Asymmetry revisited by Fujita’s stereoisogram approach. Part 1: Asymmetry under point-group symmetry and under RS-permutation-group symmetry

Asymmetry revisited by Fujita’s stereoisogram approach. Part 1: Asymmetry under point-group symmetry and under RS-permutation-group symmetry

Tetrahedron: Asymmetry xxx (2016) xxx–xxx Contents lists available at ScienceDirect Tetrahedron: Asymmetry journal homepage: www.elsevier.com/locate...

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Tetrahedron: Asymmetry xxx (2016) xxx–xxx

Contents lists available at ScienceDirect

Tetrahedron: Asymmetry journal homepage: www.elsevier.com/locate/tetasy

Asymmetry revisited by Fujita’s stereoisogram approach. Part 1: Asymmetry under point-group symmetry and under RS-permutation-group symmetry Shinsaku Fujita Shonan Institute of Chemoinformatics and Mathematical Chemistry, Kaneko 479-7 Ooimachi, Ashigara-Kami-Gun, Kanagawa-Ken 258-0019, Japan

a r t i c l e

i n f o

Article history: Received 6 February 2016 Accepted 2 June 2016 Available online xxxx

a b s t r a c t Three kinds of symmetries are derived from stereoisograms, i.e., point-group symmetry, RS-permutationgroup symmetry, and ligand-reflection-group symmetry. Among them, point-group symmetry is correlated to chirality as the first kind of handedness, while RS-permutation-group symmetry is correlated to RS-stereogenicity as the second kind of handedness. Thereby, asymmetry under point-group symmetry (denoted as asymmetry(P)) is differentiated from asymmetry under RS-permutation-group symmetry (denoted as asymmetry(RSP)), where tetrahedral derivatives are used as probes. The term asymmetry or hypo-chirality (newly coined) should be used to refer to the asymmetry(P), which is useful to discuss geometric features. For the purpose of supporting R/S-stereodescriptors, the term RS-stereogenic center (or more generally RS-stereogenic unit) based on the concept of RS-stereogenicity should be used in place of van’t Hoff’s ‘asymmetric carbon atom’ or its successor term ‘stereogenic center’, where the assignability of R/S-stereodescriptors is discussed by referring to the asymmetry(RSP) restricted to tetrahedral derivatives. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The term ‘asymmetric carbon atoms’ has been introduced by van’t Hoff,1,2 a founder of stereochemistry, to discuss tetrahedral carbon atoms at the beginning of the history of stereochemistry in 1870s. Almost at the same time, Le Bel,3,4 an another founder of stereochemistry, has discussed tetrahedral carbon atoms in terms of ‘dissymmetry’, which stems from Pasteur.5 The term ‘dissymmetry’ has been replaced by the term chirality proposed by Thomson (Lord Kelvin),6 so that the Cahn–Ingold–Prelog (CIP) system7,8 has adopted the terms ‘chirality centers’ in place of ‘asymmetric centers’ for the purpose of referring to targets for R/S-stereodescriptors. After discussing stereoisomerism and local chirality by referring to McCasland’s article,9 Mislow and Siegel10 have introduced the term ‘stereogenic atoms’ in place of the terms ‘asymmetric carbon atoms’ and ‘chirality centers’. The revised CIP system11 has adopted the term ‘stereogenic’ for specifying the capability of the assignment of R/S-stereodescriptors. This replacement has resulted in the transmutation of the distinctive connotation of the term

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‘asymmetric carbon atoms’ (or ‘chirality centers’), because the term ‘asymmetric carbon atoms’ is concerned with R/S-stereodescriptors, while the term ‘stereogenic’ has a broader meaning to cover R/S-stereodescriptors, Z/E-descriptors, and so on. Because of the transmuted feature of this replacement, ‘stereogenic centers’ are often also called ‘chirality centers’ even now,12,13 so that confused situations concerning the terms ‘asymmetric’, ‘chiral’, and ‘stereogenic’ have not been settled yet in the conventional framework of stereochemistry. On the other hand, the term ‘asymmetric’ is used to refer to an object which lacks all symmetry elements other than the identity operation (I or E), i.e., belonging to the (trivial) point group C1 , as found in textbooks (e.g., page 25 of Mislow,14 page 72 of Eliel– Wilen,15 and page 81 of Morris16) as well as in an article on the IUPAC recommendations 1996.17 The IUPAC recommendations 199617 has pointed out that the term ‘asymmetric’ for indicating the assignment to the point group C1 has been used loosely (and incorrectly) to describe the absence of an rotation-reflection axis (alternating axis) in a molecule, i.e., as meaning chiral, and that this usage persists in traditional terms ‘asymmetric carbon atom’, ‘asymmetric synthesis’, ‘asymmetric induction’, etc. However, the problem how loosely (or incorrectly) such traditional terms are used has not been solved yet by a distinct theoretical formulation.

http://dx.doi.org/10.1016/j.tetasy.2016.06.006 0957-4166/Ó 2016 Elsevier Ltd. All rights reserved.

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The term ‘asymmetric’ to mean belonging to C1 has often been used in the form ‘chiral, but not asymmetric’, as discussed detailedly in a review by Gal.18 If the meaning of the term ‘asymmetric carbon atom’ is applied to the expression ‘chiral, but not asymmetric’, it causes some confusion. To avoid such confusion, the problem described in the preceding paragraph should be solved on the basis of more rational theoretical formulations. This means that its successor terms ‘chirality center’ and ‘stereogenic center’ in addition to the traditional term ‘asymmetric carbon atom’ should be discussed on the basis of more rational theoretical formulations than the conventional stereochemistry supports. The author (Fujita) has proposed the concept of stereoisograms to reorganize theoretical foundations for discussing stereochemistry and stereoisomerism.19–21 Thereby, ‘stereogenicity’ is restricted meaningfully to the concept of RS-stereogenicity by formulating RS-permutation groups, which are integrated with point groups to generate RS-stereoisomeric groups. The resulting RSstereoisomeric groups are represented by stereoisograms as diagrammatic expressions, where the concepts of RS-diastereomers and holantimers (in addition to enantiomers) are introduced as key concepts of the integration.22 According to Fujita’s stereoisogram approach, misleading standpoints for Cahn–Ingold–Prelog’s R/S-stereodescriptors and for Hanson’s pro-R/pro-S-descriptors have been avoided in a rational fashion.23–25 In addition, Fujita’s stereoisogram approach has provided us with a remedy against oversimplified dichotomy between enantiomers and diastereomers in the conventional stereochemistry.26,27 The present article is devoted to discussions on the problem how loosely (or incorrectly) such traditional terms as ‘asymmetric carbon atom’ are used. For this purpose, the term ‘asymmetry’ is examined from the viewpoints of point-group symmetry, RS-permutation-group symmetry, and ligand-reflection symmetry, after these are formulated by Fujita’s stereoisogram approach.21 Such successor terms as ‘chirality center’ and ‘stereogenic center’ are also discussed by adopting Fujita’s stereoisogram approach as a basis for giving reliable theoretical formulations. The term RS-stereogenic center (or more generally RS-stereogenic unit) will be proposed in place of ‘asymmetric carbon atom’, ‘chirality center’, and ‘stereogenic center’ after the adoption of the proligandpromolecule model.28

2. Revisited terminology of stereochemistry and stereoisomerism 2.1. Different connotations of the term ‘asymmetry’ The term ‘asymmetry’ has been conventionally used in different meanings. For the sake of convenience, these meanings are tentatively differentiated as follows:

can be defined only after the development of Fujita’s stereoisogram approach. It should be emphasized that chirality and RS-stereogenicity are recognized to be two kinds of handedness in the present article. The discussions in the following sections will clarify the respective connotations of these tentative terms. Thereby, the ‘asymmetry(P)’ will be adopted as the meaning of the term asymmetry. 2.2. Van’t Hoff’s asymmetric carbon and RS-permutations As implied in his article,1,2 van’t Hoff’s asymmetric carbon (referred to hereafter as ‘asymmetric(H) carbon’) has been originally defined as an attribute assigned to a two-dimensional (2D) structure (graph), which is afterward extended into three-dimensional (3D) structures. This implication (the hidden intervention of a 2D structure) is shown explicitly in Figure 1. As one example, a 2D structure 2 with a set of achiral proligands A, B, X, and Y has an asymmetric(H) carbon (designated by an asterisk), so that it is extended to generate 3D structures 1 and 3. Because 1 and 3 are enantiomeric to each other if the numbering is omitted, the asymmetric(H) carbon of 2 has been linked directly with the chirality of 1 and 3. As shown by Fischer in the isomers of 2,3,4-trihydroxyglutaric acids,29–31 however, a 2D structure 5 with a set of achiral and chiral proligands A, B, p, and p (the proligands p and p are enantiomeric to each other when detached), the resulting 3D structures 4 and 6 are both achiral in spite of the presence of an asymmetric(H) carbon (designated by an asterisk). Because 4 and 6 are not enantiomeric to each other even if the numbering is omitted, an asymmetric(H) carbon has nothing to do with the appearance of chirality. This exceptional case is called ‘pseudoasymmetry’ and has been excluded from a main recipe for constructing theoretical foundations of organic stereochemistry. Logically speaking, however, the carbon denoted by an asterisk in 5 should still be regarded as a kind of an asymmetric(H) carbon, although organic chemists tend to regard this case as being different from an asymmetric(H) carbon. In other words, the asymmetric(H) carbon (C⁄ of 2) in the process between 1 and 3 is misleadingly considered to linked directly with chirality after the asymmetric(H) carbon (C⁄ of 5) in the process between 4 and 6 is ignored as an exception. According to Fujita’s stereoisogram approach,19–21 an RS-permutation represented by ð1Þð2 4Þð3Þ is considered to convert the 3D-structure 1 into the other 3D-structure 3 (Fig. 2). The carbon atom of 1 (or 3) is regarded as being RS-stereogenic as designated by a dagger (y), so that 1 and 3 construct a pair of RS-diastereomers. The pairing of 1 and 3 producing a pair of RS-diastereomers stems from the substantial restriction of the term ‘stereogenic’ to the term RS-stereogenic. As a result of the pairing, RS-stereogenicity A

1

(H)

1. The term ‘asymmetry ’ is used to refer to the situations brought about by van’t Hoff’s asymmetric carbon. The term ‘asymmetry(H)’ is accompanied by the term ‘pseudoasymmetry’ in order to specify exceptional cases. 2. The term ‘asymmetry(P)’ is used to refer to asymmetry under point-group symmetry. This means C1 -chirality, which has no higher symmetry operations other than the identity group (C1 ) under point-group symmetry. Chirality is regarded as the first kind of handedness in the present article. 3. The term ‘asymmetry(RSP)’ is used to refer to asymmetry under RS-permutation-group symmetry. This means C1 -RS-stereogenicity, which has no higher symmetry operations other than the identity group (C1 ) under RS-permutation-group symmetry. RS-Stereogenicity is regarded as the second kind of handedness in the present article. It should be noted that the ‘asymmetry(RSP)’

2

X Y

C

3

B

4

1

A

1

p p

C 4

4

3

B

3D structure

X2

2

2D structure (graph) 3B A

p2

5

1

4

Y X

C

2D structure (graph)

3

B

2

3

3D structure A

1

C* 4p

A

1

C* 4Y

3D structure

2

3B A

1

4

p p

C 2

3

B

6

3D structure

Figure 1. van’t Hoff’s asymmetric carbon (⁄) as a 2D structure, which is extended to generate 3D structures. The lower row represents so-called ‘pseudoasymmetry’, which is regarded as an exceptional case having asymmetric carbon (⁄) in the conventional stereochemistry. Figures 1.2 and 9.11 of Ref. 21 are modified to meet the present context.

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A X Y

C† 3

B

4

RS-Permutation (1)(2 4)(3) RS-Permutation (1)(2 4)(3)

1

Y X

A

1

C† 3 4

4

B

2

3D structure

A p p

C† 3

B

3D structure

RS-Permutation (1)(2 4)(3) RS-Permutation (1)(2 4)(3)

1

4

p

p

C† 3 2

B

6

3D structure

Figure 2. RS-Stereogenic carbon centers (y) of 3D structures, which are generated by a RS-permutation between numbered skeletons.

can be considered to be a kind of handedness, so that it is called the second kind of handedness after chirality as the original handedness is anew called the first kind of handedness. The RS-permutation represented by ð1Þð2 4Þð3Þ converts 4 into 6 (Fig. 2). The carbon atom of 4 (or 6) is regarded as being RS-stereogenic as designated by a dagger (y). The RS-diastereomeric pair of 4 and 6 is produced by a RS-permutation represented by ð1Þð2 4Þð3Þ, just as the RS-diastereomeric pair of 1 and 3 is produced by the same RS-permutation ð1Þð2 4Þð3Þ. It is to be noted that the conventional stereochemistry regards the pair of 1 and 3 as a pair of enantiomers (as an asymmetric(H) case), while the pair of 4 and 6 as a pair of diastereomers (as an exceptional pseudoasymmetric case). By emphasizing RS-permutations such as ð1Þð2 4Þð3Þ, in contrast, Fujita’s stereoisogram approach regards the pair of 1 and 3 as a pair of RS-diastereomers, even though it is coincident with a pair of enantiomers. This conclusion is consistent with the fact that the pair of 4 and 6 is determined to be a pair of RS-diastereomers, where this pair is not coincident with a pair of enantiomers. In other words, the RS-stereogenic carbon centers (Cy) in the RS-diastereomeric pair of 1 and 3 have the same nature as the RS-stereogenic carbon centers (Cy) in the RS-diastereomeric pair of 4 and 6, so long as we lay stress on RS-stereogenicity as the second kind of handedness (Fig. 2). Such RS-stereogenic centers are referred to as RS-stereogenic units more generally, when RS-stereogenic axes are introduced as the other examples of RS-stereogenic units in the discussions on allene derivatives in Part 2 of this series. 2.3. Le Bel’s molecular dissymmetry and reflections The dissymmetry (or chirality) adopted by Le Bel3,4 supports reflection operations in a more rational fashion than the standpoint of van’t Hoff. However, he has ignored local chiralities and considered atoms or substituents to be spheres or material points. In the present context, he has taken account of achiral proligands such as A, B, X, and Y and ignored chiral proligands such as p and p. This ignorance has unfortunately concealed the superiority of his way to van’t Hoff’s way during discussions on geometric features of stereochemistry, so that van’t Hoff’s way has been widespread as the standard of the conventional stereochemistry. According to Fujita’s stereoisogram approach,19–21 a reflection represented by ð1Þð2 4Þð3Þ is considered to convert the 3D-structure 1 into the other 3D-structure 1 (Fig. 3). The overbar represents a ligand reflection, as denoted by a mirror-numbered skeleton. Each achiral proligand remains as it is, i.e., A ¼ A, B ¼ B, X ¼ X, and Y ¼ Y, under the action of reflections. Hence, the relationship between 1 and 1 is enantiomeric. The central carbon (C ) of 1 (or 1) is characterized by the coset representation C1 ð=C1 Þ under the point-group symmetry, so that it exhibits local chirality, as designated by a circle. It should be noted that 1 (the enantiomer of 1 due to the reflection ð1Þð2 4Þð3Þ) is coincident with 3 (the RS-diastereomer of 1 due

A

1

1

4

3

3D structure

2

A

A

1

2

2

X Y

C◦ 3 4

B

Reflection (1)(2 4)(3) Reflection (1)(2 4)(3)

1

⎪ ⎪ A ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ C 3 4 ⎪ ⎪ ⎪ p B ⎪ 2 ⎪ ⎪ ⎪ p ⎪ ⎪ ⎪ ⎪ ⎩ 6 3D structure

Y X

C◦ 3

B

2

1

3D structure ⎧ A 1 ⎪ ⎪ ⎪ ⎪ 2 C 3 ⎪ ⎪ ⎪ p B ⎪ ⎪ 4 ⎪ ⎪ p ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎨ 3D structure

1

4

3D structure A Reflection (1)(2 4)(3) Reflection (1)(2 4)(3)

1

4

p p

C

3

B

2

4 (= 4)

3D structure A Reflection (1)(2 4)(3) Reflection (1)(2 4)(3)

1

2

p p

C 4

3

B

6 (= 6)

3D structure

Figure 3. Reflection accompanied with ligand reflection, where a numbered skeleton is converted into a mirror-numbered skeleton. The two skeletons correspond to 3D structures which are (self-) enantiomeric relationship to each other.

to the RS-permutation ð1Þð2 4Þð3Þ), if the numbering and the mirror-numbering are ignored. However, the misleading interpretation such as 1—(1)(2 4)(3)! 1 (the direct conversion) should be avoided. The correct interpretation is represented by 1— ð1Þð2 4Þð3Þ!3 (¼ 1), which is accompanied by 1—ð1Þð2 4Þð3Þ! 1. The reflection ð1Þð2 4Þð3Þ converts 4 into 4, which is identical with 4. Note that the chiral proligand p (or p) is converted into p (or p). It follows that 4 is self-enantiomeric (achiral). The central atom of 4 exhibits local achirality as characterized by the coset representation Cs ð=Cs Þ. In a similar way, the reflection ð1Þð2 4Þð3Þ converts 6 into 6, which is identical with 6, so that 6 is selfenantiomeric (achiral). The central atom of 6 exhibits local achirality as characterized by the coset representation Cs ð=Cs Þ. It should be emphasized that the conventional stereochemistry lacks the concept of mirror-numbered skeletons, so that the process of 1 !3 (Fig. 2) is misleadingly regarded as identical with the process of 1 ! 1 (Fig. 3). These processes should be strictly differentiated from each other in a logical fashion, even though the resultant 3 is coincident with the other resultant 1. In other words, RS-permutations (e.g., ð1Þð2 4Þð3Þ in Fig. 2) should be strictly differentiated from reflections (e.g., ð1Þð2 4Þð3Þ in Fig. 3). And furthermore, RS-permutations should be strictly differentiated from permutations. 3. Fujita’s stereoisogram approach 3.1. Aufheben between van’t Hoff’s way and Le Bel’s way by Fujita’s stereoisogram approach The discussions in the preceding sections are summarized by a statement that there exist two kinds of handedness, i.e., chirality (the first kind of handedness related to reflections under point-group symmetry) and RS-stereogenicity (the second kind of handedness related to RS-permutations under RS-permutation-group symmetry), where van’t Hoff’s way is supported by RS-stereogenicity while Le Bel’s way is supported by chirality. The next task is to clarify how the two ways interact with each other. This task has been accomplished by the development of Fujita’s stereoisogram approach,19–21 which has brought about the Aufheben between van’t Hoff’s way (supported by RS-stereogenicity as the second kind of handedness) and Le Bel’s way (supported by chirality as the first kind of handedness) in the form of stereoisograms as diagrammatic expressions.

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The pair of RS-diastereomers 1/3 shown in Figure 2 is placed in the horizontal direction (related to van’t Hoff’s way), while the pair of enantiomers 1/1 shown in Figure 3 is placed in the vertical direction (related to Le Bel’s way), so as to give a prototype diagram of Figure 4. To complete a stereoisogram by starting from this prototype diagram, it is necessary to place a new 3D structure 3 in the diagonal direction.19 Thereby, a type-I stereoisogram is created as shown in Figure 4. The new 3D structure 3 is called a holantimer of 1. The diagonal directions are characterized by equality symbols, because the holantimer 3 is identical with 1 by ignoring the numbering and the mirror-numbering. On the other hand, the pair of RS-diastereomers 4/6 shown in Figure 2 is placed in the horizontal direction (related to van’t Hoff’s way). Each pair of self-enantiomers (an achiral 3D structure)4/4 (=4) (or 6/6 (=6)) shown in Figure 3 is placed in the vertical direction (related to Le Bel’s way). The resulting prototype diagram contains holantimeric relationships between 4 and 6 (or between 6 and 4) in the diagonal positions. Thereby, Figure 5 is generated as a completed type-V stereoisogram, in which the vertical directions are characterized by equality symbols. Other types of stereoisograms (type II, type III, and type IV) can be written in a similar way.22–24 They are discussed later from the present point of view. S

A

A

◦ 2 SC 3

◦ 4 RC 3

1

X Y

1

B

4

1

(1)(2)(3)(4)

Y X

3 (= 1)

(1)(2 4)(3)

A

A

RC ◦

SC ◦

1

4

Y X

B

2

2

B

1

(1)(2 4)(3)

2

X Y

4

Let us define an RS-stereoisomeric group T b to characterize a de rI stereoisogram based on a tetrahedral skeleton.32 First, a tetrahedral skeleton belongs to the point group Td , which is represented by the following coset decomposition:

Td ¼ T þ Tr;

ð1Þ

where T represents the maximum chiral subgroup of Td and the symbol r represents a reflection (or more generally a (roto) reflection). The four positions of the tetrahedral skeleton construct an orbit governed by a coset representation Td ð=C3v Þ, where each element of the coset Tr is represented by a product of cycles with an overbar (e.g., ð1Þð2 4Þð3Þ shown in Fig. 3). The point group Td (Eq. 1) corresponds to the vertical directions of a stereoisogram (e.g., Figs. 4 and 5). Second, the tetrahedral skeleton is considered to belong to the RS-permutation group Te , which is represented by the following r coset decomposition:

e; Ter ¼ T þ T r

ð2Þ

e represents an RS-permutation, which is where the symbol r defined by starting from r of Td and by omitting the ligand reflection. The four positions of the tetrahedral skeleton construct an orbit governed by a coset representation Te ð=C3e Þ, where each eler r

e is represented by a product of cycles without ment of the coset T r an overbar (e.g., ð1Þð2 4Þð3Þ shown in Fig. 2). The RS-permutation group Te r (Eq. 2) corresponds to the horizontal directions of a stereoisogram (e.g., Fig. 4 or Fig. 5). Third, the tetrahedral skeleton is considered to belong to the ligand-reflection group Tb, which is represented by the following I coset decomposition:

Tb ¼ T þ TbI; I

1

3

3.2. RS-Stereoisomeric groups

3

ð3Þ

where the symbol bI represents a ligand reflection, which is defined by starting from an identity element I of T and by adding the ligand reflection. The four positions of the tetrahedral skeleton construct an orbit governed by a coset representation Tbð=C bÞ, where each

B

3 (= 1)

(1)(2)(3)(4)

C

I

Figure 4. Stereoisogram of type I. The promolecule 1 belongs to the RS-stereoisomeric group Cb, to the point group C1 , to the RS-permutation group C1 , and to the I ligand-reflection group Cb. This is represented by the aspect index [I, Cb; C1 ; C1 ; Cb]. A I I I pair of R/S-stereodescriptors, ‘S’ and ‘R’, is assigned to a pair of RS-diastereomers 1/3 (or 3/1), where the priority sequence A > B > X > Y is presumed.

A

1

2 sC

p p

1

3

B

4

4

(1)(2)(3)(4)

4 rC

p

p

p p

2

B

6

A

1

sC

3

2

(1)(2 4)(3)

A 4

S

A

1

3

B

4 (= 4)

(1)(2 4)(3)

2

rC

p

4

p

3

B

6 (= 6)

(1)(2)(3)(4)

C Figure 5. Stereoisogram of type V. The promolecule 4 (or 6) belongs to the RS-stereoisomeric group Cs , to the point group Cs , to the RS-permutation group C1 , and to the ligand-reflection group C1 . This is represented by the aspect index [V, Cs ; Cs ; C1 ; C1 ]. A pair of R/S-stereodescriptors, ‘s’ and ’r’, is assigned to a pair of RS-diastereomers 4/6, where the priority sequence A > B > p > p (or A > p > p > B) is presumed. The lowercase letters are used because of chirality unfaithfulness.

3I

element of the coset Tb is represented by a product of cycles with I an overbar (e.g., ð1Þð2Þð3Þð4Þ) appearing in the diagonal directions of Fig. 4 or Fig. 5). Finally, the tetrahedral skeleton is considered to belong to the RS-stereoisomeric group T b, which is represented by the followde rI ing coset decomposition:

e þ TbI; T b ¼ T þ Tr þ T r de rI

ð4Þ

where Eqs. 1–3 are combined to characterize a stereoisogram. The four positions of the tetrahedral skeleton construct an orbit governed by a coset representation T bð=C bÞ. de rI 3v e rI It should be emphasized that an RS-stereoisomeric group (e.g., T b of Eq. 4) for integrating two kinds of handedness (e.g., a point de rI group Td of Eq. 1 for chirality and an RS-permutation group Te of r Eq. 2 for RS-stereogenicity) inevitably requires a ligand-reflection group (e.g., Tb of Eq. 3) for the concept of sclerality. I

3.3. Five types of RS-stereoisomeric groups and stereoisograms A tetrahedral derivative belongs to a subgroup of the RSstereoisomeric group T b. Such subgroups of the RS-stereoisode rI meric group T b are classified into five types, as summarized in de rI Table 1. These subgroups characterize stereoisograms corresponding to the respective tetrahedral derivatives, so that there are five types of stereoisograms for characterizing tetrahedral derivatives. For

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Table 1 Five types of subgroups of the RS-stereoisomeric group T b de rI

Table 2 Three attributes and type indices as well as three relationships for stereoisograms of Type I–V

example, the type-I stereoisogram shown in Figure 4 belongs to an RS-stereoisomeric group Cb ¼ fI; bIg, which is a subgroup of Tb; and I I the type-V stereoisogram shown in Figure 5 belongs to an RSstereoisomeric group Cs ¼ fI; rd g, which is a subgroup of Td . According to the type-classification of subgroups shown in Table 1, stereoisograms of five types are characterized by the combinations of three pairs of attributes, i.e., chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality, as summarized in Table 2. Each combination is denoted briefly by the corresponding type index, in which the symbol ‘’ represents chirality, RS-stereogenicity, or sclerality, while the symbol ‘a’ represents achirality, RS-astereogenicity, or asclerality. These attributes are related to vertical, horizontal, and diagonal relationships, i.e., (self-) enantiomeric, (self-)RS-diastereomeric, and (self-) holantimeric relationships. 3.4. Misleading theoretical foundations of the conventional stereochemistry The terminology of the conventional stereochemistry has suffered from serious confusion, as pointed out by several articles.33–35,18 The confused situations more or less depend on the state-of-the-art theoretical foundations of the conventional stereochemistry, which misleadingly regard the relationship between 1 and 3 in Figure 2 as an enantiomeric relationship. In other words, the conventional stereochemistry has ignored ligand reflections implied by the reflection process between 1 and 1 listed in Figure 3, because the ligand reflections of achiral (pro) ligands, e.g., A ¼ A, B ¼ B, X ¼ X, and Y ¼ Y, seemingly provide no effects during reflection processes. The concept of ‘permutational isomers’ proposed by Ugi et al.36 and Mislow–Siegel’s approach to stereoisomerism and local chirality10 have been critically discussed from the viewpoint of Fujita’s stereoisogram approach in Chapter 9 of my recent book.21 One of main points of the argument is whether or not ligand reflections are taken into consideration during the discussions on stereoisomerism. Figure 6 shows the disregard of ligand reflections in the discussions on ‘permutational isomers’,36,10 where the disregarded ligand reflections are surrounded by the gray box. By excluding the ligand reflections in the gray box, the permutation ð1Þð2 4Þð3Þ bringing about the conversion between 1 and 3 (the

left diagram of Fig. 6) is regarded as corresponding to an enantiomeric relationship, because 3 is coincident with 1. Note that the recognition of the coincidence is implicit in the conventional stereochemistry and does not depend on the systematic scheme concealed by the gray box. Thereby, the permutation ð1Þð2 4Þð3Þ is misleadingly equalized to a reflection, so that the point group Td is misleadingly equalized to the symmetric group of degree 4 (S½4 ). On the other hand, the same permutation ð1Þð2 4Þð3Þ bringing about the conversion between 4 and 6 (the right diagram of Fig. 6) cannot be regarded as corresponding to an enantiomeric relationship, so that this case is treated as an exceptional case named ‘pseudoasymmetry’. Note that the vertical equality symbols in the gray box of the right diagram are not explicitly recognized in the conventional stereochemistry in a similar way to the diagonal equality symbols in the gray box of the left diagram. Another one of main points of the argument is whether or not the scope of stereoisomerism is restricted meaningfully to RSstereoisomerism linked to stereoisograms. Thus the hierarchy represented by (self-) enantiomers  RS-stereoisomers  stereoisomers is presumed in Fujita’s stereoisogram approach, whereas the hierarchy represented by (self-) enantiomers  stereoisomers (without the concept of RS-stereoisomers) is presumed in the conventional stereochemistry. The lack of the concept of RS-stereoisomerism in the conventional stereochemistry means that RS-stereogenicity is not recognized as the second kind of handedness (cf. Fig. 6). So long as we select a tetrahedral skeleton as a standard, we could presume RSstereoisomers = stereoisomers, so that the lack of the concept of RS-stereoisomerism has been overlooked in a rather unfortunate fashion in the conventional stereochemistry. 4. Three kinds of symmetries and two kinds of handedness 4.1. The term ‘symmetry’ The term ‘symmetry’ is used to indicate that an object is invariant to a given transformation, which may be a reflection or a transform other than a reflection. Hence, it is important to clarify which transformation participates in the symmetry at issue. According to Table 2, we should take account of three types of symmetries, i.e.,

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A

B

4

1

(1)(2)(3)(4)

Y X

2

1

p

3 (= 1)

p

B

(1)(2 4)(3)

X Y

SC ◦

4

B

4

4 rC

p

p

3

sC

p

2

p

B

B

6

A

1

3

B

4 (= 4)

3 (= 1)

(1)(2 4)(3)

(1)(2)(3)(4)

C

4

3

2

(1)(2 4)(3)

1

1

2

3

4

A

A 3

1

(1)(2)(3)(4)

(1)(2 4)(3)

1

RC ◦

2 sC

B

2

S

A

1

◦ 4 RC 3

Y X

A 4

A

1

◦ 2 SC 3

X Y

S

A

1

2

rC

p

4

p

3

B

6 (= 6)

(1)(2)(3)(4)

C Figure 6. Disregard of ligand reflections in the conventional stereochemistry.

point-group symmetry, RS-permutation-group symmetry, and ligand-reflection-group symmetry, as shown in Table 3. As shown in Table 3, the point-group symmetry (symmetry under point groups) is concerned with a pair of chirality/achirality, which corresponds to the vertical direction of a stereoisogram. Chirality as the first kind of handedness is linked with a chiral point group, which is a subgroup of the maximum chiral subgroup e.g., T ( Td , cf. Eq. 1). Hence, chirality is common to type-I ([€,; a]), type-II [€, a; ], and type-III ([€, ; ]) stereoisograms, where the symbol € emphasizes common chiral nature. On the other hand, achirality is linked with an achiral point group, which is a subgroup of the achiral group for characterizing a skeleton at issue, e.g., Td (cf. Eq. 1). Hence, achirality is common to type-IV ([a,a; a]) and type-V ([a, ; ]) stereoisograms, where the symbol a emphasizes common achiral nature. The RS-permutation-group symmetry (symmetry under RS-permutation-groups) is concerned with a pair of RS-stereogenicity/RSastereogenicity, which corresponds to the horizontal direction of a stereoisogram. RS-Stereogenicity as the second kind of handedness is linked with an RS-stereogenic RS-permutation group, which is a subgroup of the maximum RS-stereogenic subgroup (e.g., T  Te , r cf. Eq. 2). Hence, the RS-stereogenicity is common to type-I (½;€,a]), type-III ([;€, ]), and type-V (½a;€, ]) stereoisograms, where the symbol € emphasizes common RS-stereogenic nature. Note that the maximum RS-stereogenic RS-permutation group (e.g., T  Te ) is identical with the maximum chiral point group r (e.g., T  Td ). On the other hand, RS-astereogenicity is linked with an RS-astereogenic RS-permutation group, which is a subgroup of an RS-astereogenic RS-permutation group for characterizing a skeleton at issue, e.g., Te (cf. Eq. 2). Hence, the RS-astereogenicity is comr mon to type-II ([; a, ]) and type-IV ([a; a,a]) stereoisograms, where the symbol a emphasizes common RS-astereogenic nature. The ligand-reflection-group symmetry (symmetry under ligand-reflection-groups) is concerned with a pair of sclerality/ asclerality, which corresponds to the diagonal direction of a stereoisograms. Sclerality is linked with a scleral ligand-reflection group, which is a subgroup of the maximum scleral ligand-reflection subgroup e.g., T ( Tb, cf. Eq. 3). Hence, the sclerality is comI

mon to type-II ([; ;€]), type-III ([; ;€]), and type-V ([a; ;€]) stereoisograms, where the symbol € emphasizes common scleral nature. Note that the maximum scleral ligand-reflection group (e.g., T  Tb) is identical with the maximum chiral point group I (e.g., T  Td ). On the other hand, asclerality is linked with an ascleral ligand-reflection group, which is a subgroup of the ascleral ligand-reflection group for characterizing a skeleton at issue, e.g., Tb (cf. Eq. 3). Hence, the asclerality is common to type-I I ([; ; a]) and type-IV (½a; a; a) stereoisograms, where the symbol a emphasizes common ascleral nature. 4.2. Three aspects of absolute configuration and aspect indices According to Table 2, a stereoisogram is characterized by three aspects of absolute configuration, i.e., chiral, RS-stereogenic, and scleral aspects.37 These aspects are characterized by three types of symmetries, i.e., point-group symmetry, RS-permutation-group symmetry, and ligand-reflection-group symmetry, as summarized in Table 3. The three types of ‘symmetries’ are accompanied by three types of ‘asymmetries’. Thus, chirality as the first kind of handedness is accompanied by asymmetry (C1 ) under point groups; RS-stereogenicity as the second kind handedness is accompanied by asymmetry (C1 ) under RS-permutation groups; and sclerality is accompanied by asymmetry (C1 ) under ligand-reflection groups. The above-mentioned type indices (e.g., ½; ; a for a type-I stereoisogram) should be revised to discuss the three types of ‘asymmetries’, because the respective identity groups C1 are necessary to be determined. For this purpose, let us define an aspect index to show the group-theoretical feature of such three aspects, where the symbols of the RS-stereoisomeric group (for the total aspect), the point group (for the chiral aspect), the RS-permutation group (for the RS-stereogenic aspect), and the ligand-reflection group (for the scleral aspect) are aligned in a pair of square brackets. For example, the type-I stereoisogram shown in Figure 4 is denoted by an aspect index [I, Cb; C1 ; C1 ; Cb], while the type-V I I stereoisogram shown in Figure 5 is denoted by an aspect index [V, Cs ; Cs ; C1 ; C1 ]. The full symmetry of a tetrahedral skeleton is retained in a derivative of the composition CA4, which is

Table 3 Three kinds of symmetries derived from stereoisograms

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5. ‘Asymmetry’ under point-group symmetry—asymmetry(P)

characterized by a type-IV stereoisogram and by an aspect index [IV, T b; Td ; Te ; T ]. r bI de rI Among the three aspects of absolute configuration, the chiral aspect corresponds to the first kind of handedness, and RS-stereogenic aspect corresponds to the second kind of handedness. In the following sections, we will discuss ‘asymmetry’ from the viewpoint of the two kinds of handedness.

To discuss ‘asymmetry’ under the point-group symmetry (named asymmetry(P)), we should focus our attention on type-I, type-II, and type-III stereoisograms, which are collected in the chiral part (the upper part) of Figure 7.

RS-astereogenic

RS-stereogenic Type I [−, −, a]

X Y

A

p

C

C

B

q

1 ([θ ]10 ) [I, CI ; C1 , C1 , CI ]

p

q

7 ([θ ]28 ) [I, Cσ ; C1 , C1 , Cσ ]

Type II [−, a, −] p p

p

A

C

C

C

p

A A

A

p p

p

8 ([θ ]20 ) [II, Tσ ; T, Tσ , T]

9 ([θ ]3 ) [II, C3σ ; C3 , C3σ , C3 ]

10 ([θ ]15 ) [II, C3σ ; C3 , C3σ , C3 ]

q

p

A

p p

chiral

p

C

p

11 ([θ ]22 ) [II, C3σ ; C3 , C3σ , C3 ]

q q

12 ([θ ]21 ) [II, C3σ ; C3 , C3σ , C3 ]

C

p p

A

A

C

C

p

p B

A

15 ([θ ]7 ) [II, Cσ ; C1 , Cσ , C1 ]

q p

A

A

A

C

C

p p

p

18 ([θ ]16 ) [II, Cσ ; C1 , Cσ , C1 ]

p p

p

p

p

C

C

p

q q

p

21 ([θ ]26 ) [II, Cσ ; C1 , Cσ , C1 ]

r

A

A

C

C

p

B

q p

23 ([θ ]11 ) [III, C1 ; C1 , C1 , C1 ]

A p

A

A

C

C

p

r

q

B

24 ([θ ]14 ) [III, C1 ; C1 , C1 , C1 ]

q

q p

25 ([θ ]18 ) [III, C1 ; C1 , C1 , C1 ]

q

19 ([θ ]17 ) [II, Cσ ; C1 , Cσ , C1 ]

C

20 ([θ ]24 ) [II, Cσ ; C1 , Cσ , C1 ]

X

16 ([θ ]9 ) [II, Cσ ; C1 , Cσ , C1 ]

C

B

A

13 ([θ ]5 ) [II, C2σ ; C2 , C2σ , C2 ]

p

17 ([θ ]12 ) [II, Cσ ; C1 , Cσ , C1 ]

q p

p

C

14 ([θ ]25 ) [II, C2σ ; C2 , C2σ , C2 ]

p p

C

p p

Type III [−, −, −]

r

26 ([θ ]19 ) [III, C1 ; C1 , C1 , C1 ]

p

p

C

C

p

s

27 ([θ ]29 ) [III, C1 ; C1 , C1 , C1 ]

r

q

28 ([θ ]30 ) [III, C1 ; C1 , C1 , C1 ]

p

q

22 ([θ ]27 ) [II, Cσ ; C1 , Cσ , C1 ]

Type IV [a, a, a] A A

achiral

A

B

A

C

C

C

A

A A

A

B B

A

29 ([θ ]1 ) [IV, Td σ I ; Td , Tσ , TI ]

30 ([θ ]2 ) [IV, C3vσ I ; C3v , C3σ , C3I ]

31 ([θ ]4 ) [IV, C2vσ I ; C2v , C2σ , C2I ]

p

B

A

p

C

p

p

32 ([θ ]23 ) [IV, S4σ I ; S4 , C2σ , C2I ]

A A

C

X

33 ([θ ]6 ) [IV, Csσ I ; Cs , Cσ , CI ]

p

C

p

A

Type V [a, −, −] A p

C

B p 4 ([θ ]13 )

[V, Cs ; Cs , C1 , C1 ]

34 ([θ ]8 ) [IV, Csσ σ ; Cs , Cσ , Cσ ]

Figure 7. Reference promolecules of quadruplets of RS-stereoisomers (Types I–V) for tetrahedral promolecules. The symbols A, B, X, and Y represent proligands (atoms or achiral ligands). The symbols p, q, r, and s represents chiral proligands, while each symbol with an overbar represents the corresponding chiral proligand with the opposite chirality. An arbitrary promolecule is depicted as a representative of each quadruplet of RS-stereoisomers. The compound number (its partition22) and its aspect index are attached to each promolecule. A gray box is drawn to show asymmetry under an RS-permutation group (asymmetry(RSP) or hypo-RS-stereogenicity). A frame box is drawn to show asymmetry under a point group (asymmetry(P) or hypo-chirality).

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5.1. Chirality versus asymmetry(P) (or hypo-chirality) The above discussions on Table 3 indicate that there are three kinds of ‘asymmetries’ (related to chirality, RS-stereogenicity, and sclerality), logically speaking, as a result of the presence of three kinds of symmetries. However, IUPAC Recommendations 199638 has selected one kind of asymmetry (asymmetry(P)), where the term asymmetric is defined as referring to an object belonging to the point group C1 (¼ fIg). In other words, the term asymmetric of this meaning is concerned with purely geometric properties of the object to be examined, so that it denotes one extreme case of chirality. Note that the term chirality is concerned with purely geometric properties of the object belonging to a chiral point group (G P C1 ) in general. Because the confusion due to the term asymmetry (defined in IUPAC Recommendations 199638) is desirable to be avoided, the term hypo-chirality (hypo-: Greek, hupo, under, below) is here coined to connote C1 -chirality or asymmetry(P). The prefix hypo- means that there is an identity I as a single operation for fixing a given object to be examined. The expression ‘chiral, but not asymmetric’ (cf. 18) corresponds to G-chirality (G > C1 ), where the expression ðG > C1 Þ means that the chirality stems from a chiral point group higher than C1 . This is because the term ‘asymmetric’ in the expression ‘chiral, but not asymmetric’ indicates asymmetry(P) (or equivalently hypo-chirality or C1 -chirality) in the present context. For the sake of simplicity, the term hyper-chirality is coined to connote ‘chirality, but not asymmetry’ or ‘chirality, but not hypo-chirality’ (hyper-: Greek, hyper, over, beyond). The prefix hyper- means that there are two or more operations for fixing a given object to be examined. The adjective hyper-chiral is used here, although Gal18 has proposed the adjective ‘symmanumorphous’ for ‘chiral but not asymmetric’. Because discussions on hyper-chirality usually specify a point group G ð> C1 ) assigned to an object at issue, the term G-chirality is convenient for practical purposes. In conclusion, chirality (cf. Table 3) is divided into hypo-chirality (=asymmetry or C1 -chirality) and hyper-chirality (=G-chirality (G > C1 ) or ‘chirality, but not asymmetry’ or ‘chirality, but not hypo-chirality’). The adoption of the terms hypo-chirality and hyper-chirality has a merit that these terms are correlated to the common stem chirality. 5.2. Asymmetry(P) promolecules

(or

hypo-chirality)

for

tetrahedral

Tetrahedral promolecules have been enumerated under RSstereoisomeric groups32,39 and examined comprehensively on the basis of Fujita’s stereoisogram approach.22–24 Figure 9 of Ref. 32 and Figure 16 of Ref. 22 are modified to meet the present treatment, so as to give Figure 7, where the data of aspect indices and those of asymmetries(P) (surrounded by a frame box) are added to characterize respective reference promolecules of quadruplets of RS-stereoisomers (Types I–V). Among the chiral promolecules (type I, type II, and type III) listed in Figure 7, each promolecule surrounded by a frame box exhibits asymmetry(P) (=hypo-chirality or C1 -chirality), where the corresponding aspect index has C1 as the third component indicating its point group. The promolecules listed in the type-I part of Figure 7 exhibit asymmetry(P) (=hypo-chirality or C1 -chirality), as each is surrounded by a frame box. For example, the promolecule 1 is a reference promolecule of the type-I stereoisogram of Figure 4, where its aspect index [I, Cb; C1 ; C1 ; Cb] has the point group C1 (underlined). I I Hence, the promolecule 1 (paired with 1) is determined to be (P) asymmetric (=hypo-chiral or C1 -chiral). Note that the present discussion is restricted to the vertical direction of the type-I stereoisogram (Fig. 4).

Among the promolecules listed in the type-II part of Figure 7, the promolecules 17–22 (each surrounded by a frame box) are determined to be asymmetric(P) (=hypo-chiral or C1 -chiral). For example, the promolecule 17 (selected as a reference promolecule) generates a type-II stereoisogram shown in Figure 8. Each promolecule (e.g., 17 as a reference) belongs to the RS-stereoisomeric group Ce , to the point group C1 , to the RS-permutation group Ce , r r and to the ligand-reflection group C1 . This is represented by the aspect index [II, Ce ; C ; C r ; C1 ], which has the point group C1 r 1 e (underlined). Hence, the promolecule 17 (paired with 17) is determined to be asymmetric(P) (=hypo-chiral or C1 -chiral). Note again that the present discussion is restricted to the vertical direction of the type-II stereoisogram (Fig. 8). It should be noted that the two chiral proligands (p’s) are inequivalent to each other under the point-group symmetry (C1 ) of the promolecule 17, because each of the four proligands A, B, p, and p in 17 belongs to a one-membered orbit governed by a coset representation C1 ð=C1 Þ. The inequivalency between the two p’s can result in a selective reaction: If an appropriate reagent attacks 17, an intermediate due to the hypothetical attack on p at the position 2 is RS-diastereomeric to another intermediate due to the hypothetical attack on p at the position 4. As surrounded by a frame box, each of the promolecules listed in the type-III part of Figure 7 is determined to be asymmetric(P)

A 2

p p

C

1

3

B

4

17

(1)(2)(3)(4)

4

p

p

4

C

3

B

2

17 (= 17) (1)(2 4)(3)

A

A

C

r C†

1

p p

1

3

B

2

S

A

1

17

(1)(2 4)(3)

2

p

p

4

3

B

17 (= 17)

(1)(2)(3)(4)

C Figure 8. Stereoisogram of type II for determining asymmetry(P) (=hypo-chirality or C1 -chirality). Each promolecule (e.g., bf 17) is represented by the aspect index [II, Ce ; C ; C r ; C1 ]. The symbols A and B represent achiral proligands (atoms or achiral r 1 e ligands). The pair of symbols p/p represents a pair of chiral proligands, which are enantiomeric when detached.

A 2 sC

p p

1

3

q

4

25

(1)(2)(3)(4)

4 rC

p

p

p p

2

25

q

35

A

1

sC

3

2

(1)(2 4)(3)

A 4

S

A

1

1

3

q

(1)(2 4)(3)

2

rC

p

4

p

3

q

35

(1)(2)(3)(4)

C Figure 9. Stereoisogram of type III for determining asymmetry(P) (=hypo-chirality or C1 -chirality) as well as asymmetry(RSP). Each promolecule (e.g., bf 25) is represented by the aspect index [III, C1 ; C1 ; C1 ; C1 ]. A pair of R/S-stereodescriptors, ‘s’ and ’r’, is assigned to a pair of RS-diastereomers 25/35 (or 25=35), where the priority sequence A > p > p > q (or A > p > p > q) is presumed. The lowercase letters are used because of chirality unfaithfulness.

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(=hypo-chiral or C1 -chiral). For example, the promolecule 25, which generates the type-III stereoisogram shown in Figure 9, is characterized by an aspect index [III, C1 ; C1 ; C1 ; C1 ]. Hence, the promolecule 25 (paired with 25) is determined to be asymmetric(P) (=hypo-chiral or C1 -chiral). In a similar way, the promolecule 35 (paired with 35) is determined to be asymmetric(P) or hypo-chiral (or C1 -chiral). Note again that the present discussion is restricted to the vertical direction of the type-III stereoisogram (Fig. 9). Among the promolecules listed in the type-II part of Figure 7, the promolecules 8–14 are determined to be hyper-chiral (or G-chiral (G > C1 )). They are traditionally referred to as being ‘chiral, but not asymmetric’. For example, the promolecule 13 with the composition A2p2 has an aspect index [II, C2e ; C ; C r ; C2 ], so r 2 2e that it is determined to be hyper-chiral (or C2 -chiral). The underlined component C2 indicates that the promolecule 13 belongs to the point group C2 (¼ fI; C 2 g), where the twofold rotation axis (C2) bisects the angles A—C—A and p—C—p. 6. ‘Asymmetry’ under RS-permutation-group symmetry— asymmetry(RSP) In order to discuss ‘asymmetry’ under the RS-permutationgroup symmetry (named asymmetry(RSP)), we should focus our attention on type-I, type-III, and type-V stereoisograms, which are collected in the RS-stereogenic part (the right part) of Figure 7. 6.1. RS-Stereogenicity versus asymmetry(RSP) (or hypo-RSstereogenicity) Table 3 shows the presence of RS-stereogenicity under RS-permutation-group symmetry. Hence, the term ‘asymmetric’ can be defined as referring to an object belonging to the RS-permutation group C1 (¼ fIg), which exhibits one extreme case of RS-stereogenicity. In order to differentiate this extreme case from asymmetry(P) described above, such tentative terms as asymmetric(RSP) and asymmetry(RSP) are adopted for the sake of convenience, where the superscript (RSP) indicates that these terms are based on RS-permutation-group symmetry. As a result, asymmetry(RSP) is equivalent to the term C1 -RS-stereogenicity, which means RS-stereogenicity characterized by the RS-permutation group C1 as the one extreme case. Note that the identity group C1 can be a subgroup of RS-permutation groups as well as a subgroup of point groups. In a similar way to the coinage of the term hypo-chirality derived from chirality, the term hypo-RS-stereogenicity (¼ C1 -RSstereogenicity) is coined to connote asymmetry(RSP) by starting from RS-stereogenicity. The prefix hypo- means that there is an identity I as a single operation for fixing a given object to be examined under RS-permutation-group symmetry. In a similar way to the coinage of the term hyper-chirality, the term hyper-RS-stereogenicity is coined to connote ‘RS-stereogenicity, but not hypo-RS-stereogenicity’. The prefix hyper- means that there are two or more operations for fixing a given object to be examined under RS-permutation-group symmetry. As found in the RS-stereogenic part (the right part) of Figure 7, hyper-RS-stereogenic cases do not appear in case of tetrahedral promolecules, Such hyper-RS-stereogenic cases will be discussed with respect to allene derivatives in Part 2 of this series. 6.2. Asymmetry(RSP) (or hypo-RS-stereogenicity) for tetrahedral promolecules The data of aspect indices collected in Figure 7 are used to discuss asymmetry(RSP) for tetrahedral promolecules, just as they are used to discuss asymmetry(P). All of the RS-stereogenic

9

promolecules (type I, type III, and type V) listed in Figure 7 are determined to be asymmetric(PSP) (=hypo-RS-stereogenic or C1 -RS-stereogenic). As surrounded by a gray box, each aspect index contains C1 as the fourth component, which indicates the corresponding RS-permutation group. For example, the promolecules (1 and 7) listed in the type-I part of Figure 7 exhibit asymmetry(RSP) (=hypo-RS-stereogenicity or C1 -RS-stereogenicity), as each is surrounded by a gray box. For example, the promolecule 1 is a reference promolecule of the type-I stereoisogram of Figure 4, where its aspect index [I, Cb; C1 ; C1 ; Cb] has the RS-permutation group C1 (underlined). I I Hence, the promolecule 1 (paired with 3) is determined to be (RSP) asymmetric (=hypo-RS-stereogenic or C1 -RS-stereogenic). Note that the present discussion is restricted to the horizontal direction (for the pair of 1/3) of the type-I stereoisogram (Fig. 4), although 3 is coincident with 1. As surrounded by a gray box, each of the promolecules listed in the type-III part of Figure 7 is determined to be asymmetric(RSP) (=hypo-RS-stereogenic or C1 -RS-stereogenic). For example, the promolecule 25, which generates the type-III stereoisogram shown in Figure 9, is characterized by an aspect index [III, C1 ; C1 ; C1 ; C1 ]. Hence, the promolecule 25 (paired with 35) is determined to be asymmetric(RSP) (=hypo-RS-stereogenic or C1 -RS-stereogenic). In a similar way, the promolecule 25 (paired with 35) is determined to be asymmetric(RSP) (=hypo-RS-stereogenic or C1 -RS-stereogenic). Note again that the present discussion is restricted to the horizontal direction of the type-III stereoisogram (Fig. 9). The type-V promolecule 4 collected in Figure 7 generates the type-V stereoisogram shown in Figure 5. The promolecule 4 (paired with 6) is characterized by an aspect index [V, Cs ; Cs ; C1 ; C1 ], where the underlined symbol C1 indicates the identity group C1 under the RS-permutation-group symmetry. Hence, the promolecule 4 (paired with 6) is determined to be asymmetric(RSP) (=hypo-RS-stereogenic or C1 -RS-stereogenic), although it is frequently referred to as pseudoasymmetry because of its achirality. Note again that the present discussion is restricted to the horizontal direction of the type-V stereoisogram (Fig. 5). 7. More discussions on the conventional terminology The history of stereochemistry teaches us that the conventional stereochemistry has adopted the term ‘asymmetry(H)’ (extended from van’t Hoff’s ‘asymmetric carbon atom’1,2) at the early stage of the history and has later adopted the term ‘stereogenicity’.10 However, the conventional term ‘stereogenicity’ has overlooked the concepts of RS-stereogenicity and asymmetry(RSP) (=hypo-RSstereogenicity or C1 -RS-stereogenicity), which have introduced in the present article. This section is devoted to more detailed discussions on the implicit features of this history. 7.1. Asymmetry(H) and stereogenicity The fundamental idea of van’t Hoff’s ‘asymmetric carbon atom’1,2) is that a 2D structure (graph, e.g., 2) stands up into 3D structures as a pair of enantiomers (e.g., 1 and 3), although van’t Hoff’s ‘asymmetric carbon atom’ is originally restricted to tetrahedral carbons, as shown in Figure 1. His idea can be extended to any 2D structures (e.g., graphs for generating octahedral complexes) other than tetrahedral carbons. This course has seemingly been pursued in the history of stereochemistry, where the term ‘asymmetry(H)’ (tentatively coined here; the superscript (H) indicates a historical meaning based on van’t Hoff’s idea) is used to mean that a 2D structure (not restricted to tetrahedral carbons) stands up into 3D structures which are stereoisomeric (enantiomeric and diastereomeric).

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In order to avoid the confusion between asymmetry(H) and asymmetry(P), the term ‘asymmetry(H)’ has been replaced by the term ‘stereogenicity’, which has been introduced by Mislow and Siegel10 by extending McCasland’s definition.40 Thus, the term ‘stereogenic unit’ is defined as ‘a grouping within a molecular entity that may be considered as a focus of stereoisomerism’ in IUPAC Recommendations 199638. Although the term ‘stereogenicity’ is throughout concerned with 3D structures by introducing ‘permutational isomers’,36,10 such a grouping of the IUPAC definition may be a 2D or 3D object. If we presume a grouping as a 2D structure, the term ‘stereogenicity’ implicitly has the same meaning as the above mentioned term ‘asymmetry(H)’, so that a 2D structure (not restricted to tetrahedral carbons) stands up into 3D structures which are stereoisomeric (enantiomeric and diastereomeric). Although this replacement has settled the confusion between asymmetry(H) and asymmetry(P), the shortcomings of the term ‘asymmetry(H)’ remain unchanged. Thus, the term ‘stereogenicity’ turns out to cover Z/E-descriptors and R/S-stereodescriptors as well as to cover configuration indices (cf. Rule IR-9.3.3 of 41) and C/Adescriptors (cf. Rule IR-9.3.4 of 41). In other words, absolute configurations are not specifically discussed because ‘stereogenicity’ is incapable of focusing our attention specifically on R/S-stereodescriptors and C/A-stereodescriptors.

stereoisomerism  RS-stereoisomerism  fchirality; RS-stereogenicity; scleralityg;

ð7Þ

where chirality is related to enantiomerism due to an enantiomeric relationship, RS-stereogenicity is related to RS-diastereoisomerism due to an RS-diastereomeric relationship, and sclerality is related to holantimerism due to a holantimeric relationship. Note that the suffix -ism is used to refer to the status provided by a relationship, as found in the definition of the term isomerism, which is defined as the relationship between isomers.38 To solve such entangled situations comprehensively, the following group hierarchy has been taken into consideration:

isoskeletal groups  stereoisomeric groups  RS-stereoisomeric groups  fpoint groups; RS-permutation groups; ligand-eflection groupsg;

7.2. RS-Stereogenicity versus stereogenicity The term RS-stereogenicity covers R/S-stereodescriptors and does not cover Z/E-descriptors, as discussed in recent articles22–24 and a book.21 Hence, the term RS-stereogenicity is more definitive than the term ‘stereogenicity’. In fact, the term ‘stereogenic center’ (or stereocenter) have been introduced by Mislow and Siegel,10 who have described that ‘any mono- or polyatomic permutation center or skeleton may be referred to as stereogenic element or unit, or as a stereocenter, if ligand permutation produces stereoisomers’. The ligand permutation in the phrase ‘if ligand permutation produces stereoisomers’ corresponds to various permutations (e.g., permutations for cis/ trans-isomerization and for isomerization of octahedral complexes) in addition to the present RS-permutations. Hence, the following scheme is obtained: stereogenicity  RS-stereogenicity  asymmetryðRSPÞ ð¼ hypo-RS-stereogenicity or C1 -RS-stereogenicityÞ; ð5Þ

where the equality symbols are fulfilled if tetrahedral carbon centers are considered. It should be noted that the scheme represented by Eq. 5 does not take ligand reflections into consideration. Grouptheoretically speaking, Eq. 5 corresponds to the hierarchy of groups as follows:

permutation groups  RS-permutation groups  RS-permutation group ðC1 Þ;

tation groups with no ligand reflections in place of point groups with ligand reflections (cf. Fig. 6). In contrast, Fujita’s stereoisogram approach takes ligand reflections into explicit consideration after the concepts of RS-stereogenicity and RS-permutation groups are introduced. As a result, RS-stereogenicity (related to RS-permutation groups) is presumed to be involved in RS-stereoisomerism (related to RS-stereoisomeric groups and stereoisograms), while stereogenicity (related to permutation groups) is involved in stereoisomerism (related to stereoisomeric groups).42 Hence, the following scheme is obtained:

ð6Þ

where the equality symbols are fulfilled if tetrahedral carbon centers are considered. The conventional stereochemistry lacks the concept of RS-stereogenicity appearing in Eq. 5. As a result, RS-stereogenicity (as the second kind of handedness) is misleadingly equalized to chirality (as the first kind of handedness), so that Eq. 5 is misleadingly presumed to be stereogenicity  chirality (cf. 11,12). Moreover, the conventional stereochemistry lacks the concept of RS-permutation groups appearing in Eq. 6, so that RS-permutation groups are misleadingly equalized to point groups. This misleading equalization stems from the disregards of ligand reflections, where even enantiomeric relationships are misleadingly explained by using permu-

ð8Þ where isoskeletal groups are introduced to treat stereoskeletons other than tetrahedral skeletons.43,44 Thereby, combinatorial enumerations have been conducted to count inequivalent pairs of (self-) enantiomers under a point group, inequivalent quadruplets of RS-stereoisomers under an RS-stereoisomeric group, inequivalent sets of stereoisomers under a stereoisomeric group, and inequivalent sets of isoskeletomers under an isoskeletal group.42 7.3. Uniqueness of tetrahedral carbon centers Figure 7 shows that a tetrahedral skeleton has unique properties, where all of the type-I, type-III, and type-V stereoisograms exhibiting RS-stereogenicity are determined to be asymmetric(RSP), as shown by a gray box. In addition, RS-stereogenicity is coincident with stereogenicity. It follows that the equality symbols are effective in Eq. 5, so as to give the following scheme for a tetrahedral skeleton:

stereogenicity ¼ RS-stereogenicity ¼ asymmetryðRSPÞ ð ¼ hypo-RS-stereogenicity or C1 -RS-stereogenicityÞ: ð9Þ (H)

Moreover, the term asymmetry (corresponding to the extended idea stems from van’t Hoff’s ‘asymmetric carbon atom’1,2 and ‘pseudoasymmetric center’ as an exception) gives the same results as the term stereogenic center based on the conventional stereochemistry10,11 when applied to a tetrahedral skeleton. The term asymmetry(H) also gives the same results as the term RS-stereogenic center based on Fujita’s stereoisogram approach,19–21 so long as the term asymmetry(H) is based on Figure 1, when applied to a tetrahedral skeleton. In spite of the unique (not general) nature represented by Eq. 9, a tetrahedral skeleton has been regarded as a standard object for constructing the conventional terminology. As a result of starting from such uniqueness, the conventional terminology has suffered from difficulty of generalization so as to involve sources of confusion. One of the most typical causes for the difficulty of generalization is the lack of the concept of RS-stereoisomerism (Eq. 7), which

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integrates chirality (the first kind of handedness), RS-stereogenicity (the second kind of handedness), and sclerality by drawing a stereoisogram. 7.4. The term ‘chirality center’ misused in the CIP system The term ‘chirality center’ is frequently used to refer to one subcategory of the term ‘stereogenic center’ for the purpose of assigning R/S-stereodescriptors, as found in Table 1 of Ref. 12 and in Section 2.5 of Ref. 11. Although Table 1 of Ref. 12 contains the promolecule 1 (Fig. 4) as a typical example having a ‘chirality center’, such a ‘chirality center’ (and such a ‘stereogenic center’) is not a clue of the assignability of R/S-stereodescriptors. Instead, an RS-stereogenic center is a true clue of the assignability of R/S-stereodescriptors. This conclusion can be confirmed by comparing the type-I promolecule 1 (Fig. 4) with the type-II promolecule 17 (Fig. 8). The central atom of 1 is a chirality center in a geometric meaning, as found by the corresponding type index [€, ; a] for a type-I stereoisogram (Fig. 4), where the symbol € emphasizes chiral nature. The aspect index [I, Cb; C1 ; C1 ; Cb] indicates that the global chiI I rality of 1 is C1 , as underlined. The carbon center of 1 belongs to a one-membered C1 ð=C1 Þ-orbit under point-group symmetry, where the local chirality is confirmed by C1 contained in a pair of parentheses. The promolecule 1 is labeled S, where the assignability of S stems from RS-stereogenicity, which is indicated by type index [;€, a] or the aspect index [I, Cb; C1 ; C1 ; Cb]. The promolecule 1 is I I an example having a chirality center and exhibiting the assignability of R/S-stereodescriptors. The assignability due to RS-stereogenicity is diagrammatically expressed by the horizontal double-headed arrows in the corresponding stereoisogram (Fig. 4). On the other hand, the central atom of 17 is a chirality center in a geometric meaning, as found by the corresponding type index [€, a; ] for a type-II stereoisogram (Fig. 8). The aspect index [II, Ce ; C ; C r ; C1 ] indicates that the global chirality of 17 is C1 , as r 1 e underlined. The carbon center of 17 belongs to a one-membered C1 ð=C1 Þ-orbit under point-group symmetry, where the local chirality is confirmed by C1 contained in a pair of parentheses. The carbon center of the promolecule 17 cannot be labeled by the CIP system, where the unassignability stems from RS-astereogenicity, which is indicated by type index [; a; ] or the aspect index [II, Ce ; C ; C r ; C1 ] (underlined). The promolecule 17 is an example havr 1 e ing a chirality center but exhibiting no assignability of R/S-stereodescriptors. The unassignability due to RS-astereogenicity is diagrammatically expressed by the horizontal equality symbols in the corresponding stereoisogram (Fig. 8). The above comparison between 1 and 17 shows that the chirality of 17 is independent of the RS-astereogenicity (no assignability of R/S-stereodescriptors), whereas the chirality of 1 is independent of the RS-stereogenicity (the assignability of R/S-stereodescriptors). This means that the term ‘chirality center’ described in Table 1 of Ref. 12 and in Section 2.5 of Ref. 11 is misleadingly linked with the assignment of R/S-stereodescriptors, where only type-I cases (e.g., 1) are taken mainly and type-II cases (e.g., 17) are disregarded before starting the assignment. The insufficient specification due to the term ‘stereogenic center’ is not so clearly found, so long as tetrahedral promolecules are taken as probes. However, the term ‘stereogenic center’ or ‘stereogenicity’ covers more general cases (e.g., octahedral complexes or alkenes) in an undecided fashion. The term ‘stereogenic center’ or ‘stereogenicity’ should be replaced by the more distinctive term RS-stereogenic center or RS-stereogenicity even in the discussions on tetrahedral promolecules (along the horizontal direction of a stereoisogram). Note that the term RS-stereogenic center or RS-stereogenicity is paired with the term RS-astereogenic center or RS-astereogenicity, whereas the term ‘stereogenic center’

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or ‘stereogenicity’ does not exhibit such pairing. Thereby, a pair of RS-stereogenicity/RS-astereogenicity (related to the second kind of handedness) can be discussed in comparison with a pair of chirality/achirality (related to the first kind handedness). The conclusions of this subsection are that the term chirality center should be used to discuss geometric features (the vertical directions of a stereoisogram), and that the term RS-stereogenic center should be used to discuss the assignability of R/S-stereodescriptors (the horizontal directions of a stereoisogram). 8. Conclusions In order to avoid the ambiguity of the term ‘asymmetry’ in the conventional terminology, asymmetry(P) (=hypo-chirality or C1 chirality) under point-group symmetry and asymmetry(RSP) (=hypo-RS-stereogenicity or C1 -RS-stereogenicity) under RS-permutation-group symmetry are examined on the basis of the three kinds of symmetries derived from stereoisograms (Table 3). As a result, the following guidelines are obtained to comprehend stereochemistry and stereoisomerism: 1. The term asymmetry should be used to refer to asymmetry(P) (=hypo-chirality or C1 -chirality), which is determined under point-group symmetry (cf. Table 3). This usage is consistent with the definition of IUPAC Recommendations 1996.38 For the purpose of unambiguous discussions, however, it is desirable to use a set of terms having the common stem chirality, i.e., chirality, hypo-chirality (=C1 -chirality), and hyper-chirality (=G-chirality (G > C1 )). 2. The term asymmetry(H) (corresponding to van’t Hoff’s ‘asymmetric carbon atom’ and ‘pseudoasymmetric center’ as an exception) should be abandoned. Instead, the term RS-stereogenicity (corresponding to type-I, type-III, and type-V stereoisograms) should be used to support R/S-stereodescriptors under RS-permutation-group symmetry. Thus, ‘asymmetric carbon atom’ should be replaced by the term RS-stereogenic center (or more generally RS-stereogenic unit), which is more distinctive than the term ‘stereogenic center’. 3. The usage of hypo-RS-stereogenicity (=C1 -RS-stereogenicity) for connoting asymmetry(RSP) is not necessary under usual situations, because the detection of hypo-RS-stereogenicity becomes necessary only in such analytical discussions as the present article. 4. Because the term asymmetry is decided to be used to designate ‘asymmetry(P)’ (=hypo-chirality or C1 -chirality) for the purpose of avoiding ambiguity, traditional terms used loosely (and incorrectly) should obey the following guidelines: (a) The term ‘asymmetric synthesis’ is desirable to be replaced by the broader term ‘chiral synthesis’. The usage of the term ‘asymmetric synthesis’ is permitted if it represents ‘asymmetric(P)’ (=hypo-chiral or C1 -chiral). (b) The term ‘asymmetric induction’ is desirable to be replaced by the broader term ‘chiral induction’. The usage of the term ‘asymmetric induction’ is permitted if it represents ‘asymmetric(P)’ (=hypo-chiral or C1 -chiral).

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