Sphericity and Sphericity Indices. A Non-Mathematical Approach on the Mathematical Basis for Restructuring Stereochemistry
Two kinds of topicity terms have been used in stereochemistry: topicity terms ("enantiotopic", "diastereotopic", etc.) for specifying relationships between two ligands in a molecule [1, 2] and topicity terms ("chirotopic" and "achirotopic") for characterizing an attribute of each ligand (or each site) in a molecule . Although the combined usage of the two kinds of topicity terms is necessary to comprehend stereochemistry, it tends to cause undesirable misunderstanding or confusion.
To remedy this situation of stereochemistry, we have proposed sphericity terms (homospheric, enantiospheric, and hemispheric) as attributes of orbits, which are equivalence classes of ligands (or other objects) in a molecule. Although we have reported mathematical definitions of the sphericity terms[4 - 7] and non-mathematical definitions of them , the scope and limitations of the latter definitions should be discussed in detail in order to make the sphericity terms more familiar to organic chemists and biochemists.
In this paper, we will show that a proper combination of the two kinds of topicity terms requires troublesome efforts. Even if the combined usage is well-done, its scope is clarified to be restricted within qualitative applications so that it is incapable of solving such quantitative problems as combinatorial enumerations. Thereafter, we will show that the sphericity terms defined non-mathematically are able to remedy all of these problems. In addition, we will introduce sphericity indices on the basis of the non-mathematical definition of sphericities and discuss their potentiality of quantitative applications .
2 Conventional Stereochemical Terminology
2. 1 Enantiotopic Relationships
Mislow-Raban's definitions  of "enantiotopic" (Def. 1) and "diastereotopic" (Def. 2) are as follows: "If the environments of the groups are enantiomeric we shall refer to the groups as enantiotopic. If the environments of the groups are diastereomeric we shall refer to the groups as diastereotopic." . On the other hand, enantiotopic ligands and faces have been alternatively defined on page 1198 of Ref.  as follows (Def. 3): "Homomorphic ligands in constitutionally equivalent locations that are related by a symmetry plane (or center or alternating axis of symmetry) but not by a (simple) symmetry axis. Replacement of one or the other enantiotopic ligands by a new ligand produces enantiomers."
In spite of their apparent resemblance, Def. 1 and Def. 3 for enantiotopic relationship are different from each other. Strictly speaking, the definition phrase of Def. 3 is sometimes inconsistent with the subsequent sentence of Def. 3. For example, the two hydrogen atoms at positions 3 and 4 of a tetrahedral molecule CABH2 (1) are enantiotopic according to both Def. 1 and Def. 3, where A and B represent achiral ligands. On the other hand, a chiral ligand p and its enantiomeric (enantiomorphic) counterpart p at positions 1 and 2 of a tetrahedral molecule CppH2 (2) are enantiotopic according to Def. 1 but not enantiotopic according to Def. 3, since p and p are enantiomorphic (enantiomeric) but not homomorphic (homomeric) . This can be corrected simply by rewriting Def. 3 as being "Homomorphic or enantiomorphic ligands in constitutionally equivalent locations ..." .
However, this type of misunderstanding is frequently inevitable, since it comes from the first impression that the two hydrogens of 2 seem to be in a relationship similar to the two hydrogens of 1 by a simple glimpse. This means that the comparison of the two hydrogens of 1 with those of 2 tends to be taken as a main interest.
Figure 1. Topicity vs. Sphericity (1).
2. 2 Diastereotopic Relationships
In addition of Def. 2, diastereotopic ligands have been defined on page 1197 of Ref.  as follows (Def. 4): "Homomorphic ligands in consititutionally equivalent locations that are not symmetry related (i.e., not interchanged by a Cn or Sn operation). Replacement of one or other of a set of diastereotopic ligands by a new ligand produces diastereomers." Def. 2 and Def. 4 give the same results.
According to Def. 2 and Def. 4, for example, the two hydrogen atoms at positions 3 and 4 of CppH2 (2) are diastereotopic, insomuch as the two hydrogen atoms at positions 3 and 4 of CApH2 (3a) are diastereotopic. It should be noted, however, that the molecule (2) is achiral, while the molecule (3a) is chiral.
The hydrogen atom at position 3 of 2 is chemically non-equivalent to the hydrogen atom at position 4 so that they can be discriminated by an appropriate achiral or chiral reagent. On the other hand, if the corresponding enantiomer 3b is present together with 3a, the behaviors of the two hydrogen atoms at positions 3 and 4 of CApH2 (3a) should be examined distinctly under the action of achiral reagents and under the action of chiral ones. These behaviors can be explained by virtue of the topicity terms of stereochemical attributes. Thus, each of the two hydrogens of 2 is achirotopic, while each of the two hydrogens of 3a is chirotopic.
From the viewpoint of the topicity terms of stereochemical attributes, each of the two hydrogens of the achiral molecule 1 is chirotopic, where the two hydrogens are in an enantiotopic relationship. Each of the two hydrogens of the chiral molecule 3a is also chirotopic, where the two hydrogens are in a diastereotopic relationship. If the pair of enantiomers 3a and 3b is taken into consideration, the hydrogen atom at position 3 (or 4) of 3a is enantiotopic to the hydrogen atom at position 3 (or 4) of 3b.
2. 3 Homotopic Relationships
Later, homotopic ligands and faces have been defined on page 1200 of Ref.  as follows (Def. 5): "Homomorphic ligands in constitutionally and configurationally equivalent positions. Such ligands are interchanged by a Cn operation. Separate replacement of one or the other of such ligands by a new ligand (see heterotopic ligands) gives identical products. ...". According to Def. 5, the two achiral ligands A's (or the hydrogens) of an achiral molecule 4 are homotopic. The two chiral ligands p's (or p's) of another achiral molecule 5 are also homotopic. However, the two achiral ligands A's (or the hydrogens) of 4 coincide with each other also by an improper rotation (reflection), whereas the two chiral ligands p's (or p's) of 5 do not coincide with each other by such improper rotations (reflections). These results come from the fact that each ligand or hydrogen of 4 is achirotopic, while each chiral ligand of 5 is chirotopic. These two cases indicate that there exist two kinds of homotopic relationships .
It should be added that a pair of p(1) and p(3) is enantiotopic in 5 and another pair of p(1) and p(4) is also enantiotopic. Thus, there are totally four enantiotopic relationships in 5. They cannot be properly treated by the conventional terminology which is focused on the treatment of a single pair.
Figure 2. Topicity vs. Sphericity (2).
2. 4 Stereoheterotopic Relationships
The term "stereoheterotopic" coined by Eliel  is defined as follows (Def. 6): "Homomorphic ligands whose separate replacement gives rise to stereoisomers; also faces of double bond, separate addition to which gives rise to stereoisomers. If the products are enantiomeric, the ligands or faces are enantiotopic; if the products are diastereomeric, the ligands or faces are diastereotopic.", as described on page 1208 of Ref. . This coinage has been intended to give a common basis to such relationships between the two hydrogens in 1 (enantiotopic), 2 (diastereotopic), and 3a (diastereotopic), which are discriminated by the pro-R/pro-S system .
It is worthy to compare the three molecules listed in Figure 3 in order to show that there exist stereoheterotopic (strictly speaking, enantiotopic) relationships that cannot be discriminated by the pro-R/pro-S system. The two hydrogens of 2 are discriminated by the pro-R/pro-S system because of their diastereotopic relationship, whereas p and p of 2 are not discriminated in spite of their enantiotopic relationship (Def. 1). On the other hand, p and p of 6 (or 7) are discriminated by the pro-R/pro-S system because of their enantiotopic relationship (Def. 1).
Figure 3. Topicity vs. Sphericity (3).
2. 5 Limitations of Conventional Stereochemical Terminology
As clarified in the preceding discussions, the combined usage of the two kinds of topicity terms requires careful and tedious efforts to settle entangled situations that have been caused by conventional semantic transmutations .
A more critical drawback of the conventional terminology is that the combined usage is based on pairwise relationships. For example, the four enantiotopic relationships in 5 are treated separately so that the four ligands (two p's and two p's) of 5 are not explicitly recognized as being equivalent to each other in terms of the combined usage. This fact results in the incapability of quantitative applications, as described in the following sections.
3 New Stereochemical Terminology
This section is devoted to the introduction of our approach to settle such entangled situations by means of a novel terminology.
3. 1 Sphericities of Orbits
As such a new terminology, we have developed the concept of sphericity (homospheric, enantiospheric, and hemispheric), where we have emphasized the importance of orbits (equivalence classes) [4, 5]. The shift of viewpoints from equivalence relationships to equivalence classes (orbits) has provided us with a deep insight to stereochemistry. Our previous treatment, however, was heavily based on the point-group and the permutation-group theories (e.g., coset representations). This means that it had a drawback to perform qualitative applications, though it enabled us to accomplish quantitative applications such as combinatorial enumeration. For such qualitative applications, we have redefined the concept of sphericity by starting from more succinct bases, as shown in Table 1 .
Table 1. Sphericity of an orbit 
Each orbit has several properties according to its sphericity, as summarized in Table 2. Among the properties listed in Table 2, several properties are concerned with a set of conventional terms for stereochemical relationships (e.g., "homotopic", "enantiotopic", etc.) as well as for stereochemical attributes (e.g., achirotopic and chirotopic).
It should be noted here that the sphericity determined by means of the criteria (Table 1) indicates the nature of an orbit in a molecule. This means that the sphericity is an attribute of the orbit, whereas the topicity term of the first kind is concerned with relationships between two sites in a molecule. By emphasizing the importance of orbits, enantiotopic and homotopic relationships can be redefined as internal relationships in an orbit (called intra-orbit relationships here). They can be replaced or easily derived by the sphericity terms listed in Table 1. On the other hand, a diastereotopic relationship can be regarded as a relationship between two distinct orbits (called inter-orbit relationship here) so that it cannot been characterized as an equivalence class (orbit) by the concept of orbits based on point groups .
The criteria listed in Table 1 and the properties listed in Table 2 provide us with the following procedure for determining the sphericity of an orbit:
Find orbits: Select equivalent ligands (or other objects) exhaustively in a molecule by applying symmetry elements (rotations or rotoreflections) to the molecule. The selected ligands are considered to construct an orbit as an equivalence class.
Assign sphericity: Each orbit is examined by virtue of the criteria listed in Table 1 so ass to be categorized into a homospheric, enantiospheric, or hemispheric orbit.
Check properties: According to the sphericity, each orbit has properties summarized in Table 2 so that the conventional topicity terms of two kinds can be derived from the sphericity. The prochirality of a molecule is ascribed to the presence of at least one enantiospheric orbit.
3. 2 Examples
Let us now exemplify the procedure described above by using the same molecules as have been discussed on the basis of the conventional terminology in the preceding section. First, the two hydrogens of 1 are equivalent so that they construct a two-membered orbit (D1 listed in Table 3). Since each hydrogen behaves according to the second criterion for characterizing an enantiospheric orbit (Table 1), the orbit is concluded to be enantiospheric. Such topicity terms as "enantiotopic" and "chirotopic" can be easily derived from the enantiosphericity of the orbit D1 by virtue of Table 2. The achiral ligand A of 1 constructs a one-membered orbit (D2 listed in Table 3), which is determined to be homospheric according to the first criterion of Table 1. Similarly, the achiral ligand B of 1 constructs another one-membered orbit (D3 listed in Table 3). Since D1 is an enantiospheric orbit, Table 2 teaches us that 1 is prochiral.
Table 2. Properties of an Orbit
On the other hand, each of the two hydrogens contained in 2 constructs a one-membered homospheric orbit (D2 or D3 listed in Table 3). The two hydrogens are regarded as being diastereotopic, because they belong to distinct orbits (i.e., an inter-orbit relationship between D2 an D3). The chiral ligands p and p construct a two-membered orbit (D1), which is determined to be enantiospheric according to the second criterion of Table 1. It should be noted that the p and the p are equivalent under a reflection operation. Such topicity terms as "enantiotopic" and "chirotopic" can be again derived from the enantiosphericity of the orbit D1 by virtue of Table 2. It should be noted that the D1 of 2 corresponds to the D1 of 1 in spite of their different appearance. Since the D1 of 2 is an enantiospheric orbit, Table 2 teaches us that 2 is prochiral.
The four ligands of 3a (or 3b) construct distinct one-membered hemispheric orbits (D1 to D4), as listed in Table 3. Among them, the hydrogen of D3 is concluded to be diastereotopic to the hydrogen of D4 because of their inter-orbit relationship.
As found in Table 3, the two achiral ligands A's and the two hydrogens contained in 4 respectively construct two-membered orbits (D1 and D2), both of which are homospheric according to the first criterion of Table 1. The "homotopic" relationship between the A's (or the hydrogens) and their "achirotopic" nature can be easily derived from the homotopicity of the respective orbit (D1 or D2), as collected in Table 2.
As found in Table 3, the two p's and the two p's contained in 5 construct a four-membered orbit (D1), which is determined to be enantiospheric according to the second criterion of Table 1. The corresponding conventional terminology can be easily derived by virtue of the enantiosphericity of the orbit D1 (Table 2). Thus, one half (two p's) and the other half (two p's)are "enantiotopic" to each other; one p and the other p in the half (or one p and the other p in the other half) are "homotopic" to each other. Each ligand is "chirotopic" in nature. The molecule 5 is prochiral because of the presence of the enantiospheric orbit D1.
Table 3. Orbits and Their Sphericities
Since the two hydrogens of 2 are chemically non-equivalent because they belong to distinct one-membered homospheric orbits (D2 and D3 listed in Table 3. Hence, there exist two modes of replacement by achiral ligands A and B so as to produce 6 and 7, which are diastereomeric to each other. A pair of chiral ligands (p and p) in 6 (or 7) constructs a two-membered enantiospheric orbit (D1), which is symmetrically akin to the orbit D1 of 2.
The discussions in the preceding paragraphs have clarified merits of using sphericity terms over the combined usage of the topicity terms of two kinds . These merits are quite obvious by comparing the complicated discussions based on the conventional terminology (in the preceding section) with the corresponding simplified discussions based on the present terminology (in the present section). Thus, the discussions based on the present sphericities are fruitful, since general properties can be listed in such a tabular form as Table 2. Moreover, the assignment of sphericities according to Table 1 requires no more efforts than the assignment of topicities.
4 Sphericity Indices
4. 1 Sphericity Indices and USCI-CFs
To give a more systematic format to the present treatment of sphericity, a dummy variable is ascribed to each orbit according to its sphericity: ad for a d-membered homospheric orbit, cd for a d-membered enantiospheric orbit, and bd for a d-membered hemispheric orbit. For the sake of simplicity, we call such dummy variables sphericity indices. Since the concept of sphericity indices have been originally proposed on the basis on the group-theoretical definition of sphericity [20, 4], such a definition as started from the present intuitive definition of sphericity (Table 1) is highly desirable to pursue their qualitative applications.
The orbits of each molecule collected in Table 3 can be characterized by respective sphericity indices, which are also listed in Table 3. Thereby, each molecule is represented by a product of sphericity indices, which is called unit subduced cycle index with chirality fittingness (USCI-CF), as collected also in Table 3.
For example, the enantiospheric orbit D1 of 1 is characterized by a sphericity index c2. Each of the homospheric orbits (D2 and D3) of 1 is characterized by a sphericity index a1. As a result, the USCI-CF of 1 is obtained to be a12c2. The USCI-CFs of the other molecules are also collected in Table 3. Obviously, the capability of considering such sphericity indices clarifies the superiority of the sphericity concept over the topicity concept.
4. 2 Derivatives and USCI-CFs
Figure 4 depicts derivatives of methane, where the symbols A and B represent achiral ligands and a pair of p and p represents an enantiomeric ligand pair. Let us consider the derivation of 1 (CABH2, e.g., chlorofluoromethane) from methane. The derivation can be regarded as desymmetrization from Td of methane to Cs of 1, where the four hydrogens of methane (belonging to a four-membered homospheric orbit in methane) are divided and changed into two hydrogens (belonging to a two-membered enantiospheric orbit D1), one A (belonging to a one-membered homospheric orbit D2), and one B (belonging to a one-membered homospheric orbit D3) in 1.
By using the sphericity indices defined above, the orbit of four hydrogens in methane is characterized by a sphericity index a4. On the other hand, the resulting set of orbits in 1 of CABH2 is characterized by a USCI-CF, i.e. a12c2. It follows that the desymmetrization from mathane (Td) to 1 of CABH2 (Cs) is characterized by the expression, a4 ® a12c2, according to the change of USCI-CFs.
Figure 4 does not contain a derivative of Dd . This is confirmed by the fact that the D2d -group is the same USCI-CF as the supergroup Td (a4). Note that Figure 4 represents a supergroup-subgroup lattice for Td . In a similar way, a D2-derivative is not involved because the USCI-CF of D2 is the same as that of the supergroup T (b4).
Figure 4. Sphericity indices for derivatives of methane
The possible existence of derivatives of the remaining subgroups is concluded by comparing their USCI-CFs with those of respective supergroups. For example, the USCI-CF of 1 of CABH2 is a12c2, which is different from those of supergroups (a1a3 for C3v and and a22 or C2v), as found in Figure 4. It follows that 1 (having a12c2) can exist as a molecule.
The mode of accommodation of ligands in an orbit is determined by the chirality fittingness [5, 4], where an enantiospheric orbit is concluded to accommodate chiral ligands and the same number of their enantiomeric ligands in the mode of compensated chiral packing. Thereby, the c2-part of the USCI-CF a12c2 for the Cs-symmetry can accommodate a pair of p and p. It follows that there exist Cs-derivatives of other types: CH2pp (one achiral derivative, i.e., 2) and CABpp (two diastereomers, i.e., 6 and 7.
The sphericity concept, the sphericity indices, and USCI-CFs that are defined non-mathematically in this paper are equivalent to those defined previously by coset representations of point groups . Hence, they are applicable to quantitative problems such as combinatorial enumerations that have ever been solved mathematically . They are also applied to logical formulations of stereochemical nomenclatures .
The combined usage of topicity terms for stereochemical relationships (homotopic, enantiotopic, diastereotopic, etc.) and those for stereochemical attributes (chirotopic and achiotopic) has been discussed so as to show its scope and limitations in examining complicated problems of stereochemical phenomena. Sphericity terms (homospheric, enantiospheric, and hemispheric), which are based on orbits of ligands (or other objects), have been clarified to provide us with a simpler terminology for discussing the same complicated problems. Sphericity indices have been discussed to show the existence of derivatives.
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 The definition of the term "enantiotopic" is based on an equivalence relationship on the action of point groups, while that of the term "diastereotopic" is not simply based on such a point-group equivalence. This is because that Defs. 1 and 2 depend upon the definition of "enantiomeric" and "diastereomeric", where diastereomers are defined as stereoisomers that are not enantiomers [11, 23]. This definition is akin to the dichotomy of domestic people and foreign people, where the variety of foreigners are neglected. In other words, the two terms "enantiotopic" and "diastereotopic" (also "enantiomeric" and "diastereomeric") belong to distinct categories from the point-group theoretical point of view.
 E. Eliel and S. H. Wilen, Stereochemistry of Organic Compounds, John Wiley & Sons, New York (1994).
 According to the terminology of stereochemistry, the term "enantiomorphic" is used to designate the opposite handedness of two ligands in isolation. However, the handedness of two ligands in isolation is conceptually the same as that of two molecules so that the present paper preferably uses the term "enantiomeric" even for designating two ligands of opposite chiralities in isolation.
 Thus, IUPAC  has defined "enantiotopic" as "Constitutionally identical atoms or groups in molecules which are related by symmetry elements of the second kind only (mirror plane, inversion center or rotation-reflection axis). For example the two groups c in a grouping Cabcc are enantiotopic. ...."
 To discriminated these cases, the terms "holotopic" and "hemitopic" have been proposed by us .
 K. R. Hanson, J. Am. Chem. Soc., 88, 2731–2742 (1966).
 An interesting discussion has appeared with respect to semantic transmutations of the term "diastereomeric" etc. .
 A comment on the term "stereoheterotopic" should be added here. The term "stereoheterotopic" contains both the term "enantiotopic" as an intra-orbit relationship and the term "diastereotopic" as an inter-orbit relationship so that the term "homotopic" is implicitly presumed as the counterpart. This terminology is parallel with the definition of stereoisomers; "diastereomers are stereoisomers other than enantiomers", where the term "homomers" is implicitly excluded as the antonym of the term "stereoisomers". In contrast, the present appproach is based on the idea that given molecules (i.e., stereoisomers of the present extended meaning) are classified into a set (orbit) of homomers under the operations of point-group elements. Then, each relationship between such sets is determined to be diastereomeric, if they are concerned with constitutional equivalency. For detailed discussions, see Ref. .
 S. Fujita, Bull. Chem. Soc. Jpn., 73, 1979–1986 (2000).
 Let us consider a case in which we decide whether two or more persons belong to a family or not. Any pair of persons is compared by virtue of relation (e.g., father, mother, brother, etc.). If they are relatives, they are classified to belong to the family (the family name A). If they are not relatives, another set of two persons is compared, and so on. Once such classification is accomplished, the family A can be described as a set having common characteristics (behaviors, properties, etc.). Thus, a classification of objects into classes (such as "families" described above) is generally conducted on the basis of a set of criteria (e.g., relationships). One of the most critical problems in the conventional stereochemical terminology (e.g., the combined usage of the topicity terms of two kinds) is that it has no explicit targets of classification (i.e., classes). In other words, it is concerned only with such criteria (or relationships) but not with such classes. Since its concern has been focused on a pairwise relationship between only two objects (ligands, etc.), its lack of the concept of classes has not appeared as an explicit drawback, so as to be long overlooked.
 S. Fujita, J. Math. Chem., 5, 121–156 (1990).
 In the present discussion, achiral ligands such as A and B are considered to be structures. Pentaerythritol (C(CH2OH)4) belongs to D2d at its highest attainable symmetry, because each of the hydroxylmethyl ligands (CH2OH) has structure. This is explained by the mismatch of local symmetry in deriving the molecule from the promolecule of D2d-symmetry . Stereochemical phenomena such as is found in pentaerythritol have been discussed extensively [26, 6, 7].
 S. Fujita, Bull. Chem. Soc. Jpn., 63, 315–327 (1990).
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