physical chemistry

23/4/19 1

Physical Chemistry I

Chemistry Department of Fudan University

Chapter IV Molecular Symmetry and Point Group

Physical Chemistry

23/4/19 2





Reference Books:

(1) F. Albert, Cotton, Chemical Application of Group Theory, Wiley Press, New York, 1971.(中译本 :群论在化学中的应用 ,科学出版社 ,1984)

(2) David M. Bishop, Group Theory and Chemistry, Clarendon Press, Oxford, 1973.(中译本 :群论与化学 ,高等教育出版社 ,1984)

23/4/19 3




Symmetry is all around us and is a fundamental property of nature.

23/4/19 4




[A]

[B]

[C]

C31

N

H1H2

H3N

H3H1

H2

N

H2H3

H1

C3

C31C3

1

23/4/19 5




The nature and degeneracy of vibrations.

•The legitimate AO combinations for MOs.

•The appearances and absences of lines in a molecule’s spectrum.

•The polarity and chirality of a molecule.

Motivational Factors

23/4/19 6




The term symmetry is derived from the Greek word “symmetria” which means “measured together”.

We require a precise method to describe how an object or molecule is symmetric.

§4-1. Symmetry Elements and Operations

23/4/19 7




Symmetry Operation A symmetry operation is a movement of a body such that, after the movement has been carried out, every point of the body is coincident with an equivalent point (or perhaps the same point) of the body in its original orientation.

Symmetry ElementA symmetry element is a geometrical

entity such as a line, a plane, or a point, with respect to which one or more symmetry operations may be carried out.

23/4/19 8




§4-1-1. Symmetry Elements and Operations Required in

Specifying Molecular Symmetry

23/4/19 9




1. The Identity Operation

E Ê

No matter how asymmetrical a molecule is, it must have an identity operation, E

•The symbol “E” comes from the German, “eigen,” meaning “the same”

23/4/19 10




2. Proper Axes and Proper Rotations

Cn

An n-fold rotation is symbolized by the element Cn, and represents n–1 rotational operations about the axis.

mnC

23/4/19 11




Molecules may have many rotation axes. But the axis with the highest n is designated as the principal axis

•The operation C1 is merely E.

23/4/19 12




3. Symmetry Planes and Reflections

There are 3 types of planes:

Vertical, horizontal and dihedral

23/4/19 13




Vertical v

If the reflection plane contains the Principle Axis, it is called a “vertical plane.”

23/4/19 14




If the reflection plane is perpendicular to the Principle Axis, it is called a “horizontal plane.”

Horizontal plane h

NN

h

C

A molecule can have only one h

23/4/19 15




C C C

H

H

H

H

C2'

C2'

C2, S4

Dihedral d

Vertical planes which bisect the angles between adjacent pairs of C2 axes perpendicular to the principle axis

23/4/19 16




4. Improper Axes and Improper Rotations

Rotations by 2/n followed by reflection in a plane to the Sn axis.

Sn=Cn h

23/4/19 17




5. The Inversion Center

i i = S2 = C2h=hC2

(x, y, z) (-x, -y, -z)

23/4/19 18




Multiplication Table of H2OC2v E C2 xz yz

E E C2 xz yz

C2 C2 E yz xz

xz xz yz E C2

yz yz xz C2 E

Operate first

Operatesecond

§4-1-2. Multiplication Table

23/4/19 19




§4-1-3. General Relations Among Symmetry Elements and Operations

1. Products

(1) The product of two reflections, (1) The product of two reflections, intersecting at an angle of intersecting at an angle of = 2= 2/2n, is a /2n, is a rotation by 2rotation by 2 about the axis defined by about the axis defined by the line of intersectionthe line of intersection

23/4/19 20




(2) When there is a rotation axis, C(2) When there is a rotation axis, Cnn, and a plan, and a plan

e containing it, there must be e containing it, there must be nn such planes se such planes separated by angles of 2parated by angles of 2/2n;/2n;

(3)The product of two C2 rotations about axes which intersect at an angle is a rotation by 2 about an axis perpendicular to the plane of the C2 axes;

(4) A proper rotation axis of even order and a perpendicular reflection plane generate an inversion center.

23/4/19 21




2. Commutation

The identity operation and the inversion The identity operation and the inversion with any operations;with any operations;

Two rotations about the same axis;Two rotations about the same axis; Reflections through planes perpendicular Reflections through planes perpendicular

to each other;to each other; Two CTwo C22 rotations about perpendicular rotations about perpendicular

axes;axes; Rotation and reflection in a plane Rotation and reflection in a plane

perpendicular to the rotation axis. perpendicular to the rotation axis.

23/4/19 22




§4-2. Molecular Point Group

§4-2-1. Definitions and Theorems of Group Theory

1. Definitions

A group is a collection of elements which are interrelated according to certain rules.

23/4/19 23




(1)The product of any two elements in the group must be an element in the group;AB=C

(3) The associative law of multiplication must hold; A(BC)=(AB)C

(2) One element in the group must commute with all others and leave them unchanged;E----the identity element EX=XE=X

(4) Every element must have a reciprocal, which is also an element of the group AA-1= A-1A= E

23/4/19 24




2. Theorems of Group (1) For a certain element, there is only one reciprocal

in the group

(2)There is only one identity element in one group.

(3)The reciprocal of a product of two or more elements is equal to the product of the reciprocals, in reverse order.

(ABC·····XY)-1 = Y-1X-1····C-1B-1A-1

(4) If A1, A2 and A3····· are group elements, their product, says B, must be a group element, A1A2A3····· =B

23/4/19 25




3. Some Important Conceptions

Order----the number of elements in a finite group

Finite groups; Infinite groups

Subgroup--- the smaller groups,whose elements are taken from the larger group

{1， -1， i， -i } {1， -1 }

10

01

10

01

01

10

01

10矩阵乘法

23/4/19 26




A, B and X are elements of a group, if

B=X-1AX

We say B is conjugate with A.

Similarity Transform

Class---A complete set of elements which are conjugate to one another

23/4/19 27




§4-2-2. Molecular Point Groups

Point groupPoint group-----All symmetry elements in -----All symmetry elements in a molecule intersect at a common point, a molecule intersect at a common point, which is not shifted by any of the symmwhich is not shifted by any of the symmetry operations.etry operations.

Schoenflies SymbolsSchoenflies Symbols

23/4/19 28




1. Cn groups---only one Cn Axis

n Cn symmetry operations, g=n

Br

I

F

Cl

C1

CFClBrI

23/4/19 29




Cl

Fe

Cl

C2 (E, C2)

UO

O

C C

C2

23/4/19 30




C3, (E, C3, C32)

O

O

O

C3

C3

C2H3Cl3

H

Cl

H

H

ClCl

23/4/19 31




2. Cnh groups:

Cn add a horizontal plane h

g=2n

n=1, C1h= Cs

Cn h =Sn

23/4/19 32




Cl

O

HOCl

H

O

Ti

H

H2TiO

Cs

CH3OH

23/4/19 33




C2h (E, C2, h, i)Trans-C2H2Cl2 Cl

H

H

Cl

23/4/19 34




H

O O

H B

O

H

C3h (E, C3, C32, h, S3, S3

2)

B(OH)3, planar

23/4/19 35




3. Cnv groups: g=2n

Cn add a vertical plane v

Cl

Fe

Cl

C2v (E, C2, 1, 2)

HH

O

H2O

23/4/19 36




C3v (E, 2C3, 3v)

H

H

N

H

NH3

Cl Cl

Cl

staggered-C2H3F3

C3

23/4/19 37




C4v

FFXe

O

FF

OXeF4

C6v

23/4/19 38




Cv : C+v

AB type of diatomic molecules

OC

H F

C

v

23/4/19 39




4. Sn groups--- with only one Sn Axis

When n is odd, SWhen n is odd, Snn = C = Cnhnh

When n is even, the group is called SWhen n is even, the group is called Snn an an

d consists of n elementsd consists of n elements

SS22=C=Cii, S, S44, S, S66

23/4/19 40




trans-C2H2F2Cl2Br2

F

Cl

Cl

F

iCi

23/4/19 41




S4

F

F

F

F

S4

F1

F2

F3

F4

23/4/19 42




5. Dn groups

Cn axis add n C2 axes perpendicular to Cn (g=2n)

D3

23/4/19 43




6. Dnh groups

Dn group + h

nC2 Cn, h

Cn h =Sn C2 h = n v

g=4n

23/4/19 44




B B

B

B

C CO O

B4(CO)2

D2h E, 3C2, s2=i, h, 2v

H

H

H

H

ethylene

23/4/19 45




D3h

Ph(Ph)3

B

F

F

F

23/4/19 46




O OC C

Mn

O

C

O

CC

O

Mn

C

O

C CO O

D4h

Mn2(CO)10

PtCl4 2-

CAl4-

23/4/19 47




D6hD5h

23/4/19 48




23/4/19 49




Dh : C v +h

A2 type of diatomic molecules

NN

CO O

h

C

23/4/19 50




7. Dnd groups

Dn + d

d Cn n d d C2 S2n

g=4n

23/4/19 51




C C C

H

H

H

H

C2'

C2'

C2, S4

D2d (E, 2S4, C2, 2C2’, 2d)

23/4/19 52




H

HHH

HH

H

H

D2d

23/4/19 53




D3d

C2H6

23/4/19 54




O

O

C

C

OMn

C

C

C

O

O

Mn

C

O

CO

CO

D4d

23/4/19 55




H

H

H

H

C

Td — 4C3 , 3C2, 6d ;

g =24

8. T, Th, Td

?

23/4/19 56




C3C3

CCl4 C10H16 (adamantance)

Cl

Cl

Cl

Cl

C3

C3

?

23/4/19 57




Th, T,

Th

h =24

Th =12

23/4/19 58




9. O, Oh

Oh— 4C3 , 3C4, i ; g =48

UF6

F

F

F

U

F

F

F

23/4/19 59




C8H8 (Cubane) UF6

F

F

F

U

F

F

F

?

23/4/19 60




Oh =24

23/4/19 61




C60 C180

10. I, Ih

Ih — E, 6C5 , 10C3, 15C2 , i ; g =120

23/4/19 62




1. Determine whether the molecule belong1. Determine whether the molecule belongs to one of the “s to one of the “specialspecial” groups” groups::

CCvv , D , Dhh , T , Tdd , O , Ohh , I , Ihh

2. 2. No proper or improper rotation axesNo proper or improper rotation axes::

CC11, C, Css, C, Cii

3. 3. Only SOnly Snn (n even) axis: (n even) axis:

SS44, S, S66, S, S88….….

§4-2-3. A systematic procedure for symmetry classification of molecules

23/4/19 63




4. One C4. One Cnn axis, with no C axis, with no C22’s ’s C Cnn

(1) if there are no symmetry elements (1) if there are no symmetry elements eexcept the Cxcept the Cnn axis, the group is axis, the group is CCnn

(2) if there are n vertical planes, the (2) if there are n vertical planes, the group is group is CCnvnv

(3) if there is a horizontal plane, the (3) if there is a horizontal plane, the group is group is CCnhnh

23/4/19 64




5. If in addition to the principal C5. If in addition to the principal Cnn axis, th axis, th

ere are n Cere are n C22 axes lying in a plane axes lying in a plane the the

CCnn axis, the molecule belongs to D gro axis, the molecule belongs to D gro

up:up:

(1) If there are no symmetry elements bes(1) If there are no symmetry elements besides Cides Cnn and C and C22, the group is , the group is DDnn

(2) If there is a horizontal plane, the group (2) If there is a horizontal plane, the group is is DDnhnh

(3) If there is no (3) If there is no hh, but a set of , but a set of dd planes, planes,

the group isthe group is DDndnd

23/4/19 65




23/4/19 66




Illustrative Examples

Example 1. Example 1. HH22OO22

OO

H

H

23/4/19 67




Example 2. Ferrocene

A. The staggered A. The staggered configurationconfiguration

Fe

C5, S10

C2

23/4/19 68




Fe C2

C5

B. The eclipsed B. The eclipsed configurationconfiguration

23/4/19 69




Problem:一个立方体，顶点上放置八个完全相同的小球，当依次取走一个、两个、三个或四个球时，余下的球构成的图形属于什么点群？

23/4/19 70




§4-2-4 Symmetry-based Molecule Properties

Dipole Moment

ii

i rq

23/4/19 71




A molecule has a Cn axis,the dipole moment should be along the axes.

A moloecule has a symmetry plane and reflection,the dipole moment should be along the plane.

23/4/19 72




A molecule with two or more Cn rotations about perpendicular axes will show no dipole moment. e.g. Dn Dnd Dnh

A molecule with a Cn(or Sn) rotation and a plane perpendicular to it will show no dipole moment.

e.g. Cnh Sn

23/4/19 73




Only molecules with Cn,Cnv and Cs point groups may have Dipole moment.

23/4/19 74




Optical Rotation

A molecule with any symmetry planes, improper axes or inversion center will show no optical rotation.

Only molecules with Dn,O, T and I point groups may have optical rotation.

23/4/19 75




Exercises

1. What group is obtained by adding to or deleting the 1. What group is obtained by adding to or deleting the indicated symmetry operation from each of the indicated symmetry operation from each of the following groups?following groups?

CC33 plus i, C plus i, C3 3 plus Splus S66, C, C3v3v plus i, plus i,

DD3d3d minus S minus S66, S, S66 minus i, T minus i, Tdd plus i plus i

2. What is the symmetry of a octahedral UF2. What is the symmetry of a octahedral UF66 molecule molecule when one or more F is removed:when one or more F is removed:(1) 1; (2) 5; (3) 2,3; (4) 1,3(1) 1; (2) 5; (3) 2,3; (4) 1,3(5) 5,6; (6)1,2,3(5) 5,6; (6)1,2,3

41

6

U

5

32

23/4/19 76




§4-3. Representations of Groups

(x, y, z) (x, y, z) (x’, y’, z’) (x’, y’, z’)The symmetry operations can be The symmetry operations can be described by matrixdescribed by matrix

§4-3-1. Matrix notation for geometric transformations

23/4/19 77




z

y

x

z

y

x

100

010

001

z

y

x

z

y

x

100

010

001B. InversionB. Inversion

A.A. The identityThe identity

23/4/19 78




z

y

x

z

y

x

xy

100

010

001

:)(

z

y

x

z

y

x

xz

100

010

001

:)(

z

y

x

z

y

x

yz

100

010

001

:)(

C. ReflectionsC. Reflections

23/4/19 79




D. Proper RotationD. Proper Rotation

x

y

(x, y, z)(x', y', z')

r cossinrx sinsinry

cosrz

is the angle between r and z axis

23/4/19 80




'

'

'

100

0cossin

0sincos

z

y

x

z

y

x

0sincos' zyxx

0cossin' zyxy

100' zyxz

23/4/19 81




E. Improper RotationE. Improper Rotation

'

'

'

100

0cossin

0sincos

z

y

x

z

y

x

23/4/19 82




x, y, zx, y, z coordinates, coordinates, z z as the rotation as the rotation axis, Caxis, C3v 3v operations:operations:

100

010

001

:E

100

02

1

2

3

02

3

2

1

:3C

100

02

1

2

3

02

3

2

1

:23C

100

010

001

:1

100

02

1

2

3

02

3

2

1

:2

100

02

1

2

3

02

3

2

1

:3

23/4/19 83




A set of matrices, each corresponding to a single operation in the group, that can be combined among themselves in a manner parallel to the way in which the group elements combine.

§4-3-2. Representations and Characters

1. Representations of Groups1. Representations of Groups

23/4/19 84




A (R) ---A representation of the group

X A(R) X-1 = B(R)

Then, the new set of matrices B(R) is also a representation of the group.

A and B ----- A and B ----- equivalent representationsequivalent representations

23/4/19 85




100

010

001

:E

100

02

1

2

3

02

3

2

1

:3C

100

02

1

2

3

02

3

2

1

:23C

100

010

001

:1

100

02

1

2

3

02

3

2

1

:2

Representation of C3v point group (X, Y, Z)

100

02

1

2

3

02

3

2

1

:3

23/4/19 86




120

100

010

001

:E

001

100

010

:3C

010

001

100

:23C

010

100

001

:1

001

010

100

:2

100

001

010

:3

Representation of C3v point group

23/4/19 87




A (R) is a representation of the point group

X A(R) X-1 = B(R)

B(R) is also the representation of the point group

A A 和和 B ----- equivalent B ----- equivalent representation of the point group

23/4/19 88




2. Characters2. Characters

The sum of the diagonal elements of a The sum of the diagonal elements of a square matrix : square matrix :

j

jja

23/4/19 89




(1) The characters of AB and BA are equal(1) The characters of AB and BA are equal

(2) Conjugate matrices have identical (2) Conjugate matrices have identical characterscharacters

Therefore, we have:Therefore, we have:a.a. The matrices of the same class of group The matrices of the same class of group

elements have equal characterselements have equal characters

b. Two equivalent representations have the b. Two equivalent representations have the same characterssame characters

23/4/19 90




3. Reducible and irreducible 3. Reducible and irreducible representationsrepresentations

All the matrices of a representation of a group can make the same similarity transformation to be block-factored matrices. We call the set of matrices as a reducible representation, otherwise call as an irreducible representation

23/4/19 91




'

'

'

'

3

2

11

A

A

A

AAxx

'''' 321 AAAA

reducible representation

23/4/19 92




4. Character Tables4. Character Tables

C3v E 2C3 3v

A1

A2

E

1 1 1

11

-1

-1

2 0

z

Rz

(x,y), (Rx, Ry)

x2+y2, z2

(x2-y2, xy) (xz, yz)

III III IV

23/4/19 93




§4-3-3. §4-3-3. The “Great Orthogonality TThe “Great Orthogonality Theorem” and Its Consequencesheorem” and Its Consequences

23/4/19 94




1. 1. The Great Orthogonality TheoremThe Great Orthogonality Theorem

''''

*)()( nnmmij

ji

nmjR

mnill

gRR

Symmetry operation

Irreduciblerepresentation

order

dimension

23/4/19 95




C3v E C3 1

A1

A2

E

1 1 1

11 -1

C32 2 3

1 1 1

1 -1 -1

10

01

2

1

2

32

3

2

1

2

1

2

32

3

2

1

10

01

2

1

2

32

3

2

1

2

1

2

32

3

2

1

23/4/19 96




2. Important rules about irreducible 2. Important rules about irreducible representations and their charactersrepresentations and their characters

(1) The vectors whose components are the (1) The vectors whose components are the characters of two different irreducible characters of two different irreducible representations are orthogonalrepresentations are orthogonal

ijR

ji gRR )()(

23/4/19 97




The sum of the squares of the characters in any irreducible representation equals g.

gRR

i 2)(

If i=j, we have:

23/4/19 98




(2) (2) The sum of the squares of the The sum of the squares of the dimensions of the irreducible dimensions of the irreducible representations of a group is representations of a group is equal to the order of the groupequal to the order of the group

glllli 23

22

21

2

23/4/19 99




Since i(E), the character of the representation of E in the ith irreducible representation is equal to the order of the representation, this rule can also be written as:

gEi

i 2)(

23/4/19 100




(3) (3) The number of irreducible The number of irreducible representations of a group is representations of a group is equal to the number of equal to the number of classes in the groupclasses in the group

23/4/19 101




R

ii RRg

a )()(1

(4) The number of times the (4) The number of times the iith irreducith irreducible representation occurs in a reducible representation occurs in a reducible representation can be determineble representation can be determined by:d by:

23/4/19 102




§4-4.§4-4. Chemical Applications of Chemical Applications of Group TheoryGroup Theory

§4-4-1.§4-4-1. Wave functions as bases for Wave functions as bases for irreducible representations irreducible representations

The eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs

23/4/19 103




If If is nondegenerate: is nondegenerate: 1R

The representations are one-dimensional

K-fold degenerate eigenfunctions are a k-dimensional representation for the group

ERRHHR

EH

23/4/19 104




§4-4-2.§4-4-2.Symmetry-adapted linear Symmetry-adapted linear combinations(SALCS)combinations(SALCS)

LCAO-MOLCAO-MO Symmetry Symmetry matchingmatching

The same The same irreducible irreducible

representationrepresentation

23/4/19 105




H2O: C2v

H

O

H

E v(xz)

A1

A2

1 1 1

11

-1

-1

1 1

C2v

B1

B2

C2 v'(xz)

1

-1

-1

11 -1 -1

z

Rz

x,Ry

y,Rx

x2,y2,z2

xy

xz

yz

(YZ)

23/4/19 106




The irreducible representation of O atom:The irreducible representation of O atom:

2s A∈ 1 2pz A∈ 1 2px B∈ 1 2py B∈ 2

The representation of a single H atom:EE CC22 σσvv σσvv

’’

11 00 00 11

is not the irreducible representationis not the irreducible representation

23/4/19 107




Combine two H atoms together:

E C2 σv σv’

1 1 1 1)2(

1)1(

1 ss

)2(1

)1(1 ss 1 -1 -1 1

1)2(

1)1(

1 Ass 2)2(

1)1(

1 Bss

O 2s 2pz O 2py

Symmetry matching

23/4/19 108




Taking energy into considerationTaking energy into consideration

LCAO-MOLCAO-MO

)(2

'2

)2(1

)1(1

'1

)(22

)2(1

)1(11

)(

)(

opss

opss

y

z

cc

cc

23/4/19 109




SALCs refer to one or more sets of SALCs refer to one or more sets of orthonormal functions, which are generorthonormal functions, which are generally either atomic orbitals or internal coally either atomic orbitals or internal coordinations of a molecule, and to make ordinations of a molecule, and to make orthonormal linear combinations of theorthonormal linear combinations of them in such a way that the m in such a way that the combinations combinations form bases for irreducible representatiform bases for irreducible representationsons of the symmetry group of the mole of the symmetry group of the molecule. cule.

23/4/19 110




RRPj

R

j )(

§4-4-3. Projection operators§4-4-3. Projection operators

23/4/19 111




(1) Identify the molecule point group;

(2) Use the orbitals as the basis for a representation;

(3) Reduce to its irreducible components;

(4) Projections operate

ijR

ii RRP )(

The procedure to construct SALCsThe procedure to construct SALCs

23/4/19 112




Illustrative Examples

Example 1. Example 1. HH22OO , , CC2v 2v groupgroup

Use the two H 1s orbitals as the basis Use the two H 1s orbitals as the basis for a representation:for a representation:

EE CC22 σσvv σσvv’’

22 00 00 22

Reduce this to its irreducible components:

21 BA

23/4/19 113




)(2

1

)(2

1

)2(1

)1(1

Normalize

)1(1

)2(1

)2(1

)1(1

)1(1

')1(1

)1(12

)1(1

)1(1

)2(1

)1(1

Normalize

)1(1

)2(1

)2(1

)1(1

)1(1

')1(1

)1(12

)1(1

)1(1

2

1

ss

ssss

sVsVsssB

ss

ssss

sVsVsssA

CEP

CEP

23/4/19 114




NH3

EA 1

C3v E 2C3 3v

A1

A2

E

1 1 1

11

-1

-1

2 0

z

Rz

(x,y), (Rx , Ry)

x2+y2, z2

(x2-y2, xy) (xz, yz)

Γ 3 0 1

23/4/19 115




32132132113

1ˆ 11 AA P

32132111 26

12ˆ EE P

31231222 26

12ˆ EE P

Smith method EEE C 123

dCddCd EEEEEEEEE112112131

02

1114

6

1)122(

6

1 CC

3232

1 E

23/4/19 116




v3E 32C 23Ch 32S

1'A

2'A

'E

1"A

2"A

"E

1 1 1 1 1 1

1 1 -1 1 1 -1

2 -1 0 2 -1 0

1 1 1 -1 -1 -1

1 1 -1 -1 -1 1

2 -1 0 -2 1 0

3 0 -1 -3 0 1

(X, Y, Z )

Base

23/4/19 117




"2 E"A

1533222

2331

" "ˆ'ˆˆˆˆ"ˆ'ˆˆˆˆˆˆ 2 vvvhA SSCCCCCEP

2"1321

3

1 A

1533

2331

" ˆˆ2ˆˆˆ2ˆ SSCCEP hE "

132126

1 E

2533

2332

" ˆˆ2ˆˆˆ2ˆ SSCCEP hE "

231226

1 E "

1"

2"

3EEE C

dCddCd EEEEEEEEE "1

"1

"2

"1

"1

"2

"1

"3

"1

32"

32

1 E

23/4/19 118




§4-4-4. §4-4-4. The direct productThe direct product

R---an operation in the symmetry group of a R---an operation in the symmetry group of a moleculemolecule

11, , 22 --- two sets of functions which are bases --- two sets of functions which are bases for representations of the groupfor representations of the group

)()()(

)(

21

2121

RMRMRM

RMR

111 )( RMR 222 )( RMR

23/4/19 119




The characters of the representation of a direct product are equal to the products of the characters of the representations based on the individual sets of functions.

)()()( 21 RRR

23/4/19 120




The integral may be nonzero only if The integral may be nonzero only if i i and and jj belong to the same irreducible representation belong to the same irreducible representation of the molecular point group.of the molecular point group.

dji *

§§4-4-5. Identifying non zero matrix elements4-4-5. Identifying non zero matrix elements

23/4/19 121




Spectral transition probabilitiesSpectral transition probabilities

ji EEh

dI ji

An electric dipole transition will be allowed with x, y, or z polarization if the direct product of the representations of the two states concerned is or contains the irreducible representation to which x, y, or z, respectively, belongs.

23/4/19 122




C4v E 2C4 C2 2σv 2σd

A1 1 1 1 1 1

A2 1 1 1 1 1

B1 1 1 1 1 1

B2 1 1 1 1 1

E 2 2 2 0 0

对 C4v

群 Ei 2Bj

dˆ ji F 问使积分不为零时 F̂ 应属于哪些不可约表示？

physical chemistry

Documents