A New Visual Basic Software Built-up for Solving-out ......2.2 Symmetry Elements and Symmetry...

AN-Najah National University Faculty of Graduate Studies

A New Visual Basic Software Built-up for Solving-out Reduction formula in Chemical

Applications of Group Theory

By Rasha Saleh Mohammad Sabri

Supervised by Dr. Mohammad Najeeb Assa'ad

Prof. Hikmat S.Hilal

Submitted in Partial Fulfillment of the Requirements for the degree of Master Computational Mathematics, Faculty of Graduate Studies, at An-Najah National University, Nablus, Palestine.

2009

ii




This Thesis was defended successfully on 10/ 6/ 2009 and approved by

Committee Members Signature

1- Dr. Mohammad Najeeb Ass'ad (Supervisor) ..

2- Prof. Hikmat S. Hilal (Co- Supervisor) ..

3- Dr. Luai Malhis (Internal Examiner) ..

4- Dr. Mohammad Awad ( External Examiner) ..

iii

Acknowledgements

FIRST GREAT THANKS TO ALLAH RAB AL ALAMEEN

Thanks a lot to my husband Ziad who did his best to see me a successful

person.

I would like also to express my thanks to my supervisors Dr. Mohammad

Najeeb Ass ad and Prof. Hikmat S. Hilal for their fruitful support and

advice, to my father Mr. Saleh Sabri, to my father in low Sheihk

Mohammad Abd Alrahman and to whole family for their support and help

all the time.

And I would like also to express my thanks and appreciation to my thesis

committee members.

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:



.

Declaration

The work provided in this thesis, unless otherwise referenced, is the

researcher's own work, and has not been submitted elsewhere for any other

degree or qualification.

: Student's name:

: Signature:

: Date:

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Table of Contents

No. Content Page

Acknowledgment iii

Declaration

iv

Table of Contents v

List of Tables vii

List of Figures viii

Abstract xii Chapter One: General Background 1

1.1 Historical Introduction 1 1.2 What is a Group? 4 1.3 Examples of Groups 6 1.4 The Relationship Between Group Theory and Chemistry

7 Chapter Two: Molecular Symmetry and Representations of Symmetry Point Groups

9

2.1 Introduction 10 2.2 Symmetry Elements and Symmetry Operations 11 2.2.1 Symmetry Operations 11 2.2.2 Symmetry Elements 12 2.3 Symmetry Point Groups 15 2.4 Matrix Representation of a Symmetry Operation 19 2.5 Reducible and Irreducible Representations 22 2.6 Construction of Character Tables for Point Groups 22 2.6.1 Example 22 2.6.2 Example 24 2.6.3 Properties of Irreducible Representations 26

2.7 Construction of a Reducible Representation for a C v2

Molecule 27

2.8 The Relationship Between Reducible and Irreducible Representations

29

2.9 Examples on Solving out Reduction Formula 31 2.9.1 D h3 Point Group 31 2.9.2 Cubic Point Group 33 2.9.3 Point Groups with Complex Elements 36 2.10 Reducing D h and C v Point Groups 40 2.10.1 Linear XYZ 41 2.10.2 Linear XY 2 42 2.10.3 Linear Symmetrical X 2 Y 2 43 2.11 Comment 44

vi

No. Content Page Chapter Three: Methodology and Software Composition 45

3.1 Introduction 46 3.1.1 Example 46 3.2 Other Software Tools 48 3.3 Differences Between Our Software and Earlier Software

50 3.4 Languages 51 3.4.1 Visual Basic 6.0 51 3.4.2 Matlab 7.4 53 3.5 Graphical User Interface ( GUI ) 54 3.6 The Software Composition 55 3.6.1 The First Part " Visual Basic 6.0 Part " 55 3.6.1.1

Calculate Reduction Formula Directly 56 3.6.1.2

Infinite Point Groups 83

3.6.1.3

Find Reducible Representations then Reduce them for Chosen Point Groups

86

3.6.1.4

Functions Of Common Command Buttons 89 3.6.2 The Second Part " Matlab 7.4 Part " 92 Chapter Four: Results and Discussions 95

4.1 Software Applications 96 4.1.1 D nh Point Groups 96

4.1.2 C nh Point Groups 97

4.1.3 C nv Point Groups 99 4.1.4 Cubic Point Groups 100 4.1.5 Infinite Point Groups 102 4.1.5.1

Linear XYZ 102 4.1.5.2

Linear XY 2 103 4.1.5.3

Linear Symmetrical X 2 Y 2 105 4.1.6 Constructing and reducing N3 106 4.2 General Comments on the Visual Basic Software Part 110 4.3 Applications on the Matlab 7.4 Part of The Software 111 4.4 Conclusion 113 4.5 Suggestions For Future Works 114

References 115

Appendices 119

vii

List of Tables

Table Subject Page Table (1.1)

Multiplication Table for (Z 3 , 3 ) 6

Table (2.1)

A list of Symmetry Operations and their inverses, where m and n are integers, and the subscripts on the mirror planes indicates that any mirror plane is its own inverse.

16

Table (2.2)

Complete Character Table of C v2 Point Group 24 Table (2.3)

Complete Character Table of C v3 Point Group 25 Table (2.4)

Character Table of D h3 Point Group 32 Table (2.5)

Character Table of hO Point Group 34 Table (2.6)

Character Table of C h4 Point Group 37 Table (2.7)

Modified Character Table of C h4 Point Group 38

viii

List of Figures

Figure Subject Page

Figure (2.1) Identity in C 2 H 4 Molecule 13 Figure (2.2) Inversion in C 2 H 6 Molecule 13 Figure (2.3) Rotations in C 2 H 4 Molecule 14 Figure (2.4) Reflection in C 2 H 4 Molecule 14 Figure (2.5) Improper Rotation " S " in Allene Molecule 15 Figure (3.1) The Form in Visual Basic 6.0 52 Figure (3.2) The Tool Box in Visual Basic 6.0 52 Figure (3.3) The Properties Windows 53 Figure (3.4) The Main Menu Form 55 Figure (3.5) The " Calculate Reduction Formula Directly" Form 56 Figure (3.6) The Nonaxial Point Groups Form 57 Figure (3.7) The C1 Point Group Form 57 Figure (3.8) The C s Point Group Form 58

Figure (3.9) The C i Point Group Form 58

Figure (3.10) The C n Point Groups Form 59 Figure (3.11) The C 2 Point Group Form 59 Figure (3.12) The C 3 Point Group Form 60 Figure (3.13) The C 4 Point Group Form 60 Figure (3.14) The C 5 Point Group Form 61

Figure (3.15) The C 6 Point Group Form 61



Figure (3.18) The D n Point Groups Form 63 Figure (3.19) The D 2 Point Group Form 63 Figure (3.20) The D 3 Point Group Form 64 Figure (3.21) The D 4 Point Group Form 64 Figure (3.22) The D 5 Point Group Form 65

Figure (3.23) The D 6 Point Group Form 65

Figure (3.24) The C nv Point Groups Form 66

Figure (3.25) The C v2 Point Group Form 66





Figure (3.30) The C nh Point Groups Form 69

Figure (3.31) The C h2 Point Group Form 69

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Figure Subject Page





Figure (3.36) The D nh Point Groups Form 72

Figure (3.37) The D h2 Point Group Form 72






Figure (3.43) The D nd Point Groups Form 75

Figure (3.44) The D d2 Point Group Form 76




Figure (3.48) The D d6 Point Group Form 78 Figure (3.49) The Cubic Point Groups Form 78 Figure (3.50) The T Point Group Form 79 Figure (3.51) The dT Point Group Form 79

Figure (3.52) The hT Point Group Form 80 Figure (3.53) The O Point Group Form 80 Figure (3.54) The hO Point Group Form 81

Figure (3.55) The S n Point Groups Form 81 Figure (3.56) The S 4 Point Group Form 82 Figure (3.57) The S 6 Point Group Form 82

Figure (3.58) The S 8 Point Group Form 83 Figure (3.59) The Infinite Point Groups Form 83 Figure (3.60) The D h Point Group Form 84

Figure (3.61) The D h2 Subgroup Form 84

Figure (3.62) The C v Point Group Form 85

Figure (3.63) The C v2 Subgroup Form 85

Figure (3.64) The Reducible Representations For Chosen Point Groups Form.

86

Figure (3.65) Reducible Representation For C v2 Point Group Form 87


x

Figure Subject Page


Figure (3.68) Reducible Representation For D h2 Point Group Form 88

Figure (3.69) Reducible Representation For D h4 Point Group Form 89

Figure (3.70) Reducible Representation for dT Point Group Form 89 Figure (3.71) The Flowchart of the Visual Basic software Part 91 Figure (3.72) The blank GUI in the GUIDE Layout Editor 92 Figure (3.73) S 4 Point Group GUI 93 Figure (3.74) C 3 Point Group GUI 93

Figure (3.75) C h6 Point Group GUI 94

Figure (4.1) A picture showing reducible representation elements after being entered into their respective places inside software form of D h3 Point Group

96

Figure (4.2) Solution of Reduction Formula using software as applied on the reducible representation in D h3 Point Group.

97

Figure (4.3) A picture showing reducible representation elements after being entered into their respective places inside software form of C h4 Point Group

98

Figure (4.4) Solution of Reduction Formula using software as applied on the reducible representation in C h4 Point Group

98

Figure (4.5) A picture showing reducible representation elements after being entered into their respective places inside software form of C v2 Point Group

99

Figure (4.6) Solution of Reduction Formula using software as applied on the reducible representation in C v2 Point Group

100

Figure (4.7) A picture showing reducible representation elements after being entered into their respective places inside software form of Oh Point Group

101

Figure (4.8) Solution of Reduction Formula using software as applied on the reducible representation in hO Point Group

101

Figure (4.9) Number of atoms entered in C v2 subgroup form for linear molecules.

102

Figure (4.10) red and vib

in C v2 subgroup form for linear molecules based on the software

103

Figure (4.11) Number of atoms entered in D h2 subgroup form for linear molecules 104

xi

Figure Subject Page


in D h2 subgroup form for linear

molecules based on the software 104

Figure (4.13) Number of atoms entered in D h2 subgroup form for linear symmetrical molecules

105


in D h2 subgroup form for linear symmetrical molecules based on the software

106

Figure (4.15) N3 and its Reduction Formula for a 6-atomic molecule

that belongs to C v2 Point Group 107


that belongs to C v3 Point Group 107

Figure (4.17) N3 and its Reduction Formula for a 10-atomic

molecule that belongs to C v4 Point Group 108


that belongs to D h2 Point Group 108


molecule that belongs to D h4 Point Group 109


molecule that belongs to dT Point Group 109

Figure (4.21) Wrong Entry In D h2 Reducible Representation Form 110

Figure (4.22) Wrong Entry In D h2 Point Group Form 111

Figure (4.23) Solving out the Reduction Formula in C 3 GUI 112 Figure (4.24) Solving out the Reduction Formula in S 4 GUI 112 Figure (4.25) Solving out the Reduction Formula in C h6 GUI 113

xii

A New Visual Basic 6.0 Software Built-up For Solving-out Reduction Formula in Chemical Applications of Group Theory


Supervisors Dr. Mohammad Najeeb Assa ad

Prof. Hikmat S. Hilal

Abstract

A need for a computer software to solve-out the "Reduction

Formula" for different Point Groups is beyond doubt. That would save time

and effort to many chemists who are involved in different aspects of

chemical applications of group theory, and may gives a good approach to

researchers dealing with Molecular Chemistry.

This thesis presents a computer software that has been developed

using Visual Basic 6.0 as a programming language. The input and output

data are performed through software forms under Windows Vista

environment.

The software is able to perform the following functions:

1. Reducing Reducible Representations for 47 Point Groups.

2. Finding Reducible Representations " red and vib " for Infinite Point

Groups " C v and D h " and reducing them by S-L Method.

3. Finding Reducible Representation N3

and reducing it for six

chosen Point Groups " C v2 , C v3 , C v4 , D h2 , D h4 and dT ".

Solutions derived from the constructed software were tested by

comparison with manual standard methods, and showed complete

consistency.

1

Chapter One

General Background

2

1.1 Historical Introduction:

The following is an overview of Group Theory history, for more

details of this subject please refer to:

"http://en.wikipedia.org/wiki/History_of_group_theory ", and reference [1].

There are three historical roots for group theory: the theory of

algebraic equations, number theory and geometry. Euler, Gauss, Lagrange,

Abel and Galois were early researchers in the field of group theory. Galois

is honored as the first mathematician linking group theory and field theory,

with the theory that is now called Galois theory. An early source occurs in

the problem of forming an mth degree equation having as its roots m of the

roots of a given nth degree equation (m < n). For simple cases the problem

goes back to Hudde (1659). Saunderson (1740) noted that the

determination of the quadratic factors of a biquadratic expression

necessarily leads to a sextic equation, and Le Soeur (1748) and Waring

(1762 to 1782) still further elaborated the idea.

A common foundation for the theory of equations on the basis of the

group of permutations was found by Lagrange (1770, 1771), which gives a

good base for the theory of substitutions. He discovered that the roots of all

resolvents, which he examined, are rational functions of the roots of the

respective equations. To study the properties of these functions he

invented a Calcul des Combinaisons. The contemporary work of

Vandermonde (1770) also foreshadowed the coming theory. Ruffini

(1799) attempted a proof of the impossibility of solving the quintic and

http://en.wikipedia.org/wiki/History_of_group_theory

3

higher equations. He distinguished what are now called intransitive and

transitive, and imprimitive and primitive groups, and in 1801 he used the

group of an equation under the name l'assieme delle permutazioni.

Galois found that if nrrr ,......,2,1 are the n roots of an equation, there is

always a group of permutations of these r's such that:

(1) every function of the roots invariable by the substitutions of the group is

rationally known.

(2) every rationally determinable function of the roots is invariant under the

substitutions of the group.

Galois also contributed to the theory of modular equations and to

that of elliptic functions. His first publication on the group theory was

made at the age of eighteen in 1829, but his contributions attracted little

attention until the publication of his collected papers in 1846 Arthur Cayley

and Augustin Louis Cauchy were among the first to appreciate the

importance of the theory, and to the latter especially are due a number of

important theorems. The subject was popularised by Serret,

who devoted section IV of his algebra to the theory; by Camille

Jordan, whose Traité des Substitutions is a classic; and to Eugen Netto

(1882), whose was translated into English by Cole (1892). Other group

theorists of the nineteenth century were Bertrand, Charles Hermite,

Frobenius, Leopold Kronecker, and Emile Mathieu. Walther von Dyck

gave the modern definition of a group in 1882.

4

The study of what are now called Lie groups, and their discrete

subgroups, as transformation groups, started systematically in 1884 with

Sophus Lie; followed by work of Killing, Study, Schur, Maurer, and

Cartan. The discontinuous (discrete group) theory was built up by Felix

Klein, Lie, Poincaré, and Charles Emile Picard, in connection in particular

with modular forms and monodromy.

Other important mathematicians in this subject area include Emil

Artin, Emmy Noether, Sylow, and many others.The next generations of

mathematicians, developed the notions and used Group Theory as we know

today. Its importance to contemporary mathematics as a whole may be seen

from the 2008 Abel Prize, awarded to John Griggs Thompson and Jacques

Tits for their contributions to group theory.

1.2 What is a "Group"?

Before introducing the concept of " a group " we should illustrate the

concept of " a binary operation ".

Definition (1):

A binary operation " * " on a set S is an operation which assigns to any s, t

S a unique element s*t

S usually called the " product of s and t ".

The modern definition of a group is usually given by the following

way:

5

Definition (2):

A group G is a set with a binary operation G×G G which assigns

to every ordered pair of elements x, y of G a unique third element of G

denoted by xy or x*y such that the following four properties are satisfied:

1. Closure: if x, y G, then x*y G.

2. Associative: if x, y , z G, then (x*y)*z = x*(y*z).

3. Identity Element: there is a unique element E

G such that : E*x =

x*E= x. for all x G.

4. Inverses: for every x

G there is an element x 1

G such that

x*x 1 = x 1 *x = E .

An additional property for certain groups say that for x, y G if x*y

= y*x, then this group is called " Abelian Group "or " Commutative

Groups".

Definition (3):

Let S be a nonempty subset of a group G. If:

x, y

S x*y

S.

x

S x 1

S.

Then the binary operation " * " on G makes S a Subgroup of G.

6

1.3 Examples of Groups:

Example (1):

The set of real numbers R is a group under the operation " + ".

If we add two real numbers , we certainly get a real number, so we

have closure.

The usual laws of addition give us associativity.

The identity E = 0.

For any element a

R, the corresponding element a 1

is usually

denoted by

a, thus a + (

a) = 0; 0 is the identity. Therefore every

element in R has an inverse.

Example ( 2 ):

The set of integer numbers Z 3 is a group under the operation 3 .

Where

Z 3 : the set 0, 1, 2 with operation addition mod 3.

The construction of the multiplication Table is thus required. Table

(1) shows the products of any two elements belong to Z 3 .

Table (1.1): Multiplication Table for (Z 3 , 3 ).

3 0 1 2

0 0 1 2

1 1 2 0

2 2 0 1

7

From the product Table we can see that for any x, y

Z 3 , x 3 y

Z 3 . The closure property holds.

In general addition is an associative operation, and so does the

operation addition mod 3.

The identity element is 0.

Every element in Z 3 has an inverse:

0 1 = 0. Since 0 3 0 = 0.

1 1 = 2. Since 1 3 2 = 0 and 2 3 1 = 0.

2 1 = 1. Since 2 3 1 = 0 and 1 3 2 = 0. [3].

1.4 The Relationship Between Group Theory and Chemistry:

Group Theory has recently developed into a powerful tool for solving

problems in several areas of Chemistry. Since two decades ago this subject

was the domain of mathematicians and physicists. Nowadays the use of

spectroscopic techniques for structural elucidation has become very

common in Chemistry and it is known that Group Theory is closely related

to spectroscopy. [4].

For chemists the most important part of group theory is "

representation theory ". This theory and the idea of characters were

developed almost single-handedly at the turn of the century by the German

algebraist George Ferdinand Frobenius (1849

1917). One of the earliest

8

applications of Group Theory was in the study of crystal structure and with

the later development of X-ray analysis, this application was revised and

elaborated.

Of much more importance is the work of Hermann Wey (1885

1955)

and Eugene Paul Winger(1902 - ) who in the late twenties developed the

relationship between Group Theory and Quantum Mechanics. Winger's

greatest contribution was the application of Group Theory to atomic and

nuclear problems; in 1963 he shared the Nobel Prize for physics with J. H.

D. Jensen and M. G. Mayer. [5].

At more recondite level, Group Theory described the properties of an

abstract model of phenomena that depends on symmetry. The source of the

power of Group Theory is its establishment of a link between symmetries

and numbers. It helped in writing the grammars of the languages which are

used to describe the physical world. The principles of quantum mechanics

can be stated with conciseness, clarity, and confidence. Group Theory

predicted and classified the mode of vibration of a molecule, the possible

shapes of wave functions characterizing the electronic structures of atoms

and molecules and the spectroscopic properties of atoms and molecules.

Group Theory techniques are easy to apply, requiring only simple

arithmetic calculations. Quantum theory and Group Theory work in parallel

manner in solving chemical problems. While the former provides

quantitative solutions with difficult calculations, the latter provides

qualitatively accurate solutions with relatively high simplicity. [6].

9

Chapter Two

Molecular Symmetry & Representations

of Symmetry Point Groups

10

2.1 Introduction

Long ago the Greek sculptors and architects were associating the

concept of " symmetry " with the ideas of beauty and harmony. Pythagoras

developed this concept when he distinguished between evenness and

oddness. Plato felt that the world consistence of the world earth, air, fire

and water had been produced from geometrical shapes.

Today symmetry is a bridge from geometry to arithmetic. The

chemists intuitively use symmetry every time to recognize which atoms in

a molecule are equivalent. Symmetry also plays an important role in the

determination of the structures of molecules. Here, a great deal of the

evidence came from the measurement of crystal structures, infta-red

spectra, ultra-violent spectra, dipole moments and orbital activities.

Symmetry enabled chemists to apply Group Theory principles. the

significance is that molecules can be categorized on the basis of their

symmetry properties, which allow the prediction of many molecular

properties. The process of placing a molecule into a symmetry category

involves identifying all of the lines, points, and planes of symmetry that it

possesses; the symmetry categories the molecules may be assigned to are

known as point groups. [4,5].

11

2.2 Symmetry Elements and Symmetry Operations:

2.2.1 Symmetry Operations:

A symmetry operation is a certain way of moving of the molecule

such that the resulting shape of the molecule is indistinguishable from the

initial state. [7].

Symmetry operations are geometrically defined ways of exchanging

equivalent parts of a system without affecting its characteristics, Even if we

don't do any thing. Equivalent parts mean atoms in a molecule which may

all be interchanged with one another by symmetry operations, and they

must be of the same chemical species. [8].

In molecular symmetry terms, we choose to think of systems as

molecules. In general practice there are five types of molecular symmetry

operations associated with isolated molecule:

1. E, the identity operation which means doing nothing to the molecule.

2. C kn , proper rotation about an axis, where n = 1, ., k = 1, , n. n

indicates a rotation of 360 /n.

3. , reflection through a plane.

3. i, inversion through a point. each point in a molecule is

movedthrough " i " to a position on the opposite side and at the same

distance from the centre as the original point

12

4. S kn , improper rotation, which involves a hypothetical rotation C n

about an axis followed by a reflection through a plane perpendicular

to that rotation axis, where n = 1, ., k = 1, , n, thus S n =

C n .

2.2.2 Symmetry Elements:

A symmetry element is a geometrical entity such as a point, line, or a

plane about which a symmetry operation is performed.

The symmetry elements associated with a molecule are:

1. C n , the proper axis of rotation where n= 1, . This implies n-fold

rotational symmetry about this axis.

2. Plane of reflection: this implies bilateral symmetry about planes.

These planes are further classified as:

, the horizontal plane which is perpendicular to the main axis of

rotation (i.e. axis with highest value of n).

', the plane which contains the main axis of rotation and

perpendicular to plane.

3. i, the center of inversion, this is a central point through which all C n

and reflection elements pass. If no such common point exists there

is no center of inversion.

4. S n , the improper axis, this is made up of two parts C n and both .

13

If both C n and are present then S n must also exist.

5. E, the identity element, this is an element with a neutral action. That

means when the identity element is combined with any other

symmetry element, the result is always to give back the same

symmetry element. [4-10].

Each of these symmetry elements associated with symmetry

elements is performed such that the molecules orientation and position

before and after the operation are indistinguishable. Symmetry Operations

for some molecules are shown in Figures 2.2.1

2.2.5 below.

Figure (2.1): Identity in C 2 H 4 Molecule.

Figure (2.2): Inversion in C 2 H 6 Molecule.

14

Figure (2.3): Rotations in C 2 H 4 Molecule.

Figure (2.4): Reflection in C 2 H 4 Molecule.

15

Figure (2.5): Improper Rotation " S 4 " in Allene Molecule.

2.3 Symmetry Point Groups

Symmetry Elements of a molecule are vary, and each element is

accompanied by one symmetry operation or more. The set of symmetry

operations for any molecule and the product operation represent a group,

where the product operation means sequential implementation of two

symmetry operations. [4,5]

The group properties are included in the symmetry operations as

follows:

1. The product of any two symmetry operations implies a symmetry

operation, the closure property holds.

16

2. The associative property is obviously valid for product of symmetry

operation.

3. The Identity operation "E" belongs to the set of symmetry operations

of a molecule, the product of "E" and any other symmetry operation

gives the same operation.

4. Every symmetry operation has an inverse, where m and n are

integers and the subscripts on the mirror planes indicate that any

mirror plane is its own inverse. [4,5 ,7]

Table (2.1) shows the symmetry operations and their inverses:

Table (2.1): A list of symmetry operations and their inverses, where m and n are integers and the subscripts on the mirror planes indicate that any mirror plane is its own inverse.

Operation X Inverse X 1

Operation X Inverse X 1

E E dvh ,, dvh ,,

i i Skn , (n is even) S kn

n

C kn C kn

n

Skn , (n is odd) S kn

n2

Molecules differ in their symmetry elements, accordingly molecules

are classified to groups called "Symmetry Point Groups" since all the

symmetry elements in the molecule intersect at a common point and this

point remains fixed under all the symmetry operations of the molecule. It is

to be noted that the identity operation "E" is present in all point groups. [4-

14].

The most famous symmetry point groups are:

17

1) Low symmetry Systems:

C1 point group contains only the identity (single element group).

There is no point group has lower symmetry than C1 . CHFClBr

molecule C 1 point group.

C 2 point group contains only the 2-fold axis of rotation and the

identity, it is a low symmetry point group (two elements group). H2 O2

molecule C 2 point group.

C s point group: has only one reflection plane and the identity, it is a

low symmetry group (two elements group). H 2 C=CClBr molecule

C s point group.

C i point group: has only one inversion center and the identity, it is a

low symmetry group (two elements group). HClBrC-HClBr molecule

C i point group.

2) The Uni-axial C n Point Groups:

In the C n groups (cyclic groups), the n-fold axis of rotation is the

only symmetry element and there are no reflection planes, improper axes,

or inversions.

3) The C nv Point Groups:

These groups are obtained from the C n groups by adding a vertical

plane of reflection v , this addition may implies the presence of another

symmetry operations. H 2 O, NH 3 and SF 4 molecules C nv point group.

18

4) The C nh Point Groups:

These groups are obtained from the C n groups by adding a horizontal

plane of reflection h , this addition may implies the presence of another

symmetry operations like i, h and S n . Boric acid [B(OH) 3 ] and

trans- H2 O2 molecules C nh point group.

5) The S n Point Groups:

These groups have an S n

axis, this usually implies another symmetry

operations like and C n . (SiFHCl) 2 molecule S n point group.

6) The D n Point Groups:

In these groups the molecule has n 2-fold axes perpendicular to the

principal n-fold axis, there are no planes of reflection. CoN 6 and

[Cr(S 2 CN(CH 3 ) 2 ) 3 ] 3 molecules D n point group.

7) The D nd Point Groups:

The addition of the symmetry operation d to the D n groups yields

the D nd groups. The operation d is a diagonal reflection. S 8 (crown

sulfur) and H 2 C 3 H 2 molecules D nd point group.

8) The D nh Point Groups:

The addition of the symmetry operation h to the D n groups yields

the D nh groups. BF 3 and naphthalene(C10 H10 ) molecules

D nh point

group.

19

9) The Cubic Point Groups:

The cubic groups each contain certain symmetry operations of a

cube. They differ from the foregoing axial point groups in that in cubic

symmetry there is more than one axis of higher order than 2. The cubic

point groups are of two types, the tetrahedral (T h ) groups and the

octahedral (O h ) groups. SF 6 molecule

O h point group, and CF 4

molecule

T h point group.

10) The Continuous (Infinite or Linear) Point Groups:

A continuous point group consists of an infinite set of symmetry

operations that satisfy the group requirements. Special cases of infinite

groups are D h

and C v , which often arise with diatomic and linear

triatomic molecules. HCl molecule

C v

point group, and CO 2 molecule

D h point group. [9-11].

2.4 Matrix Representations of Symmetry Operations:

The consideration of how the symmetry operations affect various

coordinate systems of a molecule is possible to come up with matrices that

multiply in the same way as the symmetry operation do. The symmetry

operations can all be represented mathematically as 3 × 3 square matrices

and The x, y, z coordinates are written in vector format as a 3 × 1 column

vector:

z

y

x

20

In the following we give the matrix representations of the symmetry

operations:

The identity operation keeps general coordinates x , y , z

unchanged.

In matrix terms we would write:

100

010

001

z

y

x

= z

y

x

The inversion operation takes general coordinates x , y , z to x , y ,

z . In matrix terms we would write:

100

010

001

z

y

x

= z

y

x

The reflection in ( xy ) plane operation takes general coordinates x , y ,

z to x , y , z . In matrix terms we would write:

100

010

001

z

y

x

= z

y

x

The reflection in ( xz ) plane operation takes general coordinates x , y ,


100

010

001

z

y

x

= z

y

x

the reflection in ( yz ) plane operation takes general coordinates x , y ,


21

100

010

001

z

y

x

= z

y

x

the proper rotation operation C n through

for a clockwise about the

z axis, where

= 360°/n, changes general coordinates x , y , z

to

new coordinates 'x , 'y , 'z

and they are obtained by matrix terms as

follows:

100

0cossin

0sincos

z

y

x

= '

'

'

z

y

x

the improper clockwise rotation operation through

about the z-axis

changes general coordinates x , y , z

to new coordinates 'x , 'y , 'z and

they are obtained by matrix terms as follows:

100

0cossin

0sincos

z

y

x

= '

'

'

z

y

x

The set of these matrices that describe all of the possible symmetry

operations of a point group that can act on a point with coordinates x , y ,

z is called the total representation of that group.

Commonly the traces of these matrices can provide sufficient

information, thus the full matrices are not needed (the trace of matrix is the

sum of its diagonal elements, and its usually given by the symbol ).

In a representation the matrix A(R) corresponds to the symmetry

operation R, the trace of A(R) is called the character of R for that

representation.

22

2.5 Reducible and Irreducible Representations:

Reducible representations and irreducible representations play a

major role in obtaining solutions to problems of hybridisation, molecular

vibrations, delocalisation energies of electron systems and so on. In all

these applications, the first step involves point group determination and

formation of the reducible representation. The characters of the matrices

inthe reducible representation are used to split into the different irreducible

representations of the group. Every point group consists of a certain

number of irreducible representations. [4,7,8,10,12].

2.6 Construction of Character Tables for Point Groups:

This section introduce two examples that illustrate the construction of the

character tables of C v2 and C v3 Point Groups:

2.6.1 Example:

The set of four matrices that describe all of the possible symmetry

operations in the C v2 point group that can act on a point with coordinates

x , y , z is called the total representation of the C v2 Point Group.

100

010

001

The identity operation matrix for C v2 Point Group.

100

010

001

The C 2 operation matrix for C v2 Point Group.

100

010

001

The xz operation matrix for C v2 Point Group.

23

100

010

001

The yz operation matrix for C v2 Point Group.

each of these matrices is block diagonalized, the total matrix can be

broken up into blocks of smaller matrices that have no off-diagonal

elements between blocks. These block diagonalized matrices can be broken

down, or reduced into simpler one-dimensional representations of the 3

dimensional matrix. If we consider symmetry operations on a point that

only has an x

coordinate (e.g., x , 0, 0), then only the first row of our total

representation is required:

C v2 E 2C XZ

YZ

1 1 -1 1 -1 x

We can do a similar breakdown of the y and z coordinates to setup a table:

C v2 E 2C XZ

YZ

1 1 -1 1 -1 x

2 1 -1 -1 1 y

3 1 1 1 1 z

These three 1-dimensional representations are as simple as we can

get and are called irreducible representations. There is one additional

irreducible representation in the C v2 Point Group. Consider a rotation R z :

The identity operation and the C 2 rotation operations leave the

direction of the rotation R z unchanged. The mirror planes, however,

reverse the direction of the rotation (clockwise to counter-clockwise), so

the irreducible representation can be written as:

24

C v2 E 2C XZ

YZ

4 1 1 -1 -1 R z

Note: 4 classes of symmetry operations = 4 Irreducible representations.

Finally according to Mulliken Symbol [4], the representations are

labeled and the resulting C v2 character table is:

Table (2.2): Complete Character Table for C v2 Point Group.

C v2 E 2C XZ

YZ

A1 1 1 1 1 z 222 ,, zyx

A 2 1 1 -1 -1 R z xy

B1 1 -1 1 -1 x xz

B 2 1 -1 -1 1 y yz

2.6.2 Example

This example will illustrate a 2 dimensional irreducible

representation for C v3 point group. The symmetry operation matrices for

C v3 are:

100

010

001

The identity operation matrix for C v3 Point Group.

100

0120cos120sin

0120sin120cos

The C 3 operation matrix for C v3 Point Group.

100

010

001

The v operation matrix for C v3 Point Group.

The matrices block diagonalize to give two reduced matrices. One

that is 1 dimensional for the z coordinate, and the other that is

25

2 dimensional relating the x

and y

coordinates. Multidimensional

matrices are represented by their traces. Since cos120 = -0.5, we can write

out the irreducible representations:

C v3 E 32C v

1

1 1 1 z

2

2 -1 0 yx,

There is another irreducible representation, because (3 classes of

symmetry operations = 3 Irreducible representations) based on the R z

rotation axis.

This generates the full group representation table:

Table (2.3): Complete Character Table for C v3 Point Group.

C v3 E 32C v

A1 1 1 1 z

2z,22 yx

A 2 1 1 1 zR E 2 -1 0 yx,

),)(,( 22 yzxzxyyx

Character tables include the irreducible representations. For simple

point groups, the values are either 1 or 1: 1 means that the sign or phase

(of the vector or orbital) is unchanged by the symmetry operation

(symmetric) and 1 denotes a sign change (asymmetric), and they are

labeled by Mulliken Symbol as follows:

A, when rotation around the principal axis is symmetrical.

B, when rotation around the principal axis is asymmetrical.

Subscripts 1 and 2 associated with A and B symbols indicate whether

a C 2 axis

to the principle axis produces a symmetric (1) or anti-

symmetric (2).

26

Primes and double primes indicate representations that are symmetric

( ) or anti-symmetric ( ) with respect to h plane. They are not used

when one has an inversion center present.

when the point group has an inversion center, the subscript g (gerade)

when the corresponding character is 1, and the subscript u (ungerade)

when the corresponding character is -1 , with respect to inversion

operation.

E and T are doubly and triply degenerate representations, respectively.

with point groups C v

and D h

the symbols are borrowed from

angular momentum description: , , . [7,8,10,12,13].

2.6.3 Properties of Irreducible Representations:

Consider a point group consisting of h symmetry operations "of

order h". These operations are divided into k classes. The characters of the

irreducible representations of that point group are denoted by k,...,, 21 .

1. The number of irreducible representations in a point group is equal

to the number of classes in that group.

2. The sum of the squares of the characters of an irreducible

epresentations of a point group is equal to the order of that group.

k

ii

1

2 =h

27

3. The sum of the squares of the characters of the identity operation in

the irreducible representation is equal to the order of the point

group.

2

1

))(( EXk

ii = h

4. The characters of symmetry operations in two different irreducible

representations satisfy the relation:

)()(1

pjpi

k

pp RXRXg = h ij

(2.1)

0, if i j.

Where ij denotes the Kronecker delta symbol, ij = 1, if i = j.

And pg refers to the number of symmetry operations in the p th

class. pR is the symmetry operation in the p th class. )(),( pjpi RXRX are the

characters of the irreducible representations. [4].

2.7 Construction of a Reducible Representation for C v2 Molecule

Introducing the idea of construction a reducible representation for a

molecule belongs to a point group would be easier with providing an

example. Let's take the water molecule (H 2 O) consisting of 3 atoms, one

oxygen atom and two hydrogen atoms. The H atoms represent the

equivalent parts in the water molecule. This representation is important to

the molecular vibrations and classification of normal modes of a molecule,

and denoted by N3

.To build this representation we consider three base

vectors for each atom in the water molecule. x 1 ,y 1 and z 1 for the first atom,

28

x 2 , y 2 and z 2 for the second atom, and x 3 , y 3 and z 3 for the third atom, then

transform these vectors with the symmetry operation.

Thus the x, y, z coordinates of the total atoms are written as a 9

1

column vector:

3

3

3

2

2

2

1

1

1

z

y

x

z

y

x

z

y

x

Of course, the representing 9

9 matrices will thus be:

E =

100000000

010000000

001000000

000100000

000010000

000001000

000000100

000000010

000000001

C 2 =

100000000

010000000

001000000

000000100

000000010

000000001

000100000

000010000

000001000

29

xz =

100000000

010000000

001000000

000000100

000000010

000000001

000100000

000010000

000100000

yz =

100000000

010000000

001000000

000100000

000010000

000001000

000000100

000000010

000000001

After that we calculate the trace of each matrix, and that's will be the

character of each corresponding symmetry operation, then we will have the

reducible representation N3 as follows:

C v2 E C 2 XZ

YZ

N3 9 -1 1 3

[4,8,16,17].

2.8 The Relationship Between Reducible and Irreducible

Representations:

A representation could be used to mimic the symmetry properties of

a molecule by describing the interaction of group operations with a

30

particular basis. Any collection of basis vectors that complies with the

molecular symmetry can generate a character representation of the group,

but in most cases it will be a reducible one and so can be simplified. The

simplification of a reducible representation can be made using the data

for the set of irreducible representations available in the standard character

tables. In general, Any reducible representation can be constructed as a

linear sum of the standard irreducible representations and

we can write )(RX as :

)(RX = )(RXn jj

j

(2.2)

Where jn is the number of times that the jth irreducible

representation occurs. These jn values may be 0, and )(RX denotes the

character of the matrix corresponding to a symmetry operation R

in the

reducible representation . By multiplying each side of Eq. (2.4) by the

character of the operation in the ith irreducible representation, )(RX i . Then

summation over all the operations of the group is performed.

The sum is given by:

)(RXR

)(RX i = R

)()( RXRXnj

ijj

= j

jn )()( RXRXR

ij

(2.3)

Using property 4 of the irreducible representations:

)()( RXRXR

ij = h ij (2.4)

31

we get:

)()( RXRXR

ij = j

jn h ij (2.5)

when i =j, Eq. (2.7) becomes:

)()( RXRXR

ij = i

in h (2.6)

Therefore

in = (1/h) )()( RXRXR

ij (2.7)

If pg refers to the number of symmetry operations in the p th class of the

point group, Eq. (2.9) becomes:

in = (1/h) )()( pR

ipp RXRXgp

(2.8)

pR in Eq. (2.8) denotes the symmetry operation in the p th class. The

resulting formula In Eq. (2.8) is called the " Reduction Formula ". It is

used to determine the number of times the i th irreducible representation

occurs in the reducible representation. [4]

2.9 Examples on Solving Reduction Formula:

2.9.1 D h3 Point Group:

This example presents the calculations of the reduction formula for

point groups which character tables of real elements.

In a D h3 symmetry, apply the reduction formula to reduce the

hypothetical reducible representation:

32

D h3 E 2C 3 3C 2 h 2S 3 3 v

9 3 -1 5 5 -1

Solution:

The character Table for the D h3 point group is:

Table (2.4): Character Table of D h3 Point Group.

D h3 E 2C 3 3C 2 h 2S 3 3 v

A'1

1 1 1 1 1 1 A' 2 1 1 -1 1 1 -1 E' 2 -1 0 2 -1 0

A"1 1 1 1 -1 -1 -1 A" 2 1 1 -1 -1 -1 1 E" 2 -1 0 -2 1 0

For the previous example, = 12, and the D h3 Character Table shows

that for the A'1 irreducible representation:

)( pi RX = 1 that corresponds to E

1 that corresponds to C 3


1 that corresponds to h

1 that corresponds to S 3

1 that corresponds to v

From the hypothetical representation, shown above, the

corresponding characters are:

9 for E

33

3 for C 3

-1 for C 2

5 for h

5 for S 3

-1 for v

From the reduction formula, then:

Number of times A'1

representation occurs = 12/1 [ 9×1×1 + 3×1×2 + -

1×1×3 + 5×1×1 + 5×1×2 + -1×1×3] = 2 times

Similarly, Number of times A' 2 representation occurs = 12/1 [ 9×1×1 +

3×1×2 + -1×-1×3 + 5×1×1 + 5×1×2 + -1×-1×3] = 3 times

And Number of times E' representation occurs = 12/1 [ 9×2×1 + 3×-1×2

+ -1×0×3 + 5×2×1 + 5×-1×2 + -1×0×3] = 1 time

Similarly we find:

Number of times A"1 representation occurs = 0, Number of times

A" 2 representation occurs = 0, and Number of times E" representation

occurs = 1, then = 2 A'1 + 3 A' 2 + E' + E". [7].

2.9.2 Cubic Point Groups:


hO Point Group. In hO symmetry, apply the reduction formula to reduce the

hypothetical reducible representation:

34

hO E 8C 3 6C 2

6C 4 3C 2

i 6S 4 8S 6 3 h 6 d

6 0 0 2 2 0 0 0 4 2

Solution:

The character Table for the hO point group is:

Table (2.5): Character Table of hO Point Group.

hO E 8C 3

6C 2

6C 4 3C' 2 ( C 24 )

i 6S 4

8S 6

3 h

6 d

A g1 1 1 1 1 1 1 1 1 1 1

A g2 1 1 -1 -1 1 1 -1 1 1 -1

E g 2 -1 0 0 2 2 0 -1 2 0

T g1 3 0 -1 1 -1 3 1 0 -1 -1

T g2 3 0 1 -1 -1 3 -1 0 -1 1

A u1 1 1 1 1 1 -1 -1 -1 -1 -1

A u2 1 1 -1 -1 1 -1 1 -1 -1 1

E u 2 -1 0 0 2 -2 0 1 -2 0

T u1 3 0 1 1 -1 -3 -1 0 1 1

T u2 3 0 -1 -1 -1 -3 1 0 1 -1

For the previous example, h = 48, and the hO Character Table shows that

for the A g1 irreducible representation:

)( pi RX = 1 that corresponds to E.

1 that corresponds to C 3 .



1 that corresponds to C' 2 ( C 24 ).

1 that corresponds to i.

35

1 that corresponds to S 4 .

1 that corresponds to S 6 .

1 that corresponds to h .

1 that corresponds to d .



6 for E.

0 for C 3 .

0 for C 2 .

2 for C 4 .

0 for C' 2 ( C 24 ).

0 for i.

0 for S 4 .

0 for S 6 .

0 for h .

0 for d .

36

From the reduction formula, then: Number of times A g1

representation occurs = 48/1 [ 6×1×1 + 0×1×8 + 0×1×6 + 2×1×6 +

2×1×3 + 0×1×1 + 0×1×6 + 0×1×8 + 4×1×3 + 2×1×6] = 1 time.

Similarly, number of times A g2 representation occurs = 48/1 [

6×1×1 + 0×1×8 + 0×-1×6 + 2×-1×6 + 2×1×3 + 0×1×1 + 0×-1×6 +

0×1×8 + 4× -1×3 + 2×1×6] = 0 time.

And number of times E g representation occurs = 48/1 [ 6×2×1 +

0×-1×8 + 0×0×6 + 2×0×6 + 2×2×3 + 0×2×1 + 0×0×6 + 0×-1×8 + 4×

2×3 + 2×0×6] = 1 time.

Similarly we find: Number of times T g1 representation occurs = 0,

number of times T g2 representation occurs = 0, number of times A u1

representation occurs = 0, number of times A u2 representation occurs = 0,

number of times E u representation occurs = 0, number of times T u1

representation occurs = 1 and Number of times T u2 representation occurs =

0, then = A g1 + E g + T u1 .

2.9.3 Point Groups with Complex Elements:


point groups which character tables of real and complex elements.

When dealing with applications in groups with imaginary characters,

it is sometimes convenient to add the two complex-conjugate

representations to obtain a representation of real characters. When the

paired representations have i

and i

characters, the addition is

37

straightforward; that is, i

+ ( i ) = 0.When they have and * characters,

where = exp(2 i /n), the following identities are used in taking the sum:

p = exp( pi2 /n) = cos( p2 /n) + i sin( p2 /n) (2.11)

p* = exp( pi2 /n) = cos( p2 /n) - i sin( p2 /n) (2.12)

Combining Eqs. (2.11) and (2.12), we have

p + p* = 2 cos( p2 /n) (2.13)

Thus all complex-conjugate characters in the two irreducible

representations add to give real-number characters.

For example

the character table for C h4 Point Group:

Table (2.6): Character Table of C h4 Point Group.

iiiiE

iiii

iB

A

iiiiE

iiii

B

A

SSiCCCEC

u

u

u

g

g

g

hh

1111

1111

1111111

11111111

1111

1111

11111111

111111114

34

34244

And the modified character table for C h4 Point Group:

38

Table (2.7): Modified Character Table of C h4 Point Group.

02020202}{

11111111

11111111

02020202}{

11111111

111111114

34

34244

u

u

u

g

g

g

hh

E

B

A

E

B

A

SSiCCCEC

The reduction formula can be applied depending on the modified

character table to reduce the following hypothetical reducible

representation:

C h4 E C 4 C 2 C 34 i S 3

4 h S 4

5 -1 1 -1 3 -3 -1 -3

For this example, h = 8, and the modified character table shows that for the

A g irreducible representation:

)( pi RX = 1 that corresponds to E




1 that corresponds to i


39

1 that corresponds to h




5 for E

-1 for C 4

1 for C 2

-1 for C 34

3 for i

-3 for S 34

-1 for h

-3 for S 4

From the reduction formula, then:

Number of times A g representation occurs = [ 5×1×1 + -1×1×1 +

1×1×1 + -1×1×1 + 3×1×1 + -3×1×1 + -1×1×1 + -3×1×1] = 0.

Similarly, Number of times B g representation occurs = [ 5×1×1 +

-1×-1×1 + 1×1×1 + -1×-1×1 + 3×1×1 + -3×-1×1 + -1×1×1 + -3×-1×1] =

2 times.

40

And Number of times { E g } representation occurs = [ 5×2×1 +

-1×0×1 + 1×-2×1 + -1×0×1 + 3×2×1 + -3×0×1 + -1×-2×1 + -3×0×1] =

2 time.

Similarly we find:

Number of times A u representation occurs = 1, Number of times B u

representation occurs = 0, and Number of times { E u } representation

occurs = 0, then = 2 B g + 2{ E g } + A u . [4,12].

2.10 Reduction of D h and C v Point Groups

The need to know what irreducible representations and how many of

each are present in a reducible representation of the infinite C v

or D h

point group arises in trying to classify the symmetry of normal modes of

vibration or the electronic state of linear molecules. Since the standard

reduction formula does not hold for these point groups, because their order

is infinite " h ".

Many approaches and methods were proposed, but the most flexible

one was " S L " Method.

Dennis P. Strommen and Ellis R. Lippincott developed a method "S

L

Method" in order to reduce the reducible representations by the following

steps:

1. Assume a lower symmetry point group which corresponds to a

subgroup G of the infinite point group G .

41

2. Construct a reducible representation N3

to the linear molecule

depending on its atoms number and G symmetry elements.

3. Apply the reduction formula to reduce N3 .

4. From the character table of G write t and R .

5. Subtract t and R from N3 to get vib of the linear molecule.

6. Finally compare the basis vectors of the irreducible representations

under G with those obtained for the molecule assuming G.

The following examples will show the how to apply this method:

2.10.1 Linear XYZ:

1. G = C v ; G = C v2 .

2. The reducible representation N3 for a 3-atoms molecule belongs to

C v2 point group is:

E C 2 xz yz

N3 9 -3 3 3

By applying the reduction formula, N3 = 3A1 + 3B1 + 3B 2 .

3. From the character table of C v2 :

t = A1 + B1 + B 2 , and R = B1 + B 2 .

4. vib = N3 - t - R

vib = 2A1 + B1 + B 2 .

42

6.

yB

xBzA

CBasisC vv

2

1

1

2

Obviously the 2A1 symmetry species transform as 2 species under

the molecular symmetry C v2 , while the B1 and B 2 symmetry species

transform together as species, thus vib = 2 + .

2.10.2 Linear XY 2 :

1. G = D h ; G = D h2 .


D h2 point group is:

E C 2 )(z

C 2 )(y

C 2 )(x i xy xz yz

N3 9 -3 -1 -1 -3 1 3 3

3. By applying the reduction formula:

N3 = A g + B g2 + B g3 + 2 B u1 + 2B u2 + 2B u3 .

4. From the character table of D h2 :

t = B u1 + B u2 + B u3 , and R = B g2 + B g3 .

5. vib = N3 - t - R

vib = A g + B g2 + B u1 + B u2 + B u3 .

43

6.

yB

xB

zBzA

DBasisD

u

uu

uu

gg

hh

3

2

1

21

2

Then clearly under D h the solution is: vib = g + u + u .

2.10.3 Linear Symmetrical X 2 Y 2 :

1. G = D h ; G = D h2 .


D h2 point group is:

E C 2 )(z

C 2 )(y

C 2 )(x i xy xz yz

N3 12 -4 0 0 0 0 4 4

3. By applying the reduction formula:

N3 = 2A g1 + 2B g2 + 2B g3 + 2 B u1 + 2B u2 + 2B u3 .

4. From the character table of D h2 :

t = B u1 + B u2 + B u3 , and R = B g2 + B g3 .

5. vib = N3 - t - R

vib = 2A g1 + B g2 + B g3 + B u1 + B u2 + B u3 .

44

6.

zB

yB

xB

yzB

xzBzA

DBasisD

u

uu

uu

g

gg

gg

hh

3

2

1

3

2

21

2

Then clearly under D h

the solution is: vib = 2 g + g + u + u .

[8, 18].

2.11 Comments:

The "Reduction Formula" is useful for all point groups, but the

manual Calculations become more difficult and time consuming, especially

with high symmetry point groups like D h4 and D h6 .

45

Chapter Three

Methodology and the Software Composition

46

3.1 Introduction:

In chemical applications of group theory, the so called "Reduction

Formula" is being heavily used to find numbers of irreducible

representations (of a given symmetry point group) within a given reducible

representation. The reduction formula may be solved out via two different

methods, namely: the trial & error method, and the reduction formula

method. Both methods are clarified by the following example.

3.1.1 Example:

In a C v2 symmetry, use two different methods (trial & error and

Reduction Formula) to reduce the hypothetical reducible representation:

E C 2 xz yz

3 3 1 1

Solution:

The Character Table for the C v2 point group is:

C v2 E C 2 xz yz

A1 1 1 1 1 A 2 1 1 -1 -1 B1 1 -1 1 -1 B 2 1 -1 -1 1

1. Reduction by first method (trial & error): If we sum up 2A1 + A 2 ,

by trial and error, we can then have : = 2A1 +A 2

The trial & error method may easily work for simple cases of low

symmetry molecules. For higher symmetries, such as D h4 , D h6 or higher,

such method will be a tedious task and will not be practical .

47

2. The reduction formula method :

Number of times for an irreducible representation:

in = (1/h) )()( pR

ipp RXRXgp

For the previous example, h = 4 , then:

Number of times A1 representation occurs = ¼ [ 3×1×1 + 3×1×1 +

1×1×1 + 1×1×1 ] = 2 times

Similarly, number of times A 2 representation occurs = ¼ [ 3×1×1 +

3×1×1 + 1×(-1)×1 + 1×(-1)×1 ] = 1 time

Similarly we find:

Number of times B1 representation occurs = 0 and Number of times

B 2 representation occurs = 0, then = 2A1 +A 2 . [7].

The " Reduction Formula " is useful for all point groups, but the

manual calculations become more difficult and take long time especially

with high symmetry point groups. Therefore, there is a need to construct a

program performs reduction formula calculations with no errors.

It is assumed that Matlab & Visual Basic programming languages

can be used to construct software that perform such mathematical

operations saving time and effort while avoiding calculation errors.

48

3.2 Other Software Tools:

Through our literature search and web search we found the following

software which deal with the applications of group theory in chemistry:

1. Computer Programs to Determine the Symmetries of Vibrational

Modes of Nonlinear Molecules:

Two computer programs written in the BASIC and FORTRAN IV

languages to determine the symmetries of the vibrational modes of a

nonlinear molecules were included in K. V. RAMAN book "Group Theory

and Its Applications to Chemistry". The reducible representation is split

into the various irreducible representations using the reduction formula.

One can thus obtain the symmetries of the 3N degrees of freedom for the

nonlinear molecules. The symmetries of translations and rotations are

subtracted to determine the symmetries of the vibrational modes of the

molecule. [4].

2. Online Software to Solve the Reduction Formula for Different

Symmetry Point Groups:

This online software was constructed by Computational Laboratory

for Analysis, Modeling, and Visualization (CLAMV) in Jacobs University

In Germany, and its available on the web-site:

http://symmetry.jacobs-university.de This web-site presents:

The character tables for chemically important point groups.

http://symmetry.jacobs-university.de

49

Predefined reducible representations ( valencevibN ,,,3 ) for chosen

molecules of 3-atoms, 4-atoms and 5-atoms.

Calculation of the reduction formula for different kinds of reducible

representations for all point groups.

Additional information about every point group.

Determination point group from molecular structure. [19].

3. Finite Group Theory for Large Systems. 2. Generating Relations

and Irreducible Representations for the Icosahedral Point Group,

SCRIPTF.h:

SCRIPTF.h software generates relations and reducible

representations for the icosahedral point group that are suited to

computerize projection of symmetry-adapted bases of arbitrary spaces

invariant to the point group. It is a good prototype for symmetry-adaptation

to a large finite nonabelian group.

4. GROUPONC Software:

GROUPONC is a Turbo Pascal version for IMB PC to construct

matrices of symmetry operations.

5. Group Theory with MathCAD " Issue 9801MW for Mac OS and

Windows :

MathCAD has a powerful array of matrix manipulation commands

that make it an ideal programming environment for applying group theory

50

to chemical problems. It is used to do symmetry analyses on number of

examples involving vibrational and electronic spectroscopy and chemical

bonding.

6. BETHE Program:

BETHE program was recently developed for using point group

symmetries in various fields of chemistry, including molecular

spectroscopy, ligand-field theory and construction of molecular wave

functions. (These Software are not available in our library but they may be

obtained via the British library).

3.3 Differences Between Our Software and Earlier Software:

Our Software was developed by Visual Basic 6.0 Programming

Language, and it is the first time using this programming language in

the field of "Chemical Applications of group Theory".

Our Software functions are similar to other software functions to

some extent, but there are differences:

1. RAMAN Software deals with " Nonlinear Molecules ", but our

software deals with " Linear Molecules " and calculate vib and

its " Reduction Formula ".

2. Jacobs online software can predefine all kinds of reducible

representations, but our software can predefine only vib and 3N.

51

3.4 Languages:

Matlab & Visual Basic were used to construct our software, they

have many advantages and make programs easier to use by providing them

with " Graphical User Interfaces ".

3.4.1 Visual Basic 6.0:

Visual Basic has many advantages, and they are:

It is Easy to learn and yet it's a powerful programming language.

Windows based applications and games can be developed by It.

It's a simple language. Things that may be difficult to program with

other languages can be done in Visual Basic very easily.

Because Visual Basic is so popular, there are many good resources

(Books, Web sites) that can help to learn the language.

Visual Basic has the widest variety of tools that can be downloaded

from the internet and used in programs. A Visual Basic project is

usually made up of:

Forms - Windows that you create for user interface.

Controls - Graphical features drawn on forms to allow user

Interaction (text boxes, labels, command buttons, etc.).

Properties - Every characteristic of a form or control is specified by

a property. Example properties include names, captions, size, color,

52

position, and contents. Visual Basic applies default properties. You

can change properties at design time or run time.

Methods - Built-in procedure that can be invoked to impart some

action to a particular object.

Event Procedures - Code related to some object. This is the code

that is executed when a certain event occurs.

In General there are five primary steps involved in building a Visual

Basic application, and they are:

Creation of the user interface for a new program. The Form window

is the base of developing Visual Basic applications. Controls are

added to the form by choosing them from the Visual Basic "tool

box" with the mouse, and inserting them in the form.

Figure (3.1): The Form in Visual Basic 6.0.

Figure (3.2): The Tool Box in Visual Basic 6.0.

53

Setting the properties for each object in the user interface.

Once forms/controls are created, the programmer can change the

properties (appearance, structure etc.) related to those objects in that

particular objects properties window. From this window, the programmer

can choose the property he/she wants to change from the list and change its

corresponding setting.

Figure (3.3): The Properties Windows.

Writing program code by Visual Basic language which directs

specific tasks at runtime.

Saving and running the project.

Building an executable file of the project. [20].

3.4.2 Matlab 7.4

Matlab is an interpreted language for numerical computation. It

allows one to perform numerical calculations, and visualize the results

without the need for complicated and time consuming programming.

54

Matlab allows its users to accurately solve problems, produce graphics

easily and produce code efficiently. Matlab has many advantages such as:

Using MATLAB tools and toolboxes, makes it possible to develop a

prototype of an application for a relatively very short time.

It is easy to exchange toolboxes and parts of codes within a team

and between teams working in the same area.

MATLAB is available for a broad diversity of environments: MS-

Windows, Linux, Sun Solaris etc.

Powerful built-in math functions and extensive function libraries.

[21-23].

3.5 Graphical User Interface (GUI)

A graphical user interface (GUI) is a human-computer interface (i.e.,

a way for humans to interact with computers) that uses windows, icons and

menus and which can be manipulated by a mouse (and often to a limited

extent by a keyboard as well), called components, that enable a user to

perform interactive tasks. The user of the GUI does not have to create a

script or type commands at the command line to accomplish the tasks.

The user of a GUI needs not understand the details of how the tasks

are performed. A major advantage of GUIs is that they make computer

operation more intuitive, and thus easier to learn and use. GUIs generally

provide users with immediate, visual feedback about the effect of each

55

action. GUIs created in MATLAB and Visual Basic software can read,

write and display data. [24].

3.6 The Software Composition

Our software consists of two parts, the first one that is built by Visual

Basic 6.0, and to a lesser extent the second one that is built by Matlab.

3.6.1 The First Part " Visual Basic 6.0 Part "

The software is composed of the main form or screen shown in

Figure 3.6.1.1 that gives the user three choices to start, each of which

contains sub forms. Each sub form displays input boxes, output boxes and

command buttons. The language of this software is English and it needs

Windows Vista to run it. The software was composed on laptop with 2 GB

RAM.

Figure (3.4): The Main Menu Form.

56

3.6.1.1 Calculate Reduction Formula Directly

When the user Presses this button a new form will appear as shown in

Figure 3.5 The form gives the user 10 command buttons named by the

finite point groups: The Nonaxial Point Groups, C n Point

Groups, D n Point Groups, C nv Point Groups, C nh Point

Groups, D nh Point Groups, D nd Point Groups, Cubic Point

Groups, the Icosahedral Point Groups and S n Point Groups.

Figure (3.5): The "Calculate Reduction Formula Directly" Form.

When the user presses " The Noaxial Point Groups" button he/she

gets the following form as shown in Figure 3.6.

57

Figure (3.6): The Nonaxial Point Groups Form.

When the user presses any commands buttons he/she gets the

corresponding sub forms shown below in Figures 3.7

3.9

Figure (3.7): The C 1 Point Group Form.

58

Figure (3.8): The C s Point Group Form.

Figure (3.9): The C i Point Group Form.

When the user presses "The C n Point Groups" button he gets the

following form shown in Figure 3.10.

59

Figure (3.10): The C n Point Groups Form.

When the user presses any commands buttons he/she gets the

corresponding sub forms shown below in Figure 3.11 - 3.17.


60



61



62



When the user presses " The D n Point Groups" button he gets the


63

Figure (3.18): The D n Point Groups Form.

When the user presses any commands buttons he gets the

corresponding Sub forms shown below in Figures 3.19 - 3.23.

Figure (3.19): The D 2 Point Group Form.

64



65



When the user presses " The C nv Point Groups" he gets the


66

Figure (3.24): The C nv Point Groups Form.

When the user presses any commands buttons of ( C v2 , C v3 , C v4 , C v5

and C v6 ) he/she gets the corresponding sub forms shown below in Figures

3.25 - 3.29.

Figure (3.25): The C v2 Point Group Form.

67



68



When the user presses " The C nh Point Groups" he/she gets the


69

Figure (3.30): The C nh Point Groups Form.

When the user presses any commands buttons of (C h2 , C h3 , C h4 , C h5

and C h6 ) he/she gets the corresponding Sub forms shown below in

Figures 3.31 - 3.35.

Figure (3.31): The C h2 Point Group Form.

70



71



When the user presses " The D nh Point Groups" button he gets the


72

Figure (3.36): The D nh Point Groups Form.

When the user presses any commands buttons of ( D h2 , D h3 , D h4 ,

D h5 , D h6 and D h8 ) he gets the corresponding Sub forms shown below in

Figures 3.37 - 3.42.

Figure (3.37): The D h2 Point Group Form.

73



74



75


When the user presses " The D nd Point Groups" button he gets the


Figure (3.43): The D nd Point Groups Form.

When the user presses any commands buttons of ( D d2 , D d3 , D d4 ,

D d5 and D d6 ) he gets the corresponding Sub forms shown below in

Figures 3.44 - 3.48.

76

Figure (3.44): The D d2 Point Group Form.


77



78


When the user presses " The Cubic Point Groups" button he gets the


Figure (3.49): The Cubic Point Groups Form.

When the user presses any commands buttons of (T , dT , hT , O and

hO ) he gets the corresponding Sub forms shown below in Figures 3.50 -

3.54.

79

Figure (3.50): The T Point Group Form.

Figure (3.51): The dT Point Group Form.

80

Figure (3.52): The hT Point Group Form.

Figure (3.53): The O Point Group Form.

81

Figure (3.54): The hO Point Group Form.

When the user presses "The S n Point Groups" button he/she gets the


Figure (3.55): The S n Point Groups Form.

82

When the user presses any commands buttons of ( S 4 , S 6 and S 8 ) he

gets the corresponding Sub forms shown below in Figures 3.56 - Figure

3.58.

Figure (3.56): The S 4 Point Group Form.


83


3.6.1.2 Infinite Point Groups::

This choice deals with linear molecules belong to C v

and D h

point

groups. The user has only to enter the number of atoms of the molecule

and press the " start " button to get the reduction formula of red and vib

representations .When the user presses this button a new form will appear

shown in Figure 3.59 gives the user 4 command buttons.

Figure (3.59): The Infinite Point Groups Form.

84

When the user presses button " D inf h " the form of D h

Point

Group will appear shown below in Figure 3.60.

Figure (3.60): The D h Point Group Form.

The user will press " Use D h2 as a subgroup" button and a new form

will appear shown below in Figure 3.61

Figure (3.61): The D h2 Subgroup Form.

85

When the user presses button " C inf v " the form of C v

Point

Group will appear shown below in Figure 3.62.

Figure (3.62): The C v Point Group Form.

The user will press " Use C v2 as a subgroup" button and a new form

will appear shown below in Figure 3.63.

Figure (3.60): The C v2 Subgroup Form.

86

The programming of these forms depends on S-L method shown in

2.10.

3.6.1.3 Find Reducible Representations Then Reduce Them For Finite

Point Groups:

here the user can get the N3 reducible representation for 6 chosen point

groups ( C v2 , C v3 , C v4 , D h2 , D h4 and T d ) listed in the Form shown in

Figure 3.64 below.

Figure (3.64): The Reducible Representations For Chosen Point Groups Form.

When the user presses any commands buttons of ( C v2 , C v3 , C v4 ,

D h2 , D h4 and T d ) he gets the corresponding Sub forms shown in Figures

3.65 - 3.70.

87

Figure (3.65): Reducible Representation For C v2 Point Group Form.


88


Figure (3.68): Reducible Representation For D h2 Point Group Form.

89

Figure (3.69): Reducible Representation For D h4 Point Group Form.

Figure (3.70): Reducible Representation for dT Point Group Form.

3.6.1.4 Functions Of Common Command Buttons:

The function of button "Quit" is to quit the program, the function of

button "Main Menu" is to get back the "Main Menu Form", the function

90

of button "Finite Point Group" is to get back the "Calculate Reduction

Formula Directly Form", the function of button "Clear" is to delete

contents of the text boxes in the form, the function of button " Start " is to

run the program after inputting the reducible representation elements in

order to get the answer, the function of button "Back" is to get back the

previous form, the function of the button "Finite Point Groups" is to get

back the form of the Finite Point groups, similarly the user can get the form

of any point group by pressing the command button labeled by the name of

that point group.

The software contains the most famous 47 point groups. In brief the

software is constructed in order to perform three functions:

1. Calculation the "Reduction Formula" for Finite Point Groups for a given

reducible representation.

2. Calculation the "Reduction Formula" for Infinite Point Groups: C v

and

D h .

3. Finding the reducible representation N3 and reduce it by the "Reduction

Formula".

The functions of our software are illustrated in Figure 3.71 below.

91

Figure (3.71): The Flowchart of the Visual Basic software Part.

Start

Is the Point Group Finite?

Is the Reducible

Representation known?

Display red , vib

Enter Number of Atoms of the Linear

" Infinite " Molecule

Calculate red , vib

Enter Reducible Representation

Elements

Calculate Reduction Formula

Display the

Enter Number of

Atoms

Calculate N3

and Reduce it

Display the

End

Yes

No

Yes

No

92

3.6.2 The Second Part "Matlab 7.4 Part"

At the beginning, the researcher tried to construct the software by

using Matlab 7.4 programming language. This is because there are point

groups of imaginary and real elements, and Matlab deals with such kinds of

data.

Therefore, the researcher designed 20 GUI's for several point groups

by using " GUIDE " in Matlab "shown below in Figure 3.5.1.68 . She then

transformed them into EXE files to work as standalone applications in any

computer after installation of MCRInstaller.EXE on such computer. The

following GUI's have the same design, three of them are shown in Figures

3.72 - 3.74. [23].

Figure (3.72): The blank GUI in the GUIDE Layout Editor.

93

Figure (3.73): S 4 Point Group GUI.

Figure (3.74): C 3 Point Group GUI.

94

Figure (3.75): C h6 Point Group GUI.

These GUI's perform one function, they are designed to reduce

reducible representations. The Visual Basic software is more flexible and

its forms properties is better than Matlab GUI's.

95

Chapter Four

Results and Discussions

96

4.1 Software Applications:

In this chapter we'll introduce all the results and discussions on the

applications of our software on a variety of examples of various Point

Groups.

4.1.1 D nh Point Groups:

This example was manually solved in 2.9.1 for D h3 Point Group where:

D h3 E 2C 3 3C 2 h 2S 3 3 v

9 3 -1 5 5 -1

And the manual solution was = 2 A'1

+ 3 A' 2 + E' + E". By using

the software the user has to press " Calculate Reduction Formula

Directly " button first from the "Main Menu" form, then press "D nh Point

Group" button and choose D h3 , finally the user has to enter the reducible

representation elements in the horizontal text boxes then press "Start"

button. The form is shown in Figure 4.1 and Figure 4.2 below.

Figure (4.1): A picture showing reducible representation elements after being entered into their respective places inside software form of D h3 Point Group.

97

Figure (4.2): Solution of Reduction Formula using software as applied on the reducible representation in D h3 Point Group.

The solution is thus: = 2 A'1 + 3 A' 2 + E' + E"

The result is completely consistent that obtained using the manual

standard method outlined in 2.9.1.

4.1.2 C nh Point Groups:

This example was manually solved in 2.9.3 for C h4 Point Group

where:

C h4 E C 4 C 2 C 34 i S 3

4 h S 4

5 -1 1 -1 3 -3 -1 -3

And the manual solution was = 2 B g + 2{ E g } + A u . The solution by the

software is shown in Figure 4.3 and Figure 4.4 below.

98

Figure (4.3): A picture showing reducible representation elements after being entered into their respective places inside software form of C h4 Point Group.

Figure (4.4): Solution of Reduction Formula using software as applied on the reducible representation in C h4 Point Group.

And the solution is thus: = 2 B g + 2{ E g } + A u .

The result is completely consistent that obtained using the manual


99

4.1.3 C nv Point Groups:

This example was manually solved in 3.1.1 for C v2 Point Group

where:

C v2 E C 2 xz yz

3 3 1 1

And the manual solution was = 2A1 +A 2 . The solution by the

software is shown in Figure 4.5 and Figure 4.6 below.

Figure (4.5): A picture showing reducible representation elements after being entered into their respective places inside software form of C v2 Point Group.

100

Figure (4.6): Solution of Reduction Formula using software as applied on the reducible representation in C v2 Point Group.

And the solution is thus: = 2A1 + A 2 . The result is completely

consistent that obtained using the manual standard method outlined in

3.1.1.

4.1.4 Cubic Point Groups:

This example was manual solved in 2.9.2 for hO Point Group where:

hO E 8C 3 6C 2 6C 4 3C 2 i 6S 4 8S 6 3 h 6 d

6 0 0 2 2 0 0 0 4 2

And the manual solution was = A g1 + E g + T u1 . The solution by

the software is shown in Figure 4.7 and Figure 4.8 below.

101

Figure (4.7): A picture showing reducible representation elements after being entered into their respective places inside software form of Oh Point Group.

Figure (4.8): Solution of Reduction Formula using software as applied on the reducible representation in hO Point Group.

And the solution is thus: = A g1 + E g + T u1 . The result is

completely consistent that obtained using the manual standard method

outlined in 2.9.2.

102

4.1.5 Infinite Point Groups:

4.1.5.1 Linear XYZ:

This example was manual solved in 2.10.1 for C v

Point Group by

S-L Method and C v2 was used as a subgroup, where:

E C 2 xz yz

red 9 -3 3 3

And the manual solution was red = 3A1 + 3B1 + 3B 2 and vib =

2 + .

By using the software the user has to press " Infinite Point Groups "

button first from the "Main Menu " form, then press " C v

" button and

press " Use C v2 as a subgroup " , finally the user has to enter the number

of the linear molecule atoms in the upper text box then press " Start "

button. The Form is shown below in Figure 4.9 and Figure 4.10.

Figure (4.9): Number of atoms entered in C v2 subgroup form for linear molecules.

103

Figure (4.10): red and vib

in C v2 subgroup form for linear molecules based on the software.

The results are completely consistent that obtained using the manual


4.1.5.2 Linear XY 2 :

This example was manually solved in 2.10.2 for D h

Point Group by

S-L Method and D h2 was used as a subgroup, where:

E C 2 )(z

C 2 )(y

C 2 )(x i xy xz yz

red 9 -3 -1 -1 -3 1 3 3

And the manual solution was red = 2A g1 + 2B g2 + 2B g3 + 2 B u1 +

2B u2 + 2B u3 and vib = g + u + u .

By using the software the user has to press " Infinite Point Groups "

button first from the "Main Menu " form, then press " D h

" button and

press " Use D h2 as a subgroup ", finally the user has to enter the number

104

of the linear molecule atoms in the upper text box then press " Start "

button. The Form is shown below in Figure 4.11 and Figure 4.12.

Figure (4.11): Number of atoms entered in D h2 subgroup form for linear molecules.


in D h2 subgroup form for linear molecules based on the software.

105



4.1.5.3 Linear Symmetrical X 2 Y 2 :

This example was manually solved in 2.10.3 for D h

Point Group by

S-L Method and D h2 was used as a subgroup, where:

E C 2 )(z

C 2 )(y

C 2 )(x i xy xz yz

red 12 -4 0 0 0 0 4 4

And the manual solution was red = A g1 + 2B g2 + 2B g3 + 2 B u1 +

2B u2 + 2B u3 , and vib = 2 g + g + u + u .

The solution by the software is shown in Figures 4.13 and 4.14

below.

Figure (4.13): Number of atoms entered in D h2 subgroup form for linear symmetrical molecules.

106


in D h2 subgroup form for linear symmetrical molecules based on the software.



4.1.6 Constructing and reducing N3 :

The following examples showing how to use our software in order to

construct and reduce N3 representations. The user has only to enter the

number of atoms according to the condition stated in the form.

By using the software the user has to press "Find Reducible

Representations Then Reduce Them For Finite Point Groups" button.

This button is present in the "Main Menu" form. The user will then

press any one of buttons labeled by the names of chosen Point Groups.

Finally the user has to enter the number molecule atoms in the upper text

107

box, and press "Start" button. The user will then get the reducible

representation and its reduction Formula. The forms are shown below in

Figures 4.15

4.20.

Figure (4.15): N3 and its Reduction Formula for a 6-atomic molecule that belongs

to C v2 Point Group.

Figure (4.16): N3 and its Reduction Formula for a 6-atomic molecule that

belongs to C v3 Point Group.

108


belongs to C v4 Point Group.


to D h2 Point Group.

109


belongs to D h4 Point Group.


to dT Point Group.

110

4.2 General Comments on the Visual Basic Software Part:

It is obvious that the software gives out the same results as the manual

solution, with far less time and effort. The user can get the solution

within only a few seconds without any rigorous mathematical

calculations.

The software gives out accurate results and avoids errors.

The software does not accept wrong entries and gives a statement that

shows that like: "Wrong Entries!!!!".

Figure 4.21 and Figure 4.22 show that.

Figure (4.21): Wrong Entry In D h2 Reducible Representation Form.

111

Figure (4.22): Wrong Entry In D h2 Point Group Form.

4.3 Applications on the Matlab 7.4 Part of The Software:

Here we'll introduce applications of the second part of our software

that was built by Matlab 7.4 on S 4 , C 3 and C h6 Point Groups.

The user needs to open " Matlab Reduction Formula " folder, then

he/she must choose EXE files named by the Point Groups names and

ignore the "ctf" files.

When the user double click the EXE file the GUI of the chosen

Point Group will appear, after that the user can enter the reducible

representation elements in the horizontal edit text boxes, then he/she

presses " Start " push button in order to get the Reduction Formula.

Figures 4.23

4.25 shown below illustrate the applications on S 4 ,

C 3 and C h6 Point Groups.

112

Figure (4.23): Solving out the Reduction Formula in C 3 GUI.

Figure (4.24): Solving out the Reduction Formula in S 4 GUI.

113

Figure (4.25: Solving out the Reduction Formula in C h6 GUI.

4.4 Conclusion:

a. A new software has been constructed in this work.

b. The software is our own and independent of any other earlier software

in the field.

c. The software is based on Visual basic 6.0, and to a lesser extent on

Matlab.

d. The software is intended to be used in solving so called "Reduction

formula" which is heavily used in Chemical Applications of Group

Theory.

e. The software runs successfully with no difficulties or interruptions,

and spans most widely used 47 point groups.

114

f. The calculations based on the software are completely consistent with

standard manual solution of the "Reduction Formula".

g. The software is able to perform the following functions:

1. Reducing Reducible Representations for most known Point

Groups.

2. Finding Reducible Representations " red and vib " for Infinite Point

Groups " C v and D h " and reducing them by S-L Method.

3. Finding Reducible Representation N3

and reducing it for six

chosen Point Groups " C v2 , C v3 , C v4 , D h2 , D h4 and dT ".

4.5 Suggestions For Future Works:

The researcher suggests expanding the software functions like the

Jacobs software, by Continuing developing the it, and make it online

software.

115

References

116

References

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2007, PP 17-19.

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http://en.wikipedia.org/wiki/History_of_group_theory (Accessed on

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4. Raman, K.V.: Group Theory and Its applications to Chemistry.

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http://en.wikipedia.org/wiki/History_of_group_theory

117

9. Hochstrasser, Robin M.: Molecular Aspects of Symmetry. New

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12. Carter, Robert L.: Molecular Symmetry and Group Theory. New

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Chemists.N.Y. Kluwer Academic Publisher,2004, PP 15-19.

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Eyes of Chemists. 2 nd . N.Y. Plenum Press.1995, PP198-200.

18. Strommen, Dennis P., Ellis R. Lippincott: Comments On Infinite

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118

19. http://symmetry.jacobs-university.de/ (Accessed on December 4,

2008).

20. www.ctp.bilkent.edu.tr/~ctp108/ctp108_ln_w1.pdf (Accessed on

February 25,2009).

21. Hunt, Brian R.: A Guide to Matlab for Beginners and Experienced

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22. Eyre, David ( August 9, 1998 ). Matlab Basics and Little Beyond.

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(Accessed March 11, 2009).

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25. Visual Basic 6.0.

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(Accessed on October 28, 2008).

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http://www.math.utah.edu/~eyre/computing/matlab-intro/

http://www.linfo.org/gui.html

http://www.jcomsoft.com

http://www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html

119

Appendix

120

Appendix: Character Tables

1. Nonaxial Point Groups:

Ci E

i

Ag 1 1 Rx, Ry, Rz

x2, y2, z2, xy, xz, yz

Au

1 1

x, y, z

2. C n Point Groups: C2

E

C2

A 1 1 Rz, z x2, y2, z2, xy

B 1 1

Rx, Ry, x, y

xz, yz

C3

E

C3

C32

= e2 i /3

A 1 1 1 Rz, z x2 + y2

E 1

1

C

C

(Rx, Ry),

(x, y) (x2 + y2, xy),

(xz, yz)

C4

E

C4 C2 C43

A 1 1 1 1 Rz, z x2 + y2, z2

B 1 1

1 1 x2 y2, xy

E 1

1 i

i 1

1

i

i (Rx, Ry),

(x, y) (xz, yz)

C5

E

C5 C52 C5

3 C54 = e2 i /5

A 1 1 1 1 1 Rz, z x2 + y2, z2

E1

1

1

C

2

( 2)C

( 2)C

2

C

(Rx, Ry),

(x, y) (xz, yz)

E2

1

1

2

( 2)C

C

C ( 2)C

2 (x2 - y2, xy)

C1

E

A 1

Cs E

h

A' 1 1 x, y, Rz x2, y2, z2, xy

A''

1 1

z, Rx, Ry

yz, xz

121

C8

E

C8 C4 C83 C2 C8

5 C42

C87 = e2 i /8

A 1 1 1 1 1 1 1 1 Rz, z x2 + y2, z2

B 1 1 1 1 1 1 1 1

E1

1

1

C i

i

C

1

1

C

i

i

C

(Rx, Ry),

(x, y) (xz, yz)

E2

1

1 i i

1

1

i

i 1

1 i i

1

1 i

i (x2 y2, xy)

E3

1

1

C

i

i

C

1

1

C i

i

C

3. C nh Point Groups:

C2h

E

C2 i h

Ag 1 1 1 1 Rz x2, y2, z2, xy

Bg 1 1

1 1

Rx, Ry

xz, yz

Au 1 1 1

1

z

Bu 1 1

1

1 x, y

C3h

E

C3

C32

h S3 S35 = e2 i /3

A' 1 1 1 1 1 1 Rz x2 + y2, z2

E' 1

1

C

C

1

1

C

C

(x, y) (x2 y2, xy)

A'' 1 1 1 1

1 1 z

E'' 1

1

C

C

1

1

C

C

(Rx, Ry)

(xz, yz)

C4h E C4 C2 C43 i S4

3 h S4

Ag 1 1 1 1 1 1 1 1 Rz x2 + y2, z2

Bg 1 1 1 1 1 1 1 1 x2 y2, xy

Eg 1

1 i i

1

1 i

i 1 1

i i

1

1 i

i (Rx, Ry) (xz, yz)

Au 1 1 1 1 1 1 1 1 z

Bu 1 1 1 1 1 1 1 1

Eu 1

1 i i

1

1 i

i 1

1 i

i 1 1

i i (x, y)

C6

E

C6 C3 C2 C3

2 C65 = e2 i /6

A 1 1 1 1 1 1 Rz, z x2 + y2, z2

B 1 1 1 1

1 1

E1

1

1

C

C

1

1

C

C

(Rx, Ry),

(x, y) (xz, yz)

E2

1

1

C

C

1

1

C

C

(x2 y2, xy)

122

3. C nv Point Groups:

C2v

E

C2

v v'

A1 1 1 1 1 z x2, y2, z2

A2 1 1 1

1 Rz xy

B1 1 1 1 1 Ry, x

xz

B2 1 1 1

1 Rx, y yz

C3v

E

2 C3

3 v

A1 1 1 1 z x2 + y2, z2

A2 1 1 1 Rz

E 2 1 0 (Rx, Ry), (x, y) (x2 y2, xy), (xz, yz)

C5h E

C5 C5

2 C53 C5

4 h S5 S5

7 S53 S5

9 = e2 i /5

A' 1 1 1 1 1 1 1 1 1 1 Rz x2 + y2, z2

E1' 1

1

C

2

( 2)C

( 2)C

2

C

1

1

C

2

( 2)C ( 2)C

2

C

(x, y)

E2' 1

1

2

( 2)C

C

C ( 2)C

2 1

1

2

( 2)C

C

C ( 2)C

2 (x2 - y2, xy)

A'' 1 1 1 1 1 1

1 1 1 1 z

E1''

1

1

C

2

( 2)C

( 2)C

2

C

1

1

- C

2

( 2)C

( 2)C

2

C

(Rx, Ry)

(xz, yz)

E2''

1

1

2

( 2)C

C

C ( 2)C

2 1

1

2

( 2)C

C

C ( 2)C

2

C6h

E

C6 C3 C2 C32 C6

5 i S35 S6

5 h S6 S3 = e2 i /6

Ag 1 1 1 1 1 1 1 1 1 1 1 1 Rz x2 + y2, z2

Bg 1 1 1 1

1 1 1 1 1 1

1 1

E1g 1

1

C

C

1

1

C

C

1

1

C

C

1

1

C

C

(Rx, Ry)

(xz, yz)

E2g 1

1

C

C

1

1

C

C

1

1

C

C

1

1

C

C

(x2 y2, xy)

Au 1 1 1 1 1 1 1

1 1 1

1 1 z

Bu 1 1 1 1

1 1 1

1 1 1 1 1

E1u

1

1

C

C

1

1

C

C

1

1

C

C

1

1

C

C

(x, y)

E2u

1

1

C

C

1

1

C

C

1

1

C

C 1

1

C

C

123

C5v

E

2 C5 2 C52 5 v

= 2 /5

A1 1 1 1 1 z x2 + y2, z2

A2 1 1 1 1 Rz

E1 2 2 cos( ) 2 cos(2 )

0 (Rx, Ry), (x, y) (xz, yz)

E2 2 2 cos(2 )

2 cos( ) 0 (x2 y2, xy)

5. S n Point Groups:

S6

E

S6 C3

i C32

S65 = e2 i /6

Ag 1 1 1 1 1 1 Rz x2 + y2, z2

Eg 1

1

C

C 1 1

C

C (Rx, Ry) (x2 y2, xy),

(xz, yz)

Au 1 1 1 1

1 1 z

Eu 1

1

C

C 1

1

C

C

(x, y)

C4v

E

2 C4

C2

2 v

2 d

A1 1 1 1 1 1 z x2 + y2, z2

A2 1 1 1 1 1 Rz

B1 1 1 1 1 1 x2 y2

B2 1 1 1 1 1 xy

E 2 0 2 0 0 (Rx, Ry), (x, y) (xz, yz)

C6v

E

2 C6

2 C3

C2

3 v

3 d

A1 1 1 1 1 1 1 z x2 + y2, z2

A2 1 1 1 1 1 1 Rz

B1 1 1 1 1 1 1

B2 1 1 1 1 1 1

E1 2 1 1 2 0 0 (Rx, Ry), (x, y) (xz, yz)

E2 2 1 1 2 0 0 (x2 y2, xy)

S4

E

S4 C2

S43

A 1 1 1 1 Rz, x2 + y2, z2

B 1 1

1 1 z x2 y2, xy

E 1

1 i

i 1

1 i

i (Rx, Ry),

(x, y) (xz, yz)

124

S8

E

S8 C4

S8

3 i S85 C4

2

S8

7 = e2 i /8

A 1 1 1 1 1 1 1 1 Rz x2 + y2, z2

B 1 1 1 1 1 1 1 1 z

E1 1

1

C

i i

C

1

1

C

i

i

C

(x, y) (xz, yz)

E2 1

1 i

i 1

1 i

i 1 1

i i

1

1 i

i (x2 y2, xy)

E3 1

1

C

i

i

C 1

1

C

i i

C

(Rx, Ry)

(xz, yz)

6. D n Point Groups: D2

E

C2 (z)

C2 (x)

C2 (y)

A 1 1 1 1 x2, y2, z2

B1 1 1 1 1 Rz, z xy

B2 1 1 1 1 Ry, y xz

B3 1 1 1 1 Rx, x

yz

D3

E

2 C3

3 C2

A1 1 1 1 x2 + y2, z2

A2 1 1 1 Rz, z

E 2 1 0 (Rx, Ry), (x, y) (x2 y2, xy), (xz, yz)

D4

E

2 C4

C2

2 C2'

2 C2''

A1 1 1 1 1 1 x2 + y2, z2

A2 1 1 1 1 1 Rz, z

B1 1 1 1 1 1 x2 y2

B2 1 1 1 1 1 xy

E 2 0 2 0 0 (Rx, Ry), (x, y) (xz, yz)

D5

E

2 C5 2 C52 5 C2

=2 /5

A1 1 1 1 1 x2 + y2, z2

A2 1 1 1 1 Rz, z

E1 2 2 cos( ) 2 cos(2 )

0 (Rx, Ry), (x, y) (xz, yz)

E2 2 2 cos(2 )

2 cos( ) 0 (x2 y2, xy)

D6

E

2 C6

2 C3

C2

3 C2'

3 C2''

A1 1 1 1 1 1 1 x2 + y2, z2

A2 1 1 1 1 1 1 Rz, z

B1 1 1 1 1 1 1

B2 1 1 1 1 1 1

E1 2 1 1 2 0 0 (Rx, Ry), (x, y) (xz, yz)

E2 2 1 1 2 0 0 (x2 y2, xy)

125

7. D nh Point Groups: D2h

E

C2

C2 (x)

C2 (y)

i (xy)

(xz)

(yz)

Ag 1 1 1 1 1 1 1 1 x2, y2, z2

B1g 1 1 1 1 1 1 1 1 Rz xy

B2g 1 1 1 1 1 1 1 1 Ry xz

B3g 1 1 1 1 1 1 1 1 Rx

yz

Au 1 1 1 1 1

1 1 1

B1u 1 1 1 1 1

1 1 1 z

B2u 1 1 1 1 1

1 1 1 y

B3u 1 1 1 1 1

1 1 1 x

D3h

E

2 C3

3 C2

h 2 S3

3 v

A1' 1 1 1 1 1 1 x2 + y2, z2

A2' 1 1 1 1 1 1 Rz

E' 2 1 0 2 1 0 (x, y) (x2 y2, xy)

A1'' 1 1 1 1

1 1

A2'' 1 1 1 1

1 1 z

E'' 2 1 0 2

1 0 (Rx, Ry)

(xz, yz)

D4h

E

2 C4

C2

2 C2'

2 C2''

i 2 S4

h 2 v

2 d

A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2

A2g 1 1 1 1 1 1 1 1 1 1 Rz

B1g 1 1 1 1 1 1 1 1 1 1 x2 y2

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 1

Eu 2 0 2 0 0 2

0 2 0 0 (x, y)

D5h

E

2 C5 2 C52 5 C2 h 2 S5 2 S5

3 5 v =2 /5

A1' 1 1 1 1 1 1 1 1 x2 + y2, z2

A2' 1 1 1 1 1 1 1 1 Rz

E1' 2 2 cos( ) 2 cos(2 ) 0 2 2 cos( ) 2 cos(2 ) 0 (x, y)

E2' 2 2 cos(2 ) 2 cos( ) 0 2 2 cos(2 ) 2 cos( ) 0 (x2 y2, xy)

A1'' 1 1 1 1 1

1 1 1

A2'' 1 1 1 1 1

1 1 1 z

E1'' 2 2 cos( ) 2 cos(2 ) 0 2

2 cos( ) 2 cos(2 ) 0 (Rx, Ry) (xz, yz)

E2'' 2 2 cos(2 ) 2 cos( ) 0 2

2 cos(2 ) 2 cos( ) 0

126

D6h

E

2 C6 2 C3

C2 3 C2'

3 C2'' i 2 S3 2 S6

h 3 d

3 v

A1g 1 1 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2

A2g 1 1 1 1 1 1 1 1 1 1 1 1 Rz

B1g 1 1 1 1 1 1 1 1 1 1 1 1

B2g 1 1 1 1 1 1 1 1 1 1 1 1

E1g 2 1 1 2 0 0 2 1 1 2 0 0 (Rx, Ry) (xz, yz)

E2g 2 1 1 2 0 0 2 1 1 2 0 0 (x2 y2, xy)

A1u 1 1 1 1 1 1 1 1 1 1 1 1

A2u 1 1 1 1 1 1 1 1 1 1 1 1 z

B1u 1 1 1 1 1 1 1 1 1 1 1 1

B2u 1 1 1 1 1 1 1 1 1 1 1 1

E1u 2 1 1 2 0 0 2 1 1 2 0 0 (x, y)

E2u 2 1 1 2 0 0 2 1 1 2 0 0

D8h

E

2 C8

2 C83

2 C4

C2

4 C2'

4 C2''

i 2 S83

2 S8

2 S4

h 4 d

4 v

=21/2

A1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 x2 + y2, z2

A2g 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Rz

B1g 1 1 1 1 1 1 1 1 1 1 1 1 1 1

B2g 1 1 1 1 1 1 1 1 1 1 1 1 1 1

E1g 2

0 2 0 0 2

0 2

0 0 (Rx, Ry) (xz, yz)

E2g 2 0 0 2 2 0 0 2 0 0 2 2 0 0 (x2 y2, xy)

E3g 2

0 2 0 0 2

0 2

0 0

A1u 1 1 1 1 1 1 1 1

1 1 1 1

1 1

A2u 1 1 1 1 1 1 1 1

1 1 1 1

1 1 z

B1u 1 1 1 1 1 1 1 1

1 1 1 1

1 1

B2u 1 1 1 1 1 1 1 1

1 1 1 1

1 1

E1u 2

0 2 0 0 2

0 2 0 0 (x, y)

E2u 2 0 0 2 2 0 0 2

0 0 2 2

0 0

E3u 2

0 2 0 0 2

0 2 0 0

8. D nd Point Groups: D2d

E

2 S4

C2

2 C2'

2 d

A1 1 1 1 1 1 x2, y2, z2

A2 1 1 1 1 1 Rz

B1 1 1 1 1 1 x2 y2

B2 1 1 1 1 1 z xy

E 2 0 2 0 0 (Rx, Ry), (x, y) (xz, yz)

127

D3d

E

2 C3

3 C2

i 2 S6'

3 d

A1g 1 1 1 1 1 1 x2 + y2, z2

A2g 1 1 1 1 1 1 Rz

Eg 2 1 0 2 1 0 (Rx, Ry) (x2 y2, xy), (xz, yz)

A1u 1 1 1 1

1 1

A2u 1 1 1 1

1 1 z

Eu 2 1 0 2

1 0 (x, y)

D4d

E

2 S8

2 C4

2 S83

C2

4 C2'

4 d

=21/2

A1 1 1 1 1 1 1 1 x2 + y2, z2

A2 1 1 1 1 1 1 1 Rz

B1 1 1 1 1 1 1 1

B2 1 1 1 1 1 1 1 z

E1 2

0

2 0 0 (x, y)

E2 2 0 2 0 2 0 0 (x2 y2, xy)

E3 2

0

2 0 0 (Rx, Ry) (xz, yz)

D5d

E

2 C5 2 C52 5 C2

i 2 S10 2 S103 5 d

=2 /5

A1g 1 1 1 1 1 1 1 1 x2 + y2, z2

A2g 1 1 1 1 1 1 1 1 Rz

E1g 2 2 cos( ) 2 cos(2 )

0 2 2 cos(2 ) 2 cos( ) 0 (Rx, Ry)

(xz, yz)

E2g 2 2 cos(2 )

2 cos( ) 0 2 2 cos( ) 2 cos(2 ) 0 (x2 y2, xy)

A1u 1 1 1 1 1

1 1 1

A2u 1 1 1 1 1

1 1 1 z

E1u 2 2 cos( ) 2 cos(2 )

0 2

2 cos(2 )

2 cos( ) 0 (x, y)

E2u 2 2 cos(2 )

2 cos( ) 0 2

2 cos( ) 2 cos(2 ) 0

D6d

E

2 S12

2 C6

2 S4

2 C3

2 S125

C2

6 C2'

6 d

=31/2

A1 1 1 1 1 1 1 1 1 1 x2 + y2, z2

A2 1 1 1 1 1 1 1 1 1 Rz

B1 1 1 1 1 1 1 1 1 1

B2 1 1 1 1 1 1 1 1 1 z

E1 2

1 0 1

2 0 0 (x, y)

E2 2 1 1 2 1 1 2 0 0 (x2 y2, xy)

E3 2 0 2 0 2 0 2 0 0

E4 2 1 1 2 1 1 2 0

E5 2

1 0 1

2 0 0 (Rx, Ry)

(xz, yz)

128

9. Cubic Point Groups:

T

E

4 C3

4 C3

2

3 C2

=e2 i/3

A 1 1 1 1 x2 + y2 + z2

E 1

1

C

C

1 1

(2 z2 x2 y2,

x2 y2)

T 3 0 0 1 (Rx, Ry, Rz),

(x, y, z) (xy, xz, yz)

Th E

4 C3

4 C32

3 C2

i 4 S6

4 S65

3 h

=e2 i/3

Ag 1 1 1 1 1 1 1 1 x2 + y2 + z2

Au 1 1 1 1 1

1 1 1

Eg 1

1

C

C

1 1

1 1

C

C

1 1

(2 z2 x2 y2,

x2 y2)

Eu 1

1

C

C

1 1

1

1

C

C

1 1

Tg 3 0 0 1 3 0 0 1 (Rx, Ry, Rz) (xy, xz, yz)

Tu 3 0 0 1 3

0 0 1 (x, y, z)

O E

6 C4

3 C2 (C42)

8 C3

6 C2

A1

1 1 1 1 1 x2 + y2 + z2

A2

1 1 1 1 1

E 0 2 1 0 (2 z2 x2 y2,

x2 y2)

T1 3 1 1 0 1 (Rx, Ry, Rz),

(x, y, z)

T2 3 1 1 0 1 (xy, xz, yz)

Td

E

8 C3

3 C2

6 S4

d

A1 1 1 1 1 1 x2 + y2 + z2

A2 1 1 1 1 1

E 2 1 2 0 0 (2 z2 x2 y2,

x2 y2)

T1 3 0 1 1 1

(Rx, Ry, Rz)

T2 3 0 1 1 1 (x, y, z) (xy, xz, yz)

129

10. Icsahedral Point Groups:

Oh E

8 C3

6 C2

6 C4

3 C2 (C4

2)

i 6 S4

8 S6

3 h

6 d

A1g 1 1 1 1 1 1 1 1 1 1 x2 + y2 + z2

A2g 1 1 1 1 1 1 1 1 1 1

Eg 2 1 0 0 2 2 0 1 2 0 (2 z2 x2 y2,

x2 y2)

T1g 3 0 1 1 1 3 1 0 1 1 (Rx, Ry, Rz)

T2g 3 0 1 1 1 3 1 0 1 1 (xy, xz, yz)

A1u

1 1 1 1 1 1

1 1 1 1

A2u

1 1 1 1 1 1

1 1 1 1

Eu 2 1 0 0 2 2

0 1 2 0

T1u 3 0 1 1 1 3

1 0 1 1 (x, y, z)

T2u 3 0 1 1 1 3

1 0 1 1

I E

12 C5 12 C52 20 C3

15 C2

= /5

A 1 1 1 1 1 x2 + y2 + z2

T1

3 2 cos( ) 2 cos(3 )

0 1 (Rx, Ry, Rz),

(x, y, z)

T2

3 2 cos(3 )

2 cos( ) 0 1

G 4 1 1 1 0

H 5 0 0 1 1 (2 z2 x2 y2,

x2 y2,

xy, xz, yz)

Ih E

12 C5 12 C52 20 C3

15 C2

i 12 S10 12 S103 20 S6

15

= /5

Ag 1 1 1 1 1 1 1 1 1 1 x2 + y2 + z2

T1g 3 2 cos( ) 2 cos(3 )

0 1 3 2 cos(3 ) 2 cos( ) 0 1 (Rx, Ry, Rz)

T2g 3 2 cos(3 )

2 cos( ) 0 1 3 2 cos( ) 2 cos(3 ) 0 1

Gg 4 1 1 1 0 4 1 1 1 0

Hg 5 0 0 1 1 5 0 0 1 1 (2 z2 x2 y2,

x2 y2,

xy, xz, yz)

Au 1 1 1 1 1 1

1 1 1 1

T1u

3 2 cos( ) 2 cos(3 )

0 1 3

2 cos(3 )

2 cos( ) 0 1 (x, y, z)

T2u

3 2 cos(3 )

2 cos( ) 0 1 3

2 cos( ) 2 cos(3 )

0 1

Gu 4 1 1 1 0 4

1 1 1 0

Hu 5 0 0 1 1 5

0 0 1 1

130

10. Infinite Point Groups:

C v E 2 C

...

v

A1= +

1 1 ...

1 z x2 + y2, z2

A2=

1 1 ...

1 Rz

E1=

2 2 cos( ) ...

0 (x, y), (Rx, Ry) (xz, yz)

E2=

2 2 cos(2 )

...

0 (x2 - y2, xy)

E3=

2 2 cos(3 )

...

0

... ...

... ...

...

D h

E 2 C

...

v

i 2 S

...

C2

g+ 1 1 ...

1 1 1 ...

1 x2 + y2, z2

g

1 1 ...

1 1 1 ...

1 Rz

g 2 2 cos( ) ...

0 2 2 cos( ) .. 0 (Rx, Ry)

(xz, yz)

g 2 2 cos(2 )

...

0 2 2 cos(2 ) .. 0 (x2 y2, xy)

... ...

... ...

... ... ... ...

...

u+ 1 1 ...

1 1

1 ...

1 z

u

1 1 ...

1 1

1 ...

1

u 2 2 cos( ) ...

0 2

2 cos( ) .. 0 (x, y)

u 2 2 cos(2 )

...

0 2

2 cos(2 )

.. 0

... ...

... ...

... ... ... ...

...

.

. .

.

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A New Visual Basic Software Built-up for Solving-out ......2.2 Symmetry Elements and Symmetry...

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Transcript of A New Visual Basic Software Built-up for Solving-out ......2.2 Symmetry Elements and Symmetry...