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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
Group theory is now an essential tool for chemists. Thus, there have appeared a vast number of pedagogical articles on its applications to chemistry. [1]–[9] In addition, we can enrich our knowledge by means of excellent textbooks on this topic.[10]–[17]
Shinsaku Fujita

### Chapter 2. Symmetry and Point Groups

Abstract
Symmetry and (point) groups have been recognized as essential concepts for chemists and there have appeared several excellent textbooks on these topics.[1]–[6] In the light of these textbooks, we can obtain fundamental knowledge on symmetry and group theory. In this section, we revisit a minimum set of concepts concerning symmetry and groups by using an allene molecule (D 2d ) as an example.
Shinsaku Fujita

### Chapter 3. Permutation Groups

Abstract
Consider a set $$\Delta = \left\{ {{\delta _1}{\delta _2}, \ldots ,{\delta _{\left|\Delta \right|}}}\right\}$$. A one-to-one mapping from Δ to Δ is called a permutation. The number $$\left| \Delta \right|$$ is called the degree of the permutation. Since we here take accout only of such mappings, the elements of Δ may be any objectives. For simplicity’s sake, we consider a set of positive integers, Δ = {1,2,3,4}.
Shinsaku Fujita

### Chapter 4. Axioms and Theorems of Group Theory

Abstract
In Chapters 2 and 3, we have used some concepts of group theory in a practical fashion. In this chapter, we restate the axioms of the group theory and several ralated concepts such as subgroups and cosets in a more logical fashion. Two types of partitions of group elements are discussed to classify them. The one is a partition according to cosets. And the other is a partition based on conjugacy.
Shinsaku Fujita

### Chapter 5. Coset Representations and Orbits

Abstract
Let H be a subgroup of a group G of finite order. Then we have a (right) coset decomposition:
$$G = H{g_1} + H{g_2} + \cdots + H{g_{r,}}$$
(5.1)
where g 1= I (identity). Let us consider a set of the cosets,
$$G/H = \left\{ {H{g_{1,}}H{g_{2,}} \cdots ,H{g_r}} \right\}.$$
(5.2)
Shinsaku Fujita

### Chapter 6. Systematic Classification of Molecular Symmetries

Abstract
Point groups are frequently incomplete to afford full symmetry information about molecules; various molecules belong to the same point group. For example, a list of C 2v molecules contains water, hydrogen sulfide, formaldehyde, phosgene, chlorobenzene, fluorobenzene, 1,2- and 1,3-difluorobezenes, 1,2-benzoquinone, 1,4-dichloronaphthalene, 1,4-naphthoquinone, pyridine, 4-chloropyridine, 2,6-dichloropyridine, pyridine N-oxide, furan, thiophene, cyclopropanone, cyclobutanone, cyclopentanone, tetrahydrofuran, tetrahydrothiophene, dichloromethane, difluoromethane, oxirane, phenanthrene, bicyclo[2.2.1]heptane, basketane, adamantanone, noradamantane and so on. Such a list should be classified into several categories in the light of a rational criterion. For this purpose, Pople[1] has proposed the concept of “framework group”. By this method, difluoromethane is designated as C2v [C2(C), σ v (F2), σ ’2v (H2)]. Flurry[2] pointed out that the framework group is related to local (site) symmetries and proposed his notation based on the local symmetries. Thus, the difluoromethane is designated as C2v [C2v ,(C), C s (F2), C’ s (H2)]. Although Flurry’s method has a potential applicability, it is not so easy to determine such local symmetries especially in the cases of complex molecules. We here propose the SCR (set-of-coset-representation) notation for classifying molecular symmetry.
Shinsaku Fujita

### Chapter 7. Local Symmetries and Forbidden Coset Representations

Abstract
In the preceding chapter, we have discussed SCR notations, which are based on the correspondence between coset representations (CRs) and orbits (sets of equivalent atoms). Without taking account of this correspondence, enumeration of such orbits has been done by Brester[1], Jahn-Teller[2], Boyle[3] and Fowler- Quinn[4] for most point groups. By these conventional methods, each orbit (O A ) is usually characterized by the site symmetry group (H A) that stabilizes (or fixes) one site (A) of the orbit. This characterization is accomplished by using tables presented by Fowler and Quinn.[4] The site symmetry group (H A ) for a non-center atom is shown to be one of C1, C s , C n , or C nv (n ≥ 2);[3] several subsymmetries of G are incapable of being site symmetry groups.
Shinsaku Fujita

### Chapter 8. Chirality Fittingness of an Orbit

Abstract
In the preceding chapter, we have introduced the concept of “local symmetry” that is associated with a coset representation. This concept strictly controls symmetrical properties of any three-dimensional (3D) objects such as molecular models. The concept should be extended into a slightly different form, because most problems in stereochemistry are concerned with chirality/achirality dichotomy.
Shinsaku Fujita

### Chapter 9. Subduction of Coset Representations

Abstract
In this book, we deal with two types of desymmetrization processes, (1) asymmetric syntheses and (2) derivation of a molecule from a high-symmetry parent molecule. As a basis of solving such problems, we introduce subduction of coset representations.
Shinsaku Fujita

### Chapter 10. Prochirality

Abstract
We have discussed the concept of chirality fittingness (Chapter 8). In terms of this concept, we re-examine the concept of “prochirality”, which has been proposed by Hanson.[1, 2] Hanson’s definition is based upon permutation-group theory and plays down point-group theory, though the latter is essential to discuss stereochemistry. The prochirality concept has been restricted to molecular systems in which two ligands or faces are differentiated to create a chiral compound. By integrating the permutation- and point-group theories, the present method gives a general approach that treats two or more ligands successfully.
Shinsaku Fujita

### Chapter 11. Desymmetrization of Para-Achiral Compounds

Abstract
A homospheric orbit does not split away with and without a chiral environment. This fact implies that there is no direct method for converting a para-achiral molecule into a chiral one. However, a multistep conversion is available to solve this problem. Mislow and Siegel discussed the desymmetrization of achiral objects that contain one or more symmetry elements of the second kind (σ, i, or S 4) and proposed the concept of (pro) p -chirality.[1] Later, Halevi added some extension to this concept. [2] This section deals with these issues from the present viewpoint.
Shinsaku Fujita

### Chapter 12. Topicity and Stereogenicity

Abstract
Since the terminology in stereochemistry has been to a great extent empirically developed, it has not escaped some ambiguity. Thus, there have appeared many reviews for clarifying or getting rid of such ambiguity.[1]–[7] The term stereogenic was originally proposed by McCasland[8] in order to avoid confusions provided by so-called “asymmetric” and “pseudoasymmetric” carbon atoms. The term stereogenic has been utilized as a key concept in the revised CIP-system for describing chirality and related matters. [9] Later, Mislow and Siegel[10] have extended the definition of a stereogenic atom so as to manipulate a stereogenic element (unit); this is defined as an element (uint) bearing several groups of such nature that an interchange of two groups produces a stereoisomer. They also discussed the conceptual distinction between the stereogenicity and chirotopicity. The term stereogenic has been adopted by several authors[11] in place of the terms “asymmetric” and “chiral”. In the present chapter, we redefine topicity terms in the light of chirality fittingness of orbits and then discuss stereogenicity.
Shinsaku Fujita

### Chapter 13. Counting Orbits

Abstract
The purpose of this chaper is to introduce the Pólya-Redfield theorem, which has been widely used for enumeration of chemical compounds. This theorem is of fundamental importance, not only because of its wide applicability but also because of its conceptual depth.[1, 2] Many mathematical textbooks have dealt with the topics in various manners.[3]–[7] Chemical enumerations have been reviewed in several books[8]–[10] and articles.[11]–[13] Recent advances of chemical enumerations have been reported.[14] We have reported enumerations of organic reactions after the formulation of imaginary transition structures.[15]
Shinsaku Fujita

### Chapter 14. Obligatory Minimum Valencies

Abstract
In chemical enumerations, there are many cases in which we should take account of obligatory minimum valencies (OMVs). For example, consider an adamantane skeleton (1) of T d symmetry, in which ten carbon atoms are replaced by other atoms.
Shinsaku Fujita

### Chapter 15. Compounds with Achiral Ligands Only

Abstract
Enumeration of compounds in chemistry is one of the most important fields to which the Pólya-Redfield theorem[1] has been applied. Ruch et al.[2] and later Brocas[3] have proved the concept of double coset to be very convenient for such compound-counting problems. Sheehan[4] has applied the concept of table of marks (see Chapter 5) to enumeration of graphs under an automorphic group. More recently, Hässelbarth[5] has developed a method that is also based on the concept of table of marks. This method is capable of enumerating compounds with respect to a given symmetry. Mead[6] has discussed an alternative method that is a combination of table of marks and double cosets. These methods, however, have taken no account of intransitivity of a domain explicitly; hence, they have disregarded the restriction due to OMVs (obligatory minimum valencies).
Shinsaku Fujita

### Chapter 16. New Cycle Index

Abstract
In Chapter 15, we have discussed an application of unit subduced cycle indices (USCIs) to enumeration of compounds, which is based on a new type of generating functions. In a continuation of the work, the present chapter deals with the relationship between USCIs and Pólya’s cycle indices.
Shinsaku Fujita

### Chapter 17. Cage-Shaped Molecules with High Symmetries

Abstract
Chiral and achiral hydrocarbons of high symmetry have attracted continuous attention of synthetic and physical organic chemists because of their potential interesting properties.[1]–[5] Two strategies have been applied to the derivation of new compounds of given symmetries. The first is a vertex strategy in which substituents are placed on vertices of a parent skeleton.[6, 7]The second (edge strategy) consists of the insertion of methylene or other units into the edges (bonds) of a parent skeleton. In particular, the latter is versatile methodology used to guide synthetical studies concerning cage-shaped compounds of high symmetry. Thus, there have appeared many papers that reported successful syntheses of such compounds. For example, a selected list of achiral hydrocarbons reported contains prismane of D 3h symmetry,[8] iceane (tetracyclo[5.3.1.12,6.04,9]dodecane) of D 3h symmetry,[9]–[11],([26](1,2:3,4:5,6-tris(bicyclo[2.2.2]octa-2-eno)benzene of D 3h symmetry,[12] superphane ([26](l,2,3,4,5,6)cyclophane) of D 6h symmetry,[13] pentaprismane of D 5h symmetry,[14] a tetrahedrane of T d symmetry,[15] cubane of O h symmetry,[16] and dodecahedrane of I h symmetry.[17] Chiral hydrocarbons of high symmetry have also been investigated synthetically and physicochemically, e.g., twistane of D 2 symmetry[18, 19] and D 3-trishomocubane of D 3 symmetry.[20]–[22] Although several theoretical studies have discussed the two methodologies,[23, 24] there have emerged no comprehensive studies concerning the following problem: what symmetries are realized and how many isomers are allowed on the basis of a skeleton of a given symmetry.
Shinsaku Fujita

### Chapter 18. Elementary Superposition

Abstract
Davidson[1] has pointed out that enumeration problems in chemistry can be solved by using the Redfield-Read superposition theorem, which was originally established by Redfield,[2] rediscovered independently by Read[3] and later refocussed by several mathematicians.[4, 5] Lloyd[6] has actually applied this theorem to chemical enumerations. All of these methods are capable of enumerating chemical objects with respect only to their weights (molecular formulas).
Shinsaku Fujita

### Chapter 19. Compounds with Achiral and Chiral Ligands

Abstract
We have discussed subduction of coset representations (Chapter 9) and presented a systematic classification of molecular symmetry (Chapter 6). In addition, we have pointed out that several related concepts, e.g. unit subduced cycle indices (USCIs) and unit subduced cycle indices with chirality fittingness (USCI-CFs), are useful for qualitative discussions on molecular symmetry (Chapter 8). Chapters 8 and 9 have clarified their meanings, especially that of chirality fittingness. This chapter deals with a quantitative application of the USCI-CFs to enumeration problems.
Shinsaku Fujita

### Chapter 20. Compounds with Rotatable Ligands

Abstract
Pólya[1] has already mentioned the enumeration of non-rigid molecules in terms of “coronas”, which are equivalent to wreath products. The counting of non-rigid cyclohexane isomers was discussed by introducing a ring-flip-rotation operator along with usual symmetry operations.[2] Isomers derived from a non-rigid ethane were enumerated by means of the concept of covering groups.[3] A generalized wreath product method has been presented for enumerating stereo- and positional isomers.[4] However, systematic enumeration of non-rigid isomers that takes into account the spacial symmetries of isomers is open to further investigation. As a continuation of our studies on USCIs,[5] the present chapter will deal with this type of enumeration by using 2,2-diphenyl- and 2,2-dimethylpropane as examples. In particular, we discuss a general case in which a non-rigid molecule involves chiral ligands as well as achiral ones.
Shinsaku Fujita

### Chapter 21. Promolecules

Abstract
We have discussed the usefulness of coset representations (CRs) in enumerating various molecules (Chapters 15–20),[1]–[6] in classifying molecular symmetry (Chapter 6),[7] and in specifying stereochemical equivalency (Chapters 11 and 12).[8] These applications are based on the fact that each CR governs an orbit that consists of equivalent objects (atoms, bonds, faces and so on). We have also discussed the stereochemical relationship between global symmetry and local ones in a rigid molecule (Chapters 7–9).[9] This treatment affords a foundation for deciding how to obtain a molecule of a given symmetry. The present chapter deals with global-local relationships in non-rigid molecules, where several new concepts such as promolecules and proligands are proposed.
Shinsaku Fujita

### 22. Appendix A Mark Tables

Abstract
This is a list of mark table of several point groups.
Shinsaku Fujita

### 23. Appendix B Inverses of Mark Tables

Abstract
This is a list of the inverse matrices of mark tables for several point groups.
Shinsaku Fujita

### 24. Appendix C Subduction Tables

Abstract
This is a list of subduction tables of several point groups. A coset representation with an asterisk is forbidden.
Shinsaku Fujita

### 25. Appendix D Tables of USCIs

Abstract
This is a list of tables of unit subduced cycle indices for several point groups.
Shinsaku Fujita

### 26. Appendix E Tables of USCI-CFs

Abstract
This is a list of tables of unit subduced cycle indices with chirality fittingness for several point groups.
Shinsaku Fujita

### Backmatter

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