Symmetry and Combinatorial Enumeration in Chemistry

357 Pages·1991·31.28 MB·English

by  Dr. Shinsaku Fujita (auth.)

Shinsaku Fujita
Symmetry and
Combinatorial Enumeration
in Chemistry
With 54 Figures
Springer-Verlag
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London Paris Tokyo
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Budapest
Dr. Shinsaku Fujita
Research Laboratories, Ashigara
Fuji Photo Film Co., Ltd.
Minami-Ashigara, Kanagawaken, 250-01 Japan
ISBN-13: 978-3-540-54126-4 e-ISBN-13: 978-3-642-76696-1
DOI: 10.1007/978-3-642-76696-1
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Preface
This book is written to introduce a new approach to stereochemical problems and
to combinatorial enumerations in chemistry. This approach is based on group the
ory, but different from conventional ways adopted by most textbooks on chemical
group theory. The difference sterns from their starting points: conjugate subgroups
and conjugacy classes.
The conventional textbooks deal with linear representations and character ta
bles of point groups. This fact implies that they lay stress on conjugacy classesj
in fact, such group characters are determined for the respective conjugacy classes.
This approach is versatile, since conjugacy classes can be easily obtained by ex
amining every element of a group. It is unnecessary to know the group-subgroup
relationship of the group, which is not always easy to obtain. The same situa
tion is true for chemical enumerations, though these are founded on permutation
groups. Thus, the P6lya-Redfield theorem (1935 and 1927) uses a cycle index that
is composed of terms associated with conjugacy classes.
On the other hand, the alternative approach which I take is based on per
mutation representations and tables of marks. The latter concept has on ce been
discussed in Burnside's excellent textbook (1911) but ,unjustly neglected for a
long time. The tables of marks have been omitted even from voluminous books
on the group theory, except Oshima's textbook (1954), Sheehan's paper (1968)
and Hässelbarth's one (1985). In this approach, conjugate subgroups are of much
importancej the marks are given to the respective representatives of conjugate
subgroups. The tables of marks can be regarded as counterparts of character ta
bles. The determination of such conjugate subgroups requires the group-subgroup
relationship that is gene rally difficult of approach. However, this fact is no longer
a drawback in the light of our target, because this relationship is also necessary
to our target of understanding stereochemistry.
I have adopted the second approach, since this is suitable to our stereochem
ical problems that have discrete nature. The task of solving them, however, ne
cessitates some new elements that have never been served by the conventional
methods. Thus, I have tried to integrate point-group and permutation-group the
ories. Throughout this book, I have put special emphasis on coset representations,
which are transitive permutation representaitons. In order to treat the stereochem
ical problems comprehensively, I have introduced several new concepts: the 'SCR
VI PREFACE
(set-of-coset-representation) notation, allowed and forbidden coset representations,
chirality fittingness, subduction of coset representations, proligands, promolecules
and so forth. I have further extended them to create another set of concepts, e.g.,
unit subduced cycle indices (USeIs), subduced cycle indices (SeIs), partial cycle
indices (peIs), and elementary superposition. All of these concepts are useful to
solve combinatorial enumerations. Although examples in this book are adopted
from chemistry only; the method developed here can be utilized to solve any other
combinatorial enumerations.
This book may serve as a first introduction for students and researchers in
tending to begin their works in this field. In particular, I hope that the book would
give them some comprehensive perspective other than those given by conventional
textbooks.
I am indebted to Mr. Sosuke Hanai for a number of valuable comments on
the draft manuscript.
Minami-Ashigara, Kanagawa, Japan
January 1991 Shinsaku Fujita
Contents
1 Introd~ction 1
2 Symmetry and Point Groups 7
2.1 Symmetry Operations and Elements 7
2.2 Conjugacy Classes in Point Groups . . 10
2.3 Subgroups of Point Groups. . . . . . . 10
2.4 Conjugate and Normal Subgroups of Point Groups 12
2.5 Non-Redundant Set of Subgroups for a Point Group . 13
3 Permutation Groups 17
3.1 Permutations and Cydes . 17
3.2 Permutation Groups .. 19
3.3 Transitivity and Orbits . 20
3.4 Symmetrie Groups 21
3.5 Parity ...... . 24
3.6 Alternating Groups 25
4 Axioms and Theorems of Group Theory 29
4.1 Axioms and Multiplieation Thbles . 29
4.2 Subgroups....... 31
4.3 Cosets ........ . 32
4.4 Equivalenee Relations. 35
4.5 Conjugacy Classes .. 37
4.6 Conjugate and Normal Subgroups . 38
4.7 Subgroup Lattiees . 40
4.8 Cydic Groups . . . . . . . . . . . 40
5 Coset Representations and Orbits 45
5.1 Coset Representations ..... . 45
5.2 Transitive Permutation Representations . 48
5.3 Mark Thbles . . . . . . . . . . . . . . . . 53
5.4 Permutation Representations and Orbits 56
6 Systematic Classification of Molecular Symmetries 63
6.1 Assignment of Coset Representations to Orbits. 63
6.2 SCR Notation . . . . . . . . . . . . . . . . . . . . . . 66
7 Local Symmetries and Forbidden Coset Representations 75
7.1 Blocks and Local Symmetries .. 76
7.2 Forbidden Coset Representations ............... . 84
VIII CONTENTS
8 Chirality Fittingness of an Orbit 89
8.1 Ligands ............. . 89
8.2 Behavior of Cosets on the Action of a CR 90
8.3 Chirality Fittingness of an Orbit. . . 94
9 Subduction of Coset Representations 101
9.1 Subduction of Coset Representations 101
9.2 Subduced Mark Table ....... .. . 103
9.3 Chemical Meaning of Subduction ... . 106
9.4 Unit Subduced Cyde Indices. . . . .. . 109
9.5 Unit Subduced Cyde Indices with Chirality Fittingness . 111
9.6 Desymmetrization Lattice . . . . . . . . . . . . . . . . . 112
10 Prochirality 117
Hl.1 Desymmetrization of Enantiospheric Orbits. 117
10.2 Prochirality . . . . . . . . . . . . . . . . . . 121
10.3 Further Desymmetrization of Enantiospheric Orbits 122
10.4 Chiral syntheses . . . . . . . . . . . . . . . . . . . 127
11 Desymmetrization of Para-Achiral Compounds 135
11.1 Chhal Subduction of Homospheric Orbits ... . 135
11.2 Desymmetrization of Homospheric Orbits ... . 137
11.3 Chemoselective and Stereoselective Processes .. 140
12 Topicity and Stereogenicity 147
12.1 Topicity Based On Chirality Fittingness of an Orbit. 147
12.2 Stereogenicity ..................... . . 151
13 Counting Orbits ,163
13.1 The Cauchy-Frobenius Lemma .. 163
13.2 Configurations .......... . 167
13.3 The P6lya-Redfield Theorem .. . 171
14 Obligatory Minimum Valencies 181
14.1 Isomer Enumeration under the OMV Restriction . 182
14.2 Unit Cyde Indices ........... . . 189
15 Compounds with Achiral Ligands Only 197
15.1 Compounds with Given Symmetries ... 197
15.2 Compounds with Given Symmetries and Weight .... 205
16 New Cycle Index 215
16.1 New Cyde Indices Based On USCIs ......... . 215
16.2 Correlation of New Cyde Indices to P61ya's Theorem 217
16.3 Partial Cyde Indices . . . . . . . . . . . . . . . . 223
17 Cage-Shaped Moleeules with High Symmetries 227
17.1 Edge Strategy .................. . 228
17.2 Tricydodecanes with Td and Its Subsymmetries 229
17.3 Use of Another Ligand-Inventory 236
17.4 New Type of Cyde Index ............ . 236
CONTENTS IX
18 Elementary Superposition 241
18.1 The USCI Approach .. 241
18.2 Elementary Superposition ... 245
18.3 Superposition for Other Indices 251
19 Compounds with Achiral and Chiral Ligands 255
19.1 Compounds with Given Symmetries ..... . 255
19.2 Compounds with Given Symmetries and Weights 262
19.3 Compounds with Given Weights . 265
19.4 Special Cases . . . . . . . . . . . . . 266
19.5 Other Indices ............ . 268
20 Compounds with Rotatable Ligands 271
20.1 Rigid Skeleton and Rotatable Ligands . 271
20.2 Enumeration of Rotatable Ligands 273
20.3 Enumeration of Non-Rigid Isomers 276
20.4 Total Numbers .......... . 283
20.5 Typical Procedure for Enumeration 286
21 Promolecul~s 297
21.1 Molecular Models ...... . 297
21.2 Proligands and Promoleeules . 298
21.3 Enumeration of Promolecules 300
21.4 Moleeules Based on Promoleeules 307
21.5 Prochiralities of Promoleeules and Moleeules 313
21.6 Conduding Remarks .. 317
22 Appendix A. Mark Tables 321
A.1 Td Point Group and Its Subgroups 321
A.2 D3h Point Group and Its Subgroups . · 325
23 Appendix B. Inverses of Mark Tables 327
B.1 Td Point Group and Its Subgroups . 327
B.2 D3h Point Group and Its Subgroups . · 332
24 Appendix C. Subduction Tables 335
C.1 Td Point Group and Its Subgroups 335
C.2 D3h Point Group and Its Subgroups . · 341
25 Appendix D. Tables of USCIs 343
D.l 'Td Point Group and Hs Subgroups 343
D.2 D3h Point Group and Its Subgroups . · 349
26 Appendix E. Tables of USCI-CFs 351
E.1 T Point Group and Its Subgroups 351
d
E.2 D3h Point Group and Its Subgroups . · 357
27 Index 359
Chapter 1
Introduction
Group theory is now an essential tool for chemists. Thus, there have a~peared
a vast number of pedagogical articles on its applications to chemistry.l )-[9) In
addition, we can enrich our knowledge by means of exce1lent textbooks on this
topic.l10]-(17)
Group theory has been applied to quantum-chemical problems that are con
cemed with molecular symmetry, e.g.,
• symmetry adapted functions for molecular orbital theory,
• ligand field theory, and
• molecular vibrations.
These applications are mainly based on group-representation theory. In this the
ory, a linear representation of a point group and its reduction into irreducible
representations play significant roles, where character" tables of point groups are
used for assigning such an irreducible representation to an orbital. The textbooks
on chemical group theory have already discussed these issues in details; hence,
these are not discussed in the present book. However, we should here pay atten
tion to methodology presumed in the applications.
The applications imply a continuous molecular model in which a set of wave
functions are superposed to realize a moleeule. In other words, the concepts of
bonds and· atoms become secondary. This approach succeeds to the physical tra
dition and is against the chemical convention.
On the other hand, there exist at least two fields that require group-theoretical
investigations,
• combinatorial enumeration of chemical compounds and reactions, and
• stereochemistry,
which are the main objectives of this book. These applications require apother
molecular model in which atoms and bonds have primary meanings. In other
2 CHAPTER 1. INTRODUCTION
words, we are grounded on three-dimensional structural formulas. This model
has discrete nature, which seems to be suitable for analysis by permutation-group
theory.
Accordingly, permutation-group theory is applied to enumeration problems
of molecules and reactions.l18] Since the original introduction of P6lya's theorem
(1935)/ chemistry has supplied many problems for verifying the theorem.[19, 20]
Such results have been reviewed extensively in 1970s.[21] The concept of double
cosets has been applied to enumeration of rearrangements of phosphorus complex
es.!22] Methods based on tables of marks have recently been developed for enumera
ting graphsl23] and compounds,[24] in which their weights (molecular formulas) and
symmetries are taken into consideration. Tables of marks were originally described
in Burnside's famous textbook[25]; however, they have attracted little attention of
mathematicians as weIl as of chemists. This is because such tables of marks have
been published for a limited number of groups in comparison with the character
tables above. In the present book, we prepare mark tables for representative
(point) groups and use them thoroughly for solving our problems.
Stereochemistry is concerned with chirality and achirality of moleeules as
weIl as their interconversion.l26]-128] Since these concepts are related to molecular
symmetry, stereochemie al problems are investigated by means of group theory,
especially, of point-group theory. Important terms in stereochemistry such as top
icities are thus defined by group-theoretical consideration.l29] However, a point
group is frequently insufficient to manipulate a discrete molecular model, since
it is concerned with a continuous molecular model. For remedying this disad
vantage, the concept of framework group has been proposed.l30] An alternative
solution[31] is based on local (site) symmetries, which come from crystaIlograph
icalliteratures.!32] Algebraic aspects of chirality phenomena have been discussed
on the basis of permutation groups.!33, 34]
The short history above implies that chemical applications of point-group the
ory and those of permutation-group theory have developed separately, although
they are closely related to each other. This book aims at integrating the two the
ories in order to gain a deeper insight into stereochemistry and chemical enumera
tion. The missing link for the integration is the concept of "coset representations" .
Throughout the course of this book, the following correspondence is signifi
cant:
sets of orbits coset
equivalent e+h--e-r-n+. (e quivalence m+-a-.-t-h+. representations .
ligands classes)
A given moleeule is composed of sets of equivalent ligands (or atoms), each
of which is regarded as an orbit. This one-to-one corresponde:Q,!:e can be referred
1 We refer to this theorem a.s the P61ya.·Redfieid theorem, sinee Redfield diseovered this independently.