A New Visual Basic Software Built-up for Solving-out ......2.2 Symmetry Elements and Symmetry...
Transcript of 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
A New Visual Basic Software Built-up for Solving-out Reduction formula in Chemical
Applications of Group Theory
By Rasha Saleh Mohammad Sabri
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) ..
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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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:
A New Visual Basic Software Built-up for Solving-out Reduction formula in Chemical
Applications of Group Theory
.
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:
v
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
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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
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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.
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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
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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 (3.16) The C 7 Point Group Form 62
Figure (3.17) The C 8 Point Group Form 62
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 (3.26) The C v3 Point Group Form 67
Figure (3.27) The C v4 Point Group Form 67
Figure (3.28) The C v5 Point Group Form 68
Figure (3.29) The C v6 Point Group Form 68
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 (3.32) The C h3 Point Group Form 70
Figure (3.33) The C h4 Point Group Form 71
Figure (3.34) The C h5 Point Group Form 71
Figure (3.35) The C h6 Point Group Form 71
Figure (3.36) The D nh Point Groups Form 72
Figure (3.37) The D h2 Point Group Form 72
Figure (3.38) The D h3 Point Group Form 73
Figure (3.39) The D h4 Point Group Form 73
Figure (3.40) The D h5 Point Group Form 74
Figure (3.41) The D h6 Point Group Form 74
Figure (3.42) The D h8 Point Group Form 75
Figure (3.43) The D nd Point Groups Form 75
Figure (3.44) The D d2 Point Group Form 76
Figure (3.45) The D d3 Point Group Form 76
Figure (3.46) The D d4 Point Group Form 77
Figure (3.47) The D d5 Point Group Form 77
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
Figure (3.66) Reducible Representation For C v3 Point Group Form 87
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Figure Subject Page
Figure (3.67) Reducible Representation For C v4 Point Group Form 88
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
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Figure (4.2) Solution of Reduction Formula using software as applied on the reducible representation in D h3 Point Group.
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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
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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
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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
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Figure Subject Page
Figure (4.12) red and vib
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
Figure (4.14) red and vib
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
Figure (4.16) N3 and its Reduction Formula for a 6-atomic molecule
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
Figure (4.18) N3 and its Reduction Formula for a 9-atomic molecule
that belongs to D h2 Point Group 108
Figure (4.19) N3 and its Reduction Formula for a 13-atomic
molecule that belongs to D h4 Point Group 109
Figure (4.20) N3 and its Reduction Formula for a 7-atomic
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
By Rasha Saleh Mohammad Sabri
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
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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
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:
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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.
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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
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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
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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
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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].
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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
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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 ,
z to x , y , z . In matrix terms we would write:
100
010
001
z
y
x
= z
y
x
the reflection in ( yz ) plane operation takes general coordinates x , y ,
z to x , y , z . In matrix terms we would write:
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 C 2
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:
This example presents the calculations of the reduction formula for
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 .
1 that corresponds to C 4 .
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 .
From the hypothetical representation, shown above, the
corresponding characters are:
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:
This example presents the calculations of the reduction formula for
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 C 4
1 that corresponds to C 2
1 that corresponds to C 34
1 that corresponds to i
1 that corresponds to S 34
39
1 that corresponds to h
1 that corresponds to S 4
From the hypothetical representation, shown above, the
corresponding characters are:
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 .
2. The reducible representation N3 for a 3-atoms molecule belongs to
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 .
2. The reducible representation N3 for a 4-atoms molecule belongs to
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.
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.
Figure (3.11): The C 2 Point Group Form.
60
Figure (3.12): The C 3 Point Group Form.
Figure (3.13): The C 4 Point Group Form.
61
Figure (3.14): The C 5 Point Group Form.
Figure (3.15): The C 6 Point Group Form.
62
Figure (3.16): The C 7 Point Group Form.
Figure (3.17): The C 8 Point Group Form.
When the user presses " The D n Point Groups" button he gets the
following form shown in Figure 3.18.
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
Figure (3.20): The D 3 Point Group Form.
Figure (3.21): The D 4 Point Group Form.
65
Figure (3.22): The D 5 Point Group Form.
Figure (3.23): The D 6 Point Group Form.
When the user presses " The C nv Point Groups" he gets the
following form shown in Figure 3.24.
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
Figure (3.26): The C v3 Point Group Form.
Figure (3.27): The C v4 Point Group Form.
68
Figure (3.28): The C v5 Point Group Form.
Figure (3.29): The C v6 Point Group Form.
When the user presses " The C nh Point Groups" he/she gets the
following form shown in Figure 3.30.
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
Figure (3.32): The C h3 Point Group Form.
Figure (3.33): The C h4 Point Group Form.
71
Figure (3.34): The C h5 Point Group Form.
Figure (3.35): The C h6 Point Group Form.
When the user presses " The D nh Point Groups" button he gets the
following form shown in Figure 3.36.
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
Figure (3.38): The D h3 Point Group Form.
Figure (3.39): The D h4 Point Group Form.
74
Figure (3.40): The D h5 Point Group Form.
Figure (3.41): The D h6 Point Group Form.
75
Figure (3.42): The D h8 Point Group Form.
When the user presses " The D nd Point Groups" button he gets the
following form shown in Figure 3.43.
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.
Figure (3.45): The D d3 Point Group Form.
77
Figure (3.46): The D d4 Point Group Form.
Figure (3.47): The D d5 Point Group Form.
78
Figure (3.48): The D d6 Point Group Form.
When the user presses " The Cubic Point Groups" button he gets the
following form shown in Figure 3.49.
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
following form shown in Figure 3.55.
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.
Figure (3.57): The S 6 Point Group Form.
83
Figure (3.58): The S 8 Point Group Form.
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.
Figure (3.66): Reducible Representation For C v3 Point Group Form.
88
Figure (3.67): Reducible Representation For C v4 Point Group Form.
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
standard method outlined in 2.9.3.
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
standard method outlined in 2.10.1.
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.
Figure (4.12): red and vib
in D h2 subgroup form for linear molecules based on the software.
105
The results are completely consistent that obtained using the manual
standard method outlined in 2.10.2.
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
Figure (4.14): red and vib
in D h2 subgroup form for linear symmetrical molecules based on the software.
The results are completely consistent that obtained using the manual
standard method outlined in 2.10.3.
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
Figure (4.17): N3 and its Reduction Formula for a 10-atomic molecule that
belongs to C v4 Point Group.
Figure (4.18): N3 and its Reduction Formula for a 9-atomic molecule that belongs
to D h2 Point Group.
109
Figure (4.19): N3 and its Reduction Formula for a 13-atomic molecule that
belongs to D h4 Point Group.
Figure (4.20): N3 and its Reduction Formula for a 7-atomic molecule that belongs
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
1. Kleiner, Israel: A History of Abstract Algebra. Canada Birkhauser
2007, PP 17-19.
2. Wikipedia, The Free Encyclopedia.
http://en.wikipedia.org/wiki/History_of_group_theory (Accessed on
December 22, 2008).
3. Milne, J.S.: Group Theory( May 17, 2008). From the world wide
web: www.jmilne.org/math/CourseNotes/math594g.html (Accessed
on November 16, 2008).
4. Raman, K.V.: Group Theory and Its applications to Chemistry.
New Delhi. Tata McGraw Publishing Company Limited, 1990, PP 60-
63.
5. Bishop, David M.: Group Theory and Chemistry. New York 1973,
PP 4-6, 10-47.
6. Lesk, Arthur M.: Introduction to Symmetry and Group Theory for
Chemists. 2 nd . New York . John Wiely & Sons, Inc., PP 29-31.
7. Vincent, Alan: Molecular Symmetry and Group Theory. John
Wiely & Sons, Inc. New York, PP 53-57.
8. Cotton, F. Albert: Chemical Applications of Group Theory. 2 nd .
Wiely- Interscience, N.Y.1971, PP 62-90.
117
9. Hochstrasser, Robin M.: Molecular Aspects of Symmetry. New
York. W.A. Benjamin. INC. 1966, PP 15, 52-72, 102.
10. Willock, David J: Molecular Symmetry. John Wiely & Sons, Ltd.
2009, PP 1-9, 48-65, 105-120.
11. Hans H. Jaffe' and Orchin, Milton: Symmetry In Chemistry. New
York. John Wiely & Sons, Inc. 1967, PP 8- 11, 112-113.
12. Carter, Robert L.: Molecular Symmetry and Group Theory. New
York. John Wiely & Sons, Inc. 1998, PP 66-79.
13. Kettle, S. F. A: Symmetry and Structure Readable Group for
Chemists.N.Y. Kluwer Academic Publisher,2004, PP 15-19.
14. Russell S . Drago: Physical Methods for Chemists, 2 nd .Sannders
1992, PP 18-47 & 149-202.
15. www.reciprocalnet.org (Accessed on January 16, 2009).
16. Davidson, George: Group Theory for Chemists. Nottingham. Mac
Millan Press Ltd. 1990.
17. Hargittai, Istva'n and Hargittai, Magdolna: Symmetry Through the
Eyes of Chemists. 2 nd . N.Y. Plenum Press.1995, PP198-200.
18. Strommen, Dennis P., Ellis R. Lippincott: Comments On Infinite
Point Groups. Journal of Chemical Education 5/ 1972, 341- 342.
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
Users. PP 127-132.
22. Eyre, David ( August 9, 1998 ). Matlab Basics and Little Beyond.
From the world wide web:
http://www.math.utah.edu/~eyre/computing/matlab-intro/ (Accessed
on March 12, 2009).
23. Dechevsky, Lubomir T. and Laks , Arne (April, 2004). Brief Report
on Matlab. From the world wide web:
www.cost285.itu.edu.tr/tempodoc/TD_285_04_03_Dechevsky_Laks
(Accessed March 11, 2009).
24. http://www.linfo.org/gui.html (Accessed on April 6, 2009).
25. Visual Basic 6.0.
26. www.jcomsoft.com (Accessed on October 28, 2008).
27. Matlab 7.4.
28. www.staff.ncl.ac.uk/j.p.goss/symmetry/Molecules_pov.html
(Accessed on October 28, 2008).
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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2. ), vibred(
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3. )N3 ( )vC2,vC3,
vC4, hD2, hD4, dT(.
..
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