grouped as under.
D4h
E 2C4 C
22C
22C
2i 2S
4
h2
v2
d
1 1 1 -1 -1 1 1 1 -1 -1
When this is compared with D4h character table, it is found that this has got A2g
symmetry.
-
7/23/2019 Group Theory Nptel
21/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 21 of 24Joint Initiative of IITs and IISc Funded by MHRD
D4h E 2C4 C2 2C2 2C2 i 2S4 h 2v 2d
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz
Similarly, it can be shown that the rotation about x- and y-axis have E gsymmetry.
4 Applications of Irreducible RepresentationsIt is a representation which can be further reduced to irreducible form. At first, the
reducible representation for a molecule is derived and then it is reduced. From
this irreducible representation we can find out the representations covering the
translation, rotation and vibration and from this we can find out the IR active and
Raman active vibrations.
Example: Trans-N2F2
Step 1: Structure of the molecule
Step 2:Symmetry elements present:
1. C2 axis
2. hplane
3. i
Step 3: Hence, the point group is C2h
Step 4:The C2hcharacter table is given below
C2h E C2 i h
Ag 1 1 1 1 Rz x , y , z , xy
Bg 1 -1 1 -1 Rx, Ry xz, yz
Au 1 1 -1 -1 z
Bu 1 -1 -1 1 x, y
-
7/23/2019 Group Theory Nptel
22/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 22 of 24Joint Initiative of IITs and IISc Funded by MHRD
Step 5:The number of operations in this group is four:
E, C2, i, h (as shown by the character table).
Step 6: The characters of the different operations are found out as follows:
Identity operation, E
All the 12 vectors (x,y,z) of the four atoms of the molecule are not disturbed.
Hence, the character, (E) = 12
C2operation
All the four atoms are disturbed from their original places and occupy new
positions.
Hence, the character, (C2) = 0
i operation (inversion)
All the four atoms are displaced from their original positions to their new
positions.
Hence, (i) = 0
hoperation (reflection in the horizontal plane of symmetry)
All the four atoms retain their original positions. Nothing is changed.
Let us consider the three x, y and z vectors (arrows) of one fluorine atom. When
reflected in the hplane (i.e. plane of paper), x and z arrows are not affected,
while the y-arrow is inverted.
Thus,
Old x = new x ; character = +1
Old y = - new y; character = -1
-
7/23/2019 Group Theory Nptel
23/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 23 of 24Joint Initiative of IITs and IISc Funded by MHRD
Old z = new z; character = +1
Net character for one atom = +1
Hence, for four atoms, the total character will be equal to 4(+1) = +4.
Hence, (h) = 4.
Step 7: Hence the reducible representation is:
C2h E C2 i h 12 0 0 4
Step 8: This is reduced to get the components of this reducible representation
Ag= 1/4 [ (12)(1)(1) + 0 + 0 + (4)(1)(1) ] = 4
Bg= 1/4 [ (12)(1)(1) + 0 + 0 + (4)(-1)(1) ] = 2
Au= 1/4 [ (12)(1)(1) + 0 + 0 + (4)(-1)(1) ] = 2
Bu= 1/4 [ (12)(1)(1) + 0 + 0 + (4)(1)(1) ] = 4
Thus, = 4Ag + 2Bg+ 2Au+ 4Bu
From the character table, Aurepresents translation along the z-axis and Bu
represents that along the x- and y-axes. Thus, translation is given by Au+ 2Bu.
Similarly, rotations are covered by the representations, Ag+ 2Bg.
Translation + rotation are covered by Au+ 2Bu+ Ag+ 2Bg.
This is subtracted from the total representation to find out the normal vibrations:
(4Ag+ 2Bg+ 2Au+ 4Bu) . (Au + 2Bu+ Ag+ 2Bg) = 3Ag+ Au+ 2Bu
Of these,
IR active vibrations are Au+ 2Bu= 3
Raman active vibrations are 3Ag= 3
Total vibrations = 6
The molecule is non-linear.
Hence, the number of expected vibrations =(3N-6) = (3x4.6) = 6.
Hence, this is correct.
-
7/23/2019 Group Theory Nptel
24/24
NPTEL Chemistry and Biochemistry Coordination Chemistry (Chemistry of transitionelements)
Page 24 of 24Joint Initiative of IITs and IISc Funded by MHRD
5. References1. 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