[1970] [1971] [1972] [1973] [1974]
[1975] [1976] [1977] [1978] [1979]
[1980] [1981] [1982] [1983] [1984]
[1985] [1986] [1987] [1988] [1989]
[1990] [1991] [1992] [1993] [1994]
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1967 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The reaction of carbethoxynitrene generated by the photolysis of ethyl azidoformate has been investigated by using cyclic ethers and acetals as substrates. A main reaction is the insertion of the nitrene at a C-H bond ajacent to the ether oxygen of each substrate.
1968 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The reaction of carbethoxynitrene generated by the thermolysis of ethyl azidoformate has been investigated by using olefins as substrates. A main reaction is the stereoselective addition of the nitrene to the double bond of each olefin.
[Abstract] The photolysis of 1,2,3-tripheneylaziridene in lower alcohols takes two competitive routes: (1) benzealdehyde acetal and N-benzylaniline and (2) benzyl alkyl ethers and benzalaniline. In cyclohexane, 1,2,3-tripheneylaziridene carries out an addition reaction to give 1,2,3-triphenyloctahydroisoindole.
1969 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The thermolysis of ethyl cis- and trans--2-methyl-3-phenylaziridine-1-carboxylate produces alpha-carbethoxyaminopropiophenone and alpha-carbethoxyamino-alpha-phenylacetone in 97:3 and 40:60 ratio, respectively. The oxidative ring-openig reaction are applied to various azidine-1-carboxylates.
[Abstract] The photolysis of 1-phenylcyclohexene in acetic acid yields phenylcyclohexane，1-methyl-1-phenylcyclohexane, and 1-acetoxy-1-phenylcyclohexane. The mechanism of the generation of these compounds has been discussed.
[Abstract] The first syntheses of [7](2,6)pyridinophane and [7](2,6)pyrylophanium perchlorate have been reported.
1970 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The reactions of 9-ethoxycarbonyl-9-aza-bicyclo[6.1.0]nonane under acidic conditions have been shown to give transannular products along with normal ring-opening products.
[Abstract] The heating of aziridine-1-carboxylates in sulphoxides resulted in the oxidative opening of the aziridine rings. This is a general method for producing alpha-carbalkoxyamino ketones, since the aziridine-1-carboxylates are available via the addition of iodine isocyanate to olefines or of carboalkoxy nitrenes.
[Abstract] [9](3,5)Pyrazolophane and 11-methyl[9](2,4)furanophane were synthesized by starting from 2-cyclododecenone.
[Abstract] The oxidation of cyclododecane-1,5,9-triol has been shown to give an intramolecular hemiacetal of the corresponding dione.
1971 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The hydration of 3-ethynyl-2-cyclononenol gave 3-acetyl-2-cyclononenol, which was in turn a key compoound to synthesize 8-methyl-[6](2,4)thiophenophane and N-aryl-8-methyl-[6](2,4)pyrrolophanes. The strain of the heterocyclic rings was discussed by means of their electronic spectral data.
[Abstract] The photolysis of 1-phenylbicyclo[n.1.0]alkanes in acidic media has been investigated. 1-Phenylbicyclo[3.1.0]cyclohexane is photolyzed in acetic acid to give 1-acetoxy-1-phenylcyclohexane, 3-acetoxy-1-phenylcyclohexane, 1-methyl-1-phenylcyclohexane, and others.
[Abstract] The photo-irradiation of six- to eight-membered cycloalkenes conjugated with a phenyl ring have been photo-irradiated in acetic acid and propionic acid to give addition products (esters), 1-phenylcycloalkanes and 1-phenyl-1-alkylcycloaklanes. The mechanism of yielding these products has been discussed.
[Abstract] The absolute configuration of 2-phenylaziridine has been determined. The addition of iodine isocyante to styrene and the subsequent reaction with (-)-menthol gave a mixture of two diastereomeric esters, which were separated by recrystalization. On one hand, each of the diastereomeric ester was converted into the corresponding antipode of 2-phenylaziridine. On the other hand, the iodine of each diasteroemeric ester was reduced to give an antipode of phenenylamine with a known absolute configuration. Thereby, the absolute configuration of 2-phenylaziridine was concluded to be (R)-(-) and (S)-(+).
[Abstract] [7](2,6)Pyridinophane and its derivatives have been prepared as the first examples of heterocyclic nucleus with a heptamethylene linking 2- and 6-positions. The flipping of the heptamethylene of each compound has been studied by means of temperature-variable NMR spectroscopy. Two protons of each heptamethylene have been observed to exhibit high field shift. They are determined to be the middle methylene protons by synthesizing a derivative substituted by a deuterium.
[Abstract] The photoreaction of a 1,3-dioxolen-2-one (a vinylene carbonate) has been investigated, where a bicyclo[5.1.0]decane has been obtained as a transannular product.
1972 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The photochemical reaction of ethyl azidoformate with ketones has given an alpha-insertion product, while the thermolytic reaction has yielded a beta-insertion product.
[Abstract] The 1-methylene derivative of [7](2,6)pyridinophane has been synthesized and its conformation has been investigated by NMR spectroscopy.
[Abstract] Photolysis of benzhydryl esters in various solvents has been studied, where 1,1-diphenylalkanes, diphenylmethane, and 1,1,2,2-tetraphenylethane have been isolated as decarboxylative alkylation products. The ratios of the products have been shown to depend on solvent polarities.
[Abstract] 4,5-Disubstituted derivatives of 1,2-Dioxolen-2-one have been synthesized and its photochemical reactions have been investigated. Photoreduction of double bonds and transannular hydrogen abstraction have been observed as main reactions.
[Abstract] The photochemical and thermal reaction of ethyl azidoformate with cyclohexanone ethylene acetal have been found to give nitrene-insertion products to the alpha-CH of the acetal ring as well as to the CH's of the six-membered ring.
[Abstract] [7]Metacyclophane and its 13-bromo derivative, each of which has a heptamethylene linkage between the meta-positions of a benzene nucleus, have been synthesized as metacyclophanes with the shortest chain-length. The middle protons of each heptamethylene have been observed to exhibit high-field shift by means of NMR spectroscopy.
[Abstract] [7](3,5)Pyrazolophanes with a heptamethylene linkage between the 3,5-positions of a pyrazole nucleus have been synthesized as heterophanes with the shortest chain-length. The middle protons of each heptamethylene have been observed to exhibit high-field shift by means of NMR spectroscopy.
[Abstract] [6]Metacyclophane has been synthesized as a metacyclophane with the shortest chain-length. The conformation of the chain is discussed by means of its NMR spectrum.
1973 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] 2,2-Dimethyl-[7](2,6)pyridinophane has been synthesized. The flipping of its heptamethylene chain has been discussed on the basis of the NMR signals of the two methyl groups and as well as of middle methylene protons.
[Abstract] The reaction of N-alkoxycarbonylaziridines with acetonitrile or benzonitrile under acidic conditions has been found to give 1-alkoxylcarbonyl-2-imidazoline derivatives.
1974 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Various [n](2,4)Heterophanes along with [7](3,5)pyrazolophane have been synthesized and thier conformations have been discussed by virtue of high-field NMR signals.
1975 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The conformations of various metacyclophanes have been discussed by means of high-field NMR signals. The effect of chain lengths on benzen-ring strain has been discussed.
[Abstract] N,N-Diethylhydroxylamine has shown to be a versatile and selective reagent for reducing quinoes to quinols.
1979 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] N,N-Diethylhydroxylamine has shown to be a versatile and selective reagent for reducing quinoes into quinols as well as for reducing quinone monosulfonamides into ortho-sulfonamidophenols. The reagent has been applied to the synthesis of dye developers for instant color photography.
1981 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The reaction of 4-alkoxy-o-quinone monosulfonimides with alcoholic alkali has been found to give p-quinone monoacetals. This reaction has been shown to be a model reaction for investigating a side reaction observed in instant color films.
1982 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The Beckmann rearrangement by means of phosphoryl chloride/N,N-dimethylacetamide has been reported as a novel and convenient method of preparing benzoxazoles, which are important intermediates for synthesizing dye releasers used in instant color photography.
[Abstract] Phosphoryl chloride/N,N-dimethylacetamide has been found to be a convenient reagent for synthesizing arenesulfonyl chlorides, which are important intermediates for synthesizing dye releasers used in instant color photography.
[Abstract] Phosphoryl chloride/N,N-dimethylacetamide has been found to be a convenient reagent for synthesizing sulfonyl chlorides with a dye moiety. This reagent has been shown to be more chemoselective than phosphoryl chloride-N,N-dimethylformamide.
1983 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The isolation of o-benzoquinone monosulfonimides has been reported. Their reactions under various conditions have been investigated in order to clarify a side reaction observed in instant color films.
[Abstract] Reactions of magenta dyes used in instant color have been investigated under dark conditions in order to show the effect of molecular structures on dark stability. The main reaction under dark conditions has been established to be a Michael addition of azo dyes to vinyl compounds.
1986 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The concept of imaginary transition structures has been proposed as a computer-oriented representation of organic reactions, where an organic reaction is regarded as a structure with par-bonds, in-bonds and out-bonds. This concept provides us with a set of new concepts for characterizing organic reactions, e.g. projection to a starting stage, projection to a product stage, reaction strings, bridges of ring opening, bridges of ring closure and bridges of rearrangement.
[Abstract] Organic reactions have been classified by means of stringity (the number of reaction strings) and reaction-center graphs. The latter concept has been extended into the concept of reactions graphs. Then, even-membered reaction graphs with a single string (one-string reaction graphs) have been enumerated by using Polya's theorem.
[Abstract] Odd-membered reaction graphs with a single string (one-string reaction graphs) have been enumerated by using Polya's theorem.
[Abstract] Thee-nodal and four-nordal subgraphs extracted from imaginary transition structures have been shown to be useful descriptors for characterizing substitution reactions, C-C bond formations, additions, eliminations, oxidations and reductions.
[Abstract] Various modes of recombination of reaction strings have been discussed to treat multi-step reactions or reaction schemes. The codification of each mode by complex bond numbers has enabled us to manipulate such multi-step reactions.
1987 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Two-string reactions with a spiro-type reaction graph have been discussed and enumerated by Polya's theorem.
[Abstract] Two-string reactions with a fused-type reaction graph have been discussed and enumerated by Polya's theorem.
[Abstract] To treat formal charges for characterizing nitro, sulfo and related groups, charge spaces have been proposed as a new concept. Connection tables with three-dimensional data have been used to describe stereoselective reactions.
[Abstract] Bridges of rearrangement have been formulated as rings appearing in imaginary transition structures. They have been shown to be versatile descriptors for characterizing rearrangement reactions.
[Abstract] The description of organic reactions is discussed from the viewpoint of a structure-reaction type (SRT) paradigm, in which structural information and reaction types are stored and manipulated more or less independently. This paradigm must be overcome in order to construct an integrated system that will support both retrieval of organic reactions and synthetic design. The concept of imaginary transition structures (ITS) is introduced as a unitary representation free from the SRT paradigm.
1988 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The essential set of essential rings (ESER) has been proposed to characterize ring-opening reactions, ring-closure reactions, rearrangements, etc. Ring structures appearing in imaginary transition structures (ITSs) are classified into five categories: bridges of ring-opening (BO), bridges of ring closure (BC), bridges of rearrangement (BR), invariant rings (IR) and trivial rings. From all of the ring detected in a given ITS, the ESER is selected ato afford necessary and sufficient information on changes of the ring structures. The ESER contains rings other than tied, multi-tied or dependent rings. The tied rings are defined as rings which have one transannular type A (TATA) bond attached by at least one par-subring. The multi-tied rings contains two or more TATA bonds, at least one of which has a par-subring. Dependent rings are covered rings which are covered by tied rings under covering conditions introduced. An algorithm for selecting ESER from ring structures in imaginary transition structures has been developed. This algorithm is applicable also to the detection of ESER of an organic compound, which is more effective than conventional methods.
[Abstract] A new systematic classification of organic reactions is presented in terms of imaginary transition structures (ITS) and their hierarchical n-nodal subgraphs. The ITS corresponding to an individual organic reaction is a structural formula which has par-bonds (invariant bonds during the reaction), out-bonds (bonds appearing in the starting stage), and in-bonds (bonds appearing only in the product stage). The n-nodal subgraph indicates the corresponding reaction type and is divided into two parts: reaction kernels (RK) and terminal descriptors (TD). The RK is a set of adjacent carbon reaction centers to which out- and in-bonds are incident. The RK affords information on the changes of carbon skeletons, e.g., substitution, construction (C-C bond formation), cleavage, addition, and elimination. The TD is a set of terminal non-carbon atoms and imaginary bonds incident to the terminals. Odd- and even-nodal subgraphs have common TDs, respectively, which afford information on the changes of oxidation stages, e.g. oxidative, reductive, and isohypsic. This fact stems from the alternant character of reaction strings.
[Abstract] The concept of tied rings, multi-tied ring, and dependent rings is introduced, wherein transannular bonds and heterogeneity and abnormality of a ring are key classifiers. The essential set of essential rings (ESER) is defined as a set of rings other than tied, multi-tied, and dependent rings. An algorithm for detection of the ESER and its scope and limitations are discussed.
[Abstract] The canonical numbering and coding of an imaginary transition structure (ITS) are described. The nodes of an ITS are partitioned partially into (pseudo)equivalent classes in light of four kinds of extended connectivities. Each of the nodes of the highest class is selected as a root, to which possible spanning trees are constructed. The nominated sets of canonical numbering are obtained from the respective spanning trees. Then the canonical code is obtained by comparing newly defined lists based on the sets of numbering. The concept of a reduced ITS is proposed. The canonical numbering and coding of the reduced ITS are also discussed.
[Abstract] A reaction-center (RC) graph is the subgraph of an imaginary transition structure. Procedures for abstracting the RC graph and for canonical coding are developed in order to give the unambiguous description of the reaction type. The concept of reduced RC graph is also introduced and canonized in the seme line.
[Abstract] A novel method of enumerating reaction types is presented. The reaction types are formulated as reaction-center graphs (RCGs) that are derived from a parent basic reaction graph (BRG) or from a parent reaction graph (RG) by the substitution of various atoms on the vertices and/or by the attachment of single, double, or triple par-bonds to the edges. The counting of the isomeric RCGs is solved by appliying Polya's theorem to the derivation of a cycle index, in which different variables are introduced to the different orbits of the permultation group of a parent BRG (or RG). Various atom-figure and bond-figure inventories are described in order to count the isomers of RCGs. The numbers of the RCGs are given in the form of the coefficents of a generating function called an RCG-counting polynomial series.
1989 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] An imaginary ring that appears in an imaginary transition structure (ITS) is a descriptor for a ring-opening, ring-closure, or rearrangement reaction. An n-nodal subgraph of the ITS is a descriptor of a reaction-site change such as substitution, construction, cleavage, addition, and elimination. The n-nodal subgraph is subdivided into a reaction kernel and a terminal descriptor, the latter of which is useful to classify oxidation and reduction. The ITS that corresponds to an individual reaction is represented uniquely an unambiguously by a newly-defined canonical code.
[Abstract] A method for enumerating reaction-center graphs (Bull. Chem. Soc. Jpn., 61, 4189--5206 (1988).) has been generalized. Reaction-center graphs (RCG), reaction graphs (RG), and basic reaction graphs (BRG), all of which are defined as substructures of imaginary transition structures (ITS), represent reaction types of various levels in ascending order of generality. The concept of isomeric RCGs is introduced for the systematic enumeration of the reactions types. The enumeration of six-electron pericyclic reactions is thereby translated into the counting of hexagonal and pentagonal RCGs. This issue is solved by regarding the hexagonal or pentagonal RCG as a derivatives of a parent RG and by using Polya's theorem generalized to be applicable to the cases in which the vertices are obliged to have different modes of substitution.
[Abstract] Five-center organic reactions are formulated in the form of pentagonal reaction-center graphs (RCGs) that are subgraphs of imaginary transition structures. This formulation is based on the concept of a charge space that affords rational represetations of coordinate bonds. Polya's Theorem is applied to the enumeration of these RCGs, in which the obligatory minimum valencies of vertices restrict the mode of substitution. (cf. Bull. Chem. Soc. Jpn., 61, 4189--5206 (1988).)
[Abstract] Enumerations of compounds based on a parent skeleton with and without influence of obligatory minimum valency (OMV) are reported. The effect of the OMV is formulated by assigning different weights to the respetive orbits of the parent skeleton. This type of enumeration requires introduction of several new concepts that are derived from the subduction of coset representations, e.g., a unit subduced cycle index, a subduced cycle index and the number of suborbits.
[Abstract] Cage-shaped hydrocarbons are enumerated, starting from the tetrahedron skeleton (T_{d} symmetry). This enumeration stems from an edge strategy in which six edges of the skeleton are substituted by m methylene and n ethylene units. This is accomplised by a new method based on unit subduced cycle indices (Theor. Chim. Acta,76, 247--268 (1989)). This is a versatile methodology fro conunting cage-shaped hydrocarbons that have a give subsymmetry and an index term, x^{m}y^{n}, the latter of which corresponds to the molecular formula C_{4+m+2n}H_{4+2m+4n}.
1990 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A new method based on imaginary transition structures (ITSs) is compared with conventional ways for characterization of organic reactions. In particular, such substructures of ITSs as three-nodal substructures and four-nodal substurectures are discussed.
[Abstract] Subduction of the coset representations and the related concepts such as unit subduced cycle indices and subduced cycle indices yeild the foundation for a new type of generating functions for enumerating chemical structures. This method is related to Polya's theorem.
[Abstract] Molecules derived from a parent skeleton are enumerated where both achiral and chiral ligands are allowed. Chirality fittingness of an orbit is proposed in order to permit chiral ligands. The enumeration is conducted with and without consideration of obligatory minimum valency (OMV). The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. The importance of coset representations and their subduction by subgroups is discussed. The subduced representations are classified into three classes through their chirality fittingness, which detemines the mode of substitution with chiral and achiral ligands. Several novel concepts such as a unit subducted cycle index and a subduced cycle index are given in general forms.
[Abstract] A novel method of enumerating isomers is presented by using adamantane as a parent skeleton. This method is based on unit subduced cycle indices with and without chirality fittingness, the indices being derived from the subduction of coset representations. The chirality fittingness, which is determined by examinin the relationship between a pointo group and its subgroup, controls the mode of substitution by achiral and chiral substituents. The method provides detailed enumerations concerning symmetries and molecular formulas, whereas Polya's theorem takes only the latter into consideration.
[Abstract] Tricyclic isomers of adamantane (C_{10}H_{16}) are characterized by polymethylene indices (PMIs) and molecular symmetry. The PMI is a partition denoted as [1^{m1}, 2^{m2},..., 6^{m6}] (m_{1}+2m_{2}+...+6m_{6} = 6), in which each integer is the length of a polymethylene unit and the power (m_{r}) denotes the number of the units. The isomers are then enumerated by starting from tetrahedrane (T_{d}) and cyclobutadiene (D_{2h}) as parent skeletons, in which the edges are considered to be substituted by polymethylenes. From the tetrahedrane skeleton, there emerge 32 isomers, which are classified in terms of PMIs and subsymmetries of T_{d}. The cyclobutadiene skeleton yields 89 isomers classified by PMIs and subsymmetries of D_{2h}. Isomer enumerations are also discussed regarding noradamantane and homoadamantane.
[Abstract] The notation based on the subduction of coset representations (the SCR notation) is presented for the systematic characterization of molecular symmetry. The positions of a parent skeleton are divided into orbits on which the corresponding coset representaions (CRs) act. Each of the CRs is then subduced to the subsymmetry that characterizes a molecule derived from the skeleton. The resulting subduced representaions are further reduced into CRs, which provide a basis of the SCR notation. The SCR notation is more discariminative than the point-group notation as well as than the framework-group notation. A new concept ``unit subduced cycle index'' is introduced in order to determin which subsymmetry is realized if we begin with a parent skeleton of a given symmetry.
[Abstract] Unit subduced cycle indices (USCIs) are applied to the enumeration of isomers derived from a non-rigid parent molecule, the non-rigidity of which stems from bond-rotations. The parent is divided into a rigid skeleton of G-symmetry and mobile moieties of H-symmetries, where each mobile moiety is linked to a root that is a terminal vertex of the skeleton. In the first step, isomeric mobile moieties are enumerated with respect to the H-symmetry in terms of USCIs for H. Second, the mobile moieties counted are regarded as substituents on the vertices of the rigid skeleton. This formulation allows us to enumerated mobile isomers by means of USCIs for the G-symmetry.
[Abstract] The table of unit subduced cycle indices (USCIs) for D_{3h} is constructed and applied to the enumerations of chemical structures by starting from various parent skeletons of D_{3h} symmetry, i.e., a trigonal bipyramid, an iceane skeleton, and a prismane skeleton. The substitution positions of these skeletons are divided into orbits governed by coset representations. According to the division of such a skeleton, subduced cycle indices (SCIs) are constructed to enumerate chemical structures with respect to their symmetries. The introduction of figure inventories into the SCIs and the subsequent expansion give generating functions for counting fixed points. The resulting matrix of fixed points is multiplied by the inverse of mark table to afford the number of isomers with each molecular formula and each subsymmetry.
[Abstract] Inevitable use of different figure-inventories is discussed in the enumeration of non-ridgid isomers derived from 2,2-diphenyl- (1) and 2,2-dimethylpropane (2) by means of unit subduced cycle indices (USCIs). Such a parent non-rigid molecule is divided into a rigid skeleton and mobile moieties. Thus, the molecule (1) consists of a rigid skeleton of C_{2v} symmetry, a methyl moiety of C_{3v} symmetry and a phenyl moiety of C_{2v} symmetry. The molecule (2) contains a T_{d} rigid skeleton and a C_{3v} moiety. The USCIs are used in the first enumeration of mobile moieties as well as the seubsequent enumeration based on a rigid skeleton. A general procedure for enumerating non-rigid isomers is discussed. as an extenstion of the method described in S. Fujita, Theor. Chim. Acta, 77, 307--321 (1990).
[Abstract]
Subduction of the coset representation of
I_{h} point group produces tables of
unit subduced cycle indices (USCIs) with and without chirality
fittingness (CF). These indices are applied to
the enumeration of dodecahedrane derivatives with
respect to their molecular formulas as well as to their
symmetries. The USCIs are effective to the enumeration of
such derivatives as having only achiral substiturents on
the vertices of a dodecahedrane skeleton.
On the other hand, the USCI-CFs are used for
enumerating dodecahedrane derivatives with achiral and chiral
substituents. Substitutions on the edges of the dodecahedrane
skeleton are also discussed in terms of the present USCI approach.
[Corrections]
[Abstract] A combination of the concept of unit subduced cycle indices (USCIs) and the Redfield-Read superposition theroem affords a new methodology of enumerating geometrical object such as molecules and polyhedra. This method allows us to enumerate such objects with respect not only to their constitutions but also to their symmetries, in which subduced cycle indices (SCIs) derived from USCIs play an important role. Further derivation of a cycle index from the SCIs and its incorporation with the superposition theorem are also discussed. The inverse of the mark table and the table of USCIs for D_{2d} point group are presented for further applications of the USCI approach.
[Abstract] A coset representation (G(/G_{i})), which is defined algebraically by a coset decomposition of a finite group G by its subgroup G_{i}, is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of the G(/G_{i}) orbit belongs to the G_{i} sit-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and the k-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.
[Abstract] Local chirality and prochirality are discussed by integrating point-group and permutation-group theories. Thereby, a compound of G symmetry is considered to consist of several orbits that are subject to coset representation (CRs). Such a CR is denoted by the symbol G(/G_{i}) which comes from a coset decomposition of the group G by its subgroup G_{i}. The local chirality for a member of a G(/G_{i}) orbit is determined to be G_{i}. The concept of chirality fittingness is proposed to indicate symmetrical properties of G(/G_{i}) orbit, in which the orbit is classified into one of three categories, i.e., homospheric, enantiospheric, and hemispheric. This terminology allows us to define a prochiral compound as an achiral compound having at least one enentiospheric orbit. This membership criterion for prochirality is compared with the conventional substitution and symmetry criteria. The subduction of CRs affords a desymmetrization lattice for examining the existence and nonexistence of subgroups. Chemoselective achiral processes, chemoselective chiral proscesses, and steroselective chiral processes are discussed in terms of the chirality fittingness of orbits.
[Abstract] The chirality fittingness of orbits (homospheric, enantiospheric, and hemispheric) provides a new scheme of determining topicities (homotopic, enantiotopic, diasterotopic, and heterotopic). Thereby, transpositions of ligands on a skeleton are classified into four categories: homotopic, enantiotopic, diastereotopic, and heterotopic transpositions. The homotopic transpositions are concluded to be non-stereogenic. On the other hand, the other transpositions are proved to be stereogenic.
1991 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Proligands are defined as hypothetical ligands which are structureless but have chirality. A promolecule consists of a skeleton and such proligands. Promolecules based on a methane and an allene skeleton are enumerated. A molecule can be constructed by a method in which proligands on such a promolecule are replaced by ligands that have three-dimensional sturctures. This construction is controlled by a coset representation that governs an orbit of such proligands. Thereby, the resulting molecules are classified into matched and mismatced molecules. The matched molecules retain the symmetries of the starting promolecules; on the other hand, the mismatched molecules do not. Modes of such desymmetrizations are rationalized by subduction of coset representations. The concept of prochirality is also discussed.
[Abstract] Two methods for the enumeration of organic reactions are presented in order to take obligatory minimum valencies of a given skeleton into consideration. The first method is a generalization of Polya's theorem, in which the transitivity of the positions of the skeletons is explicitly considered. Thus, a permutaion representation acting on the positions is reduced into coset representations (CRs). In accord with this reduction, unit cycle indices derived from the CRs construct a generalized cycle index. The second method is based on the subduction of coset representations. This contains useful concepts such as unit subduced cycle indices and subduced cycle indices that afford a new type of generating functions.
[Abstract] A general method of characterizing symmetry equivalence and non-equivalence in a molecule is developed on the basis of the one-one correspondence between an orbit and a coset representation. This method provides us with a theoretical foundation to understand ansiochrony in rotatable ligands. A given molecule is regarded as a combination of a three-dimensional skeleton and ligands. The ligand is referred to as a segment or a fragment according to its environment. Thus, the fragment is defined as a ligand-in-isolation; and the segment denotes a ligand bulit into a molecule. The symmetry of the ligand as a fragment is restricted to the local symmetry. This restriction is examined by using methyl and methylene ligands as examples. A methyl fragment has a homospheric C_{3v}(/C_{s}) orbit that cosnsists of three hydrogen atoms. On the other hand, a methylene fragment has an enantiospheric C_{s}(/C_{1}) orbit that contains two hydrogen atoms. Such an orbit is incorporated into various molecules to afford restricted orbits, which are characterized by subduction of coset representations. The restricted orbits are discussed in terms of global and local orbits. This discusssion provides a general approach to rationalize various types of anisochronies.
[Abstract] Two types of derivations of molecules, subductive and inductive, are presented to design molecules of high symmetry. A subductive derivatiion consists of substituting a parent molecule with another set of atoms; on the other hand, an inductive derivation is composed of substituting a parent skeleton with a set of ligands that have three-dimensional structure. These derivations are discussed on the basis of subduction and induction of coset representations.
[Abstract] Prochirality is one of the inherent properties of an enantiospheric orbit that is designated by a coset representation G(/G_{i}). Desymmetrization of the G(/G_{i}) orbit is discussed to prsent a new concept ``H-chirogenic site''. The chirogenic site in the G(/G_{i}) orbit is capable of producing a chiral H-molecule under the influence of an appropriate chiral reagent. The condition for the existence of such a chirogenic site is discussed. A convenient method using a desymmetrization lattice is presented for determining chirogenic sites.
[Abstract] Symmerical properties of soccerane and its derivatives are discussed in the light of subduction of coset representation (CRs). The 60 vertices of soccerane are equivalent and construct a single orbit governed by the CR I_{h}(/C_{s}). This orbit is divided into serval orbits in the respective derivative, where the mode of the division is controlled by the subduction. The number of derivatives with a given formula and a given subsymmetry is calculated in terms of unit subduced cycle indices, which are derived from the subduction.
1992 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Promolecules having a D_{\infty h} skeleton are enumerated by means of a method using unit-subduced cycle indices, which is modified so as to be based on factor groups. Molecules derived from the promolecules are also enumerated by combining the method with a proevious method reported for enumerating rotatable molecules. The stereochemical properties of the promolecules are related to those of the corresponding molecules in terms of infraorbits and local orbits. In particular, the prochirality of a promolecule and of a molecule is ascribed to an appropariate enantiospheric orbit. This determination is mathematically founded so that it requires no tentative consideration of a fixed conformer of the highest attainable symmetry.
[Abstract] The elementary superposition theorems are presented for enumerating chemical compounds that contain achiral and chiral ligands. Subduced cycle indices (SCI-CF), partial cycle indiced (PCI-CF), and cycle indices (CI-CF) with chirality fittingness are defined by starting from unit subduced cycle indices with chirality fittingness (USCI-CF). All of these indices afford generating functions that are proved to be applicable to combinatorial enumeration. In addition, the concept of elementary superposition with and without chirality fittingness is proposed to provide the elementary superposition theorems. These theorems provide us with a new mehtodology of enumerating compounds, in which the numbers of isomers are obtained without relying on generating functions and are itemized with respect to molecular formulas (weights) and symmetries. The encircled-asterisk operation is defined on the basis of the elementary superposition. Thereby, we derive superposition theorems concerning the PCI-CFs and CI-CFs. These are applicable to combinatorial enumeration.
1993 [Top of List of Articles] [Bottom of List of Articles]
[Abstract]
The concept of promolecules is presented for enumerating
octahedral complexes, where achiral and chiral proligands
are considered as substutients for an octahedral
O_{h} skeleton. After the table of marks and its inverse
are calculated, octahedral promolecules are enumerated using the
USCI (unit-subduced-cycle-index) method. The numbers of
isomers are obtained in an itemized manner with respect to their fomulas
and symmetries. The stereochemical relationships between the isomers
are discussed in the light of the SCR (set-of-coset-representation)
notation, which is also effective in clarifying stereochemical
equivalency and non-equivalency of proligands and specifying
the prochirality of octahedral promolecules and molecules.
[Corrections]
[Abstract] Four methods are described for enumerating digraphs with a given automorphism group: (1) a generating-function method based on subduced cycle indices, (2) a generating-function method based on partial cycle indices, (3) a method based on the elementary-superposition theorem, and (4) a method based on the partial superposition theorem. All of these methods are based on the concept of unit subduced cycle indices and construct a set of versatile tools for combinatorial enumeration. They are applied to the enumeration of five-vertex digraphs with a given automorphism group. The table of marks and its inverse for the symmetric group of degree 5 are recalled. The table of USCIs of this group is obtained.
1994 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Combinatorial enumeration by means of unit subduced cycle indices (USCIs) is discussed by using the group I (A_{5}) and the related groups as examples. A modified method for the derivation of USCIs is presented, where a subduced mark table is a key concept. Several properties of USCIs are discussed by clarifying the relationship between double cosets and the subduction of coset representaitons, which gives an alternative formulation of the USCIs. This treatment provides us with a new prospect concerning the properties of double cosets. The usefulness of subduced cycle indices and cycle indices derived from USCIs is exemplified by enumeration based on an icosahedran.
[Abstract] The concept of equi-axial transformations is proposed to characterize conformational changes of flexible compound such as the inversion of ammonia and the ring-flipping of 1,3,5-trioxane. In order to discuss the symmetries of the equi-axial transformations, pseudo-point groups are introduced after the definition of a pseudo-dihedral rotation. Pseudo-point groups are classified into anisoenergetic and isoenergetic groups. In the light of this classification, orbits generated by a pseude-point group are characterized by the concept of chronality, which is defined by examining coset representations. Thus, the terms homochronal, enantiochronal, and hemichronal are coined for the characterization of orbits. The enumeration methods of the USCI (unit-subduced-cycle-index) approach are applied to the ammonia and 1,3,5-trioxane cases by using the pseudo-point group \hatD_{3h}.
[Abstract] The symmetry of a pair of two chair-forms of cyclohexane is represented by the pseudo-point group \hatD_{6h} of order 24. Preparation of the mark table of the group \hatD_{6h} shows the twelve substitution positions of the pair to be governed by the coset representation \hatD_{6h}(/C_{s}). After the calculation of subduction of the \hatD_{6h}(/C_{s}), cyclohexane derivatives are combinatorially enumerated by the USCI (unit-subduced-cycle-index) approach. A generating-function method and the elementary superposition therorem are used, giving values itemized with respect to molecular formulas and subsymetries of \hatD_{6h}. Since the pseudo-point groups can be classified into iso- and anisoenergetic groups as well as into achiral and chiral groups, four categories (isoenergetic-achiral, isoenergetic-chiral (Type II), anisoenergetic-achiral (Type III), and anisoenergetic-chiral (Type IV)) are generated. The isoenergetic-achiral case is further subdivided into two cases (Type I and I'). Several pairs are illustrated and discussed in the light of this classification. The concept of chronality is also discussed.
[Abstract] Methods for typesetting chemical structural formulea are discribed, where the picture environment of the well-known text fromatter TeX/LaTeX is used to incorporated the formulae into manuscripts. Macros were designed for typesetting carbocycles and heterocycles, in which the designation of substituents and bonds is simplified by applying the list-treating mechanism of TeX/LaTeX. These macros extend the range of structures printed with TeX/LaTeX, since they permit 10 or more substituents which have not been treated by previous methods.
1995 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] XyMTeX, a macro package of combined LaTeX style files, has been developed for drawing a wide variety of chemical structural formulas. The commands of XyMTeX have a set of systematic arguments for specifying substituents and their positions, endocyclic double bonds, and bond patterns. In some cases, they have an additional argument for specifying hetero-atoms on the vertices of heterocycles. As a result of this systematic feature, XyMTeX works effectively as a practical tool within the device-independent concept of TeX.
[Abstract] The concept of markaracter is proposed to discuss marks and characters for a group of finite oreder on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominat representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers fo fixed points corresponding to each DR are collected to form a row vector called a dominant markaracater (mark-character). Such dominant markaracters for the SDR are collected as a markaracter table. The markaracter table is related to a subdominant markaracter table of its subgroup so that the corresponding row of the former table is constructed from the latter. The data of the markaracter table are in turn used to construct a character table of the group, after each character is regarded as markaracter and transformed into a multiplicity vector. The concept of orbit index is proposed to classify multiplicity vectors; thus, the orbit index of each DR is proved to be equal to one, while that corresponding to an irreducible representaion is equal to zero.
[Abstract] A new method for giving cycle indices is presented for combinatorial enumeration. Thus, cyclic groups are characterized by markaracter tables, the elements of which are determined by the orders of their subgroups. A set of such cyclic groups (defined as dominant subgroups) is used to characterize a group G of finite order, where the markaracter table for the group G is constructed with respect to dominant representations (DRs), which are defined as coset represetation corresonding to the dominant subgroups. By starting from the markaracater table, we propose an essential set of subdominant markaraacter tables and magnification set for the group G; the latter concept clarifies the relationship between each subdominant markaracter table and the markaracter table of dominant subgroup. The subduction of DRs is obtained by the markaracter table to produce a dominant subduction table and a dominant USCI (unit-subduce cycle index) table. The latter is used to evaluate a cycle index to be applied to combinatorial enumeration. The cycle index is shown to be equivalent to the counterpart of our previous approach concerning both cyclic and non-cyclic subgroups. The latter, in turn, has been proved to be equivalent to the cycle index obtained by the Redfield-Polya theorem.
1996 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The pseudo-point group \tildeD_{6h} is constructed to characterize the symmetry of a basic pair of hexagonal reaction graphs having no par-bonds on its edges. Any pair of reaction graphs (reaction pairs) are considered to be obtained by adding par-bonds to the edges of the basic pair; they are counted by the USCI (unit-subduced-cycle-index) approach. Thus, the six edges of the basic pair are assigned to the coset representation \tildeD_{6h}(/C_{2v}). After the subduction of \tildeD_{6h}(/C_{2v}) is calculated, the partial-cycle-index method of the USCI approach is applied to the combinatorial enumeration of reaction pairs. Reaction pairs are classified to two categories, i.e. isoenergetic and anisoenergetic. An isoenergetic pair is concluded to be a self-reaction pair, while an anisoenergetic pair corresponds to a non-self-reaction pair. The concept of chronality is also discussed to clarify the symmetrical nature of the resulting orbits.
[Abstract] After preparing the subduction table of I_{h}-group, buckminsterfullerene derivatives are combinatorially enumerated in an itemized manner concerning symmetries and molecular formulas. The symmeterical properties of the derivatives are discussed in terms of the sphericity concept. Bond-differentiating reactions are formulated in order to clarify possible routes to derivatives of several subsymmetries of I_{h}. Such bond-defferentiating reactions are classified into four categories, i.e., chemoselective achiral, chemoselective chiral, stereoselective chiral, and stereoselective achiral reactions. Chirogenic sites in a homospheric orbit are examined to show the possibiity of direct chiral reactions. The terms mesolocative, equilocative, dialocative, and semilocative are proposed to characterize bond pairs in a homospheric orbit.
1997 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A new screening process based on tree-structured data has been constructed for an in-house chemical substance database. The data structure and algorithm have improved the selectivitity of screening and reduced search time. Thus, a prescreeing step selecting an appropriate key is introduced to accelerate the total screening process. Its effectivenesss has been proved by the comparison with the search without such prescreening. The present tree-structured system has been compared with a fragment-oriented screening system along with the retrieval of 87 query structures. The result shows excellent performance of the present screening system; it is because the tree-structureed data include structural information of all nodes of the structures.
1998 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A new method of combinatorial enumeration is presented. The subduction of Q-conjugacy representations gives a characteristic subduction table and a characteristic monomial table. A cycle index is defined on the basis of such monomials and used for combinatorial enumeration of isomers.
[Abstract] Q-Conjugacy character tables for cyclic groups are obtained by starting from character tables. Thus, irreducible representations for a cyclic group are classified into primitive and non-primitive ones. They are collected to form a matrix corresponding to each subgroup. Such a matrix is shown to be a representation (called Q-conjugacy representation) for characterizing Q-conjugacy and dominant classes. The traces of the Q-conjugacy representation are collected to form a Q-conjugacy character table, which is shown to be a square matrix. The elements of such a Q-conjugacy character table for a cyclic group are shown to be integers, which are related to the values of the corresponding character tables. They are also correlated to the markaracter tables for the cyclic group. Characteristic monomial tables for cyclic groups are obtained by starting from the Q-conjugacy character tables and dominant unit-subduced-cycle-index tables. They are applied to combinatorial enumeration of isomers derived from a skeleton belonging to a cyclic group.
[Abstract] The pseudo-point group \hatC_{2v} is defined for characterizing the symmetries of tetrahydropyran and 1,3-dioxane. Then, their derivatives with a given formula and a given symmetry are enumerated by the unit-subduced-cycle-index approach. Each derivative enumerated belongs to a subsymmetry of \hatC_{2v} so as to be classified into isonergetic (Type I, I' or II) or anisoenergetic (Type III or IV). The orbits in the enumerated derivative are discussed by the sphericity and chronality terms.
[Abstract] The pseudo-point group \hatD_{2h} is defined for characterizing the symmetries of 1,4-dioxane derivatives. The pseudo-point group \hatD_{2v}' is defined as a subgroup of \hatD_{2h} and is applied to the characterization of the symmetries of 1,4-oxathiane derivatives. The 1,4-dioxane derivatives with a given formula and a given symmetry are enumerated by the unit-subduced-cycle-index approach and compared with the 1,4-oxathiane derivatives. Each derivative enumerated belongs to a subsymmetry of \hatD_{2h} or \hatD_{2v}' so as to be classified into isoenergetic (Type I, I' or II) or anisoenergetic (Type III or IV). The orbits in the derivative are discussed by the sphericity and chronality terms.
[Abstract] The maturity of a finite group G is defined by examining how a dominant class (Q-conjugacy class) corresponding to a cyclic subgroup H contains conjugacy classes. If the ingeter q = |N _{G}(H)|/|C_{G}(H)| (called the maturity discriminant) is less than an Euler function phi(|H|), the group G is concluded to be unmatured concerning H, where N_{G}(H) and C_{G}(H) respectively denote the normalizer and the centralizer of H within G. A matured representation defined for a matured or an unmatured group is subduced into cyclic subgroups to give the corresponding monomials, which are applied to the combinatorial enumeration of isomers.
[Abstract] Q-Conjugacy character tables are proposed for finite groups and applied to combinatorial enumeration. Thus, the maturity of an irreducible representation is related to the matuarity of a finite group by means of the relashionship between the inherent automorphism of the group and its inner portion. As a result, a character table is transformed into a more concise form called a Q-conjugacy character table. Matured characters are defined as dominant-class functions on the basis of such a Q-conjugacy character table. Thereby, a matured character is represented by a linear combination of Q-conjugacy characters. By staring from Q-conjugacy character tables, characteristic monomial tables for finite groups are obtained and applied to combinatorial enumeration of isomers.
[Abstract] The pseudo-point group \hatD_{2d} is defined for characterizing the flipping of the two cyclohexane rings in spiro[5.5]undecane. Spirane derivatives with a given formula and a given symmetry are enumerated by the unit-subduced-cycle-index approach on the basis of the spiro[5.5]undecane skeleton. The symmetry of each derivative corresponds to one of the subgroups of \hatD_{2d}, which are classified into isoenergetic (isoenergetic-achiral and isoenergetic-chiral) or anisoenergetic (anisoenergetic-achiral and anisoenergetic-chiral) ones. The orbits in the derivative are discussed by the sphericity and chronality terms.
[Abstract] Characteristic monomials for a finite group are obtained by direct subductions of Q-conjugate representations. They are shown to give a generating function that is capable of solving enumeration problems.
1999 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The characteristic monomial table of an achiral group is applied to combinatorial enumeration of the following types: (1) achiral isomers and enatiomeric pairs, (2) achiral isomers and chiral isomers, (3) enantiomeric pairs, and (4) achiral isomers. The cycle index of each case is obtained by using the same set of subduced cycle indices derived from characteristic monomials, where the coefficients of the subduced cycle indices are given in advance.
[Abstract] The unit-subduced-cycle-index (USCI) approach is applied to the enumeration of isomers based on D_{6h}-skeletons, where the inverse of a mark table and a USCI table are pre-calculated for the point group D_{6h}. Thus, benzene derivatives with achiral substituents are enumerated by using partial cycle indices (PCIs) in an itemized manner concerning formulas and symmetries. The formulation of partial cycle indices with chirality fittingness (PCI-CFs) enables us to enumerate benzene derivatives with chiral and achiral substituents. The USCI approach is also applied to the enumeration of isomers derived from a coronene skeleton that also belongs to the point group D_{6h}. The PCIs (or PCI-CFs) of zero or non-zero expressions are discussed in terms of a selection rule for judging the existence of subsymmetries. The symmetries of the resulting derivatives are discussed in terms of sphericities.
[Abstract] After the definitions of amplified representations and number-theoretical vectors, the markaracter table of a cyclic subgroup is converted into the corresponding \bfQ-conjugacy character table. The conversion is shown to necesitate an interconversion matrix that contains the extended M\"{o}bius functions as elements. Since the interconversion matrix gives characteristic monomials for cyclic groups, all the powers appearing in each of the characteristic monomials are shown to be integers. Then, characteristic monomials for finite groups are built up by starting from those of cyclic groups. This procedure clarifies that all the powers appearing in each characteristic monomial for finite groups are shown to be integers. The relationship between characteristic monomial tables and USCI tables is discussed with respect to their application to isomer enumeration.
[Abstract] The ring flipping (F) and the N-inversion (I) of piperidine are discussed by presuming a quadruplet of conformers as a model. After an extended pseudo-point group \hatC_{{2\tilde{I}v}} is defined, the concept of FI-energeticity is proposed. The flexible piperidine derivatives are enumerated on the basis of the model by the unit-subduced-cycle-index approach. The enumerated derivatives are classified into FI-isoenergetic (Types I, I', and II) and FI-anisoenergetic (Types III and IV) by means of their extended pseudo-point groups.
[Abstract] The XyM notation system is proposed as a linear notation for the electronic communication of structural formulas. Each XyM notation consists of a skeleton with such arguments as SUBSLIST, ATOMLIST, and BONDLIST. The arguments are designed to be capable of carrying out substitution derivation for placing large substituents, atom derivation for spiro fution, and bond derivation for ring fusion. Additional arguments, SKBONDLIST and OMITLIST, are discussed for stereochemical and ring-opening information. The XyM notation system is implemented as a LaTeX2e application.
[Abstract] The XyM Markup Language (XyMML) is proposed as a tool for the electronic communication of chemical structural formulas and reaction schemes. The XyMML system regards a structural formula as a derivative of a skeleton, where substitution and replacement are designated by using a substitution list (a subslist tag), an atom list (an atomlist tag), and a bond list (a bondlist tag). Structural formulas generated by XyMMLs can be collected to give a reaction scheme, in which cross-references to every XyMMLs are available by using the technique of SGML attributes.
[Abstract] A new and simple procedure for counting isomers derived from non-rigid parent molecules has been developed by the combination of the promolecule concept with the characteristic-monomial method. This procedure has been applied to the enumeration of diphenylmethanes, 2,2-diphenylpropanes and 2,2-dimethylpropanes, after the characteristic monomial tables for the point groups $\bfC_{2v}$ and $\bfC_{3v}$ have been prepared.
[Abstract] The unit-subduced-cycle-index method and the characteristic-monomial method have been applied to the combinatorial enumeration of ferrocene derivatives. The enumeration has been itemized with respect to formulas as well as to symmetries, where the symmetry of each derivative is characterized by a factor gorup $\bfD_{\infty h}/\bfC_{\infty}$ and its subgroups. The chirality/achirality of ligands and of ferrocene derivatives has been discussed.
2000 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A new method for enumerating non-rigid isomers with rotatable ligands has been developed so as to take the symmetries of the ligands into consideration. The method has been based on extended partial cycle indices and has been applied to the enumeration of methyl ether derivatives, tetramethylallene derivatives, and 2,2-dimethylpropane derivatives. These results have been compared with the enumeration results of the corresponding promolecules. The factorization of terms in generating functions has been discussed so that the new method is capable of examining the relationship between promolecules and molecules quantitatively.
[Abstract] To enumerate non-rigid isomers with given ligand symmetries on the basis of a D_{\infty h} skeleton, the concept of extended partial cycle indices (extended PCIs) proposed newly has been combined with the concept of promolecules proposed previously. The infinite nature of the D_{\infty h}-group is concealed by adopting the factor group of finite order, D_{\infty h}/C_{\infty} (= K). Thus, the partial cycle indices with chirality fittingness (PCI-CFs) for the factor group K are calculated and combined with the PCIs for ligand symmetries so as to give the extended PCIs for various itemized enumurations. This method has been successfully applied to the enumeration of ethane derivatives, where the full enumeration based on K has been compared with partial enumerations based on K as well as with those based on the factor group C_{\infty h}/C_{\infty} (= K_{3} \subset K). Each term of resulting generating functions has been factorized into a pair of factors to represent ligand consitutions. Thereby, the depiction of resulting molecules can be conducted systematically so as to provide the maps of ethane derivatives corresponding to all of the substitution types.
[Abstract] Generalized partial cycle indices have been proposed as a generalization of partial cycle indices (PCIs) in order to enumerate isomers with a given set of subsymmetries. To evalulate the coefficient of each term appearing in such a generalized PCI, the nature of mark tables and of its inverse has been examined after the proposal of (modified) bisected mark tables and their inverses as well as of (modified) bisected tables of unit subduced cycle indices with chirality fittingness (USCI-CFs). The inverse of a (modified) bisected mark table and the (modified) bisected USCI-CF table have been applied to the enumeration of chiral isomers and achiral isomers.
[Abstract] A systematic method for characterizing prochirality, prostereogenicity, and stereogenicity is described. Any set of equivalent ligands is regared as an orbit governed by a coset represention, by which the orbit is classified into homospheric, enantiospheric, or hemispheric. A molecule containing at least one enatiospheric orbit is defined to be prochiral (Rule A). After the definition of the topicity terms, a prostereogenic center is defined as a center or atom having two ligands that are indistinguishable in isolation and not homotopic (i.e. either enantiotopic, diastereotopic or heterotopic) in a molecule (Rule B). Then, Rule C for characterizing a stereogenic center is defined subsidiarily from Rule B. These rules are discussed by using several molecules of stereochemical interest.
[Abstract] A chiral replacement criterion has been proposed to characterize stereochemical equivalence. This criterion is effective to determine holotopic and hemitopic relationships, which are sub-relationships of a homotopic relationship. The criterion has been shown to be complementary to the membership criterion based on the sphericity concept.
[Abstract] The concept of size-invariant subductions has been proposed to design prochiral molecules. Thus, an even-membered homospheric orbit has been proved to be desymmetrized into an enantiospheric orbit, where the sizes of the relevant orbits remain invariant. The concept has been applied to methanes, allenes, adamantanes, and biphenyls. It has proved effective in designing a wide variety of prochiral molecules. These prochiral molecules have been shown to be extrinsic meso compounds. Intrinsic meso compounds have also been discussed as another type of prochiral molecule.
[Abstract] A new method of combinatorial enumeration based on characteristic monomials with chirality fittingness (CM-CFs) has been proposed in order to enumerate isomers with chiral ligands as well as with achiral ones. The CM-CFs have been defined as monomials that consist of three kinds of dummny variables in the light of the subduction of the \bfQ-conjugacy representations for chiral and achiral cyclic groups. A procedure of calculating CM-CFs for cyclic groups and finite groups has been discribed so as to tabulate them as CM-CF tables. Then the CM-CF method has been applied to the enumeration of isomers with achiral ligands as well as chiral ones.
[Abstract] The XyMTeX command system has been refined and extended to give a new linear-notation system, which is now called the XyM Notation. The abstract nature of the XyM Notation means that XyMTeX is now regarded as an application software for TeX/LaTeX printing, where the XyM Notation is parsed by virtue of TeX/LaTeX. Moreover, the XyM Notation can be used as an intermediate language, into which another language for representing structural formulas (e.g. the XyM Markup Language (XyM{}ML)) is translated so as to print out the formulas. A mechanism for the adjustment of substitution positions (or for concealing layout data) has been developed in XyMTeX version 2.00 to support the XyM Notation and XyMML. Thus, the version 2.00 of XyMTeX (1998 and 1999) supports the yl-function introduced by the XyM Notation, where a complicated substituent is treated as a modification of a substitution list (SUSBLIST). As an extention of this mothodology, a bond list (BONDLIST) can be modified to treat ring fusion, since each ring fusion is considered to be a kind of substitution on a bond. In addition, an atom list (ATOMLIST) can also be used to treat spiro rings, since each spiro ring is a kind of atom replacement at an appropriate vertex.
[Abstract] For the design of chiral molecules of high symmetry, a set of substitution positions in an achiral skeleton of G-symmetry are replaced by chiral ligands or proligands of the same kind so that there emerge three criteria of generating chiral molecules or promolecules. Thus, Criterion 1 controls the desymmetrization of a homospheric orbit in an achiral skeleton, where the resulting molecule or promolecule is determined to belong to the maximum chiral subgroup of G in agreement with a size-invariant subduction. Criterion 2 for the desymmetrization of an enantiospheric orbit shows that the resulting molecule or promolecule is determined to belong to the maximum chiral subgroup of G and that the original enantiospheric orbits are divided into two hemispheric orbits of equal size. Criterion 3 deals with a chiral skeleton, where subsitution by chiral ligands or proligands of the same kind is examined as the \break transformation of the hemispheric orbit in the skeleton. Although no change of symmetry occurs from a group-theoretical point of view, the transformation is shown to be important chemically, since relevant ligands or proligands alter the stereochemical properties.
2001 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A set of equivalent positions in an achiral skeleton is regarded as an orbit assigned to a coset representation G(/G_{i}), where the subgroup G_{i} is C_{nv} or C_{n}. Desymmetrization of the achiral skeleton by substituting a single ligand is examined by virtue of the subduction of the coset representation G(/G_{i})\downarrowG_{i}, which is determined to be $\alpha$G_{i}(/G_{i})+ $\beta$G_{i}(/G_{i}^{(i)})$. The multiplicities ($\alpha$ and $\beta$) of the resulting coset representations are calculated from the data of the participating groups and the normalizer of G_{i}.
[Abstract] The sphericity concept proposed for specifying stereochemistry in a molecule (Fujita, S. J. Am. Chem. Soc., 1990, 112, 3390) has been extended to investigate stereoisomerism among molecules. The new matter of the present approach is to characterize the global symmetries of molecules as the ``local symmetries of stereoisomerism''. Thereby, stereochemistry and stereoisomerism have been discussed on a common basis. Promolecules, which have been generated as stereochemical models of molecules by placing proligands (structureless ligands with chirality/achirality) on the vertices of a tetrahedral skeleton, have been analyzed by a permutation-group approach as well as by a point-group one. The skeleton has been considered to belong to the symmetric group of degree 4 (S^{[4]}) as well as to the isomorphic point group T_{d}. The chirality fittingness derived from the sphericity concept has been applied to the characterization of local symmetries of a promoleule, where two types of Young's tableaus have been compared. Thus, Young's tableaus of symmetry have been introduced to treat the ligand packing based on the chirality fittingness. These tableaus have been compared with Young's tableaus of permutation, which have taken no account of such chirality fittingness. The two types of Young's tableaus have been applied to the enumeration of tetrahedral isomers under the observance and the violation of chirality fittingness. This enumeration has enabled us to clarify the quantitative aspect of the sphericity concept in characterizing isomer equivalence. Thereby, equivalent isomers under a point-group symmetry have been shown to construct an orbit of stereoisomers that is ascribed to a coset representation. Homomeric, enantiomeric, and diastereomeric relationships between stereoisomers have been discussed by means of homospheric, enantiospheric and hemispheric orbits of stereoisomers. Skeleton-based and ligand-based categories for enantiomers and diastereomers have been discussed. The stereogenicity and the prostereogenicity of the Chan-Ingold-Prelog system have been related to Young's tableaus of permutation.
[Abstract] Among the four methods of the unit-subduced-cycle-index (USCI) approach, the subduced-cycle-index (SCI) method and the partial-cycle-index (PCI) method have been discussed by using adamantane of T_{d}-symmetry as a probe for enumeration problems, where USCIs are derived on the basis of permutaion representations, coset representations (CRs) and marks. After the examination of the SCIs and PCIs, P\'{o}lya's theorem that is a standard method of chemical combinatorics has been derived from the USCI approach. As another approach, a new method called the characteristic-monomial (CM) method has been developed by virtue of charactereistic monomials (CMs). The CMs have been derived from Q-conjugacy representations and Q-conjugacy characters, which have been related to irreducible representations and irreducible characters of the standard repertoire of chemical group theory. The two approaches have been compared to discuss group-theoretical tools for chemical combinatorics on a common basis. (This paper was presented in part before the Division of Medicinal Chemistry (Symposium #140, Mathematical and Computational Aspects of Molecular Design) at 2000 International Chemical Congress of Pacific Basin Societies, Honolulu, Hawaii, December 15, 2000; Abstract No. MEDI 136.)
[Abstract] A pseudo-point group $\widehat{\bmD}_{6h}$ and its subgroups are applied to the symmetry characterization of tri- and tetra-substituted cyclohexane derivatives under flexible and fixed conditions. After the subgroups are classified into chiral and achiral ones as well as into isoenergetic and anisoenegetic ones, the conformational changes of the derivatives are categorized into isoenergetic-achiral (Types I and I$^{\prime}$), isoenergetic-chiral (Type II), anisoenergetic-achiral (Type III), and anisoenergetic-chiral cases (Type IV). These cases are rationalized by the subduction from $\widehat{\bmD}_{6h}$ to $\bmD_{3d}$. The sphericity concept based on the subduction of coset representations is applied to the characterization of the prochirality of the cyclohexane derivatives. The chronality concept is proposed to discuss fixation processes of the cyclohexane derivatives on the basis of the subduction of coset representations.
[Abstract] The error of the epic package has been analyzed and avoided by a slight modification of the pakage program. This has been applied to the XyMTeX system so as to realize the size reduction of chemial structural formulas.
2002 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The XyMJava system for drawing chemical structural formulas has been developed by using the Java language in order to enhance Worldwide-Web communication of chemical information, where the XyM notation system proposed previously has been adopted as a language for inputting structural data. The object-oriented technique, especially the design-pattern approach, is applied to parse a XyM notation in the XyMJava system. A chemical model is introduced to encapsulate information on chemical structural formulas and used to draw the formula on a CRT display. Thereby, an HTML document containing a XyM notation can be browsed by the XyMApplet of the XyM Java system.
[Abstract] A pseudo-point group D^{^}_{6h} has been applied to the design of high-symmetry cyclohexane derivatives, which are assigned to the subgroups of D^{^}_{6h}. The existence/nonexistence of these derivatives has been prediced by comparing partial cycle indices with and without chirality fittingness (PCI-CFs and PCIs). The PCI-CFs and PCIs stem from the unit-subduced-cycle-index (USCI) approach, where the subduction of coset representations (CRs) are precalculated by using mark tables. These derivatives have been classified into isoenergetic and anisoenergetic derivatives. Energetical and symmetric equivalency of ligands have been discussed by virtue of CRs, where the concepts of chronality and sphericity have been used to classify the CRs.
[Abstract] Two graphical approaches have been presented to obtain marks of a group, where one approach is based on a set of homomers and the other is based on a set of equivalent ligands. Their procedures have been illustrated by using the point group D_{3} as a common example. Comparison of these approaches has revealed that their foundation comes from the one-to-one correspondence among a homomer set H[G(/G_{i})], a ligand set L[G(/G_{i})] , and a set of cosets G/G_{i}, which are all assigned to the coset representations G(/G_{i}), where G is a group and G_{i} is a subgroup of G.
[Abstract] Allene derivatives have been combinatorially enumerated under a point group D_{2d} as well as under a permutation group S_{9}^{[4]}, which is a subgroup of the symmetric group of degree 4 and is isomorphic to the point group $\bfD_{2d}$. These enumerations have been compared in terms of the observance and the vilolation of chirality fittingness, where two types of Young's tableaus (point-group symmetry and permutation-group symmetry) have been used to clarify the relationship between stereochemistry and permutability of allene derivatives. The concept of chirality/achirality for stereochemistry has been compared with the concept of stereogenicity/astereogenicity for permutability. Enantiomers, which have been characterized by equivalency under point-group symmetry, have been classified into two classes according to whether or not they have been assigned to orbits (equivalence classes) under permutation-group symmetry. On the other hand, diastereomers have been related to equivalence classes (orbits) under permutation-group symmetry only. The terms {\em enantiostereogenic} and {\em diastereogenic} have been coined to discuss the behavior under permutaiton-group symmetry. The CIP (Cahn-Ingold-Prelog) system is examined in terms of stereogenicity/astereogenicity. A new combination of chirality and stereogenicity described in the present paper provides us with a tool for restructuring stereochemistry.
[Abstract] A tool for displaying and communicating chemical structural formulas has been developed on the basis of XyMML (XyM Markup Language), where a XyMML document according to the XML (Extensible Markup Language) specification has been transformed into an HTML (HyperText Markup Language) document by means of a translator program due to XSLT (Extensible Stylesheet Language Transformations). During this process, XyMML data written in such a XyMML document have been converted into XyM notations embedded in such an HTML document, which is browsed by virtue of a World Wide Web (WWW) browser including the XyMJava system. Another tool for printing chemical structural formulas has been developed so that the same XyMML document has been transformed into a XyMTeX document by means of XSLT. The resulting XyMTeX document has been used to print a document containing structural formulas through the TeX/LaTeX typesetting system. Thereby, the XyMML and the related techniques have been shown to have the potentiality of serving as a kernel for integrating WWW communication, electronic publishing, and conventional publishing in chemistry.
[Abstract] Square-planar complexes with achiral and chiral ligands have been enumerated exhaustively under the point-group D_{4h} and under the symmetric group S^{[4]} of degree 4, where they have been classified in terms of their symmetries and permutabilities. Thereby, their stereochemical properties and relationships have been discussed in detail. In particular, equivalency under point-group symmetry (e.g., enantiomeric relationships for chiral complexes and prochirality for achiral complexes) and that under permutation-group symmetry (e.g., proper and improper permutations, stereogenic and astereogenic groups, and enantiostereogenic and diastereogenic groups) have been characterized to give a systematic format for stereochemistry and stereoisomerism.
[Abstract] The importance of orbits (equivalence classes) has been stressed in order to restructure stereochemistry. The concept of sphericity has been formulated by a new scheme that requires a minimum set of knowledge on point-group theory. The concept of prochirality has been revisited from the viewpoint based on the new formulation of sphericity. A variety of objects such as atoms, ligands, and faces are discussed as the members of such an orbit. In particular, orbits for atoms in tetrahedral molecules and for the faces of carbonyl compounds, ethylene derivatives, and allene derivatives have been examined in terms of the concept of sphericity. The conventional topicity terms of two categories (``topic relationship between two sites'' and ``topic attribute of a site'') have been effectively replaced by the sphericity terms.
[Abstract] To restructure stereochemistry into a systematic format, enantiomeric and diastereomeric relationships have been investigated by using ethylene derivatives as examples in the light of a new group-theoretical and combinatorial approach. On one hand, enantiomeric relationship for ethylene derivatives has been characterized by means of a point group of order 8 (D_{2h}), where chirality fittingness based on the sphericity concept has been applied to the enumeration of stereoisomers. On the other hand, diastereomeric relationship for ethylene derivatives has been examined by a permutation group of order 8 (S^{[4]}_{9}), which is a subgroup of the symmetric group of order 4 (S^{[4]}) and isomorphic to a point group (D_{2h}). The subgroups of (S^{[4]}_{9} have been classified into stereogenic and astereogenic ones. A stereogenic subgroup corresponds to a pair of diastereomers, while an astereogenic subgroup is assigned to a self-diastereomer. The relationship between diastereomers and constitutional isomers have been also discussed.
2003 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The versatility of the USCI (unit-subduced-cycle-index) approach is demonstrated in characterizing octahedral complexes. Edge configurations on a regular octahedron have been combinatorially enumerated by the PCI (partial-cycle-index) method, which is one of the four methods of the USCI approach. Thereby, a complete set of edge configurations has been obtained, where every edge configurations are classified by virtue of two criteria, i.e., the numbers of edges and the point-group symmetries. The latter criterion enables us to examine chiral and achiral edge configurations, where complementary configurations are discussed in terms of the subductions of coset representations.
[Abstract] Molecules of ligancy 4 that have been derived from an allene, an ethylene, a tetrahedral, and a square-planar skeleton have been investigated to show that their symmetries are dually and distinctly controlled by point groups and permutation groups. Insomuch as the point-group symmetry was exhibited to control the chirality/achirality of a molecule, sphericity in a molecule, and enantiomeric relationship between molecules (Fujita, S. {\em J. Am. Chem. Soc.}, {\bf 1990}, {\it 112}, 3390), the permutation-group symmetry has been now clarified to control the stereogenicity of a molecule, tropicity in a molecule, and diastereomeric relationship between molecules. To characterize permutation groups, proper and improper permutations have been defined by comparing proper and improper rotations. Thereby, such permutation groups are classified into stereogenic and astereogenic ones. After a coset representation (CR) of a permutation group has been ascribed to an orbit (equivalence class), the tropicity of the orbit has been defined in term of the global stereogenicity and the local stereogenicity of the CR. As a result, the conventional ``stereogenicity'' has now been replaced by the concept {\em local stereogenicity} of the present investigation. The terms {\em homotropic, enantiotropic,} and {\em hemitropic} are coined and used to characterize prostereogenicity. Thus, a molecule is defined as being prostereogenic if it has at least one enantiotropic orbit. Since this definition has been found to be parallel with the definition of prochirality, relevant concepts have been discussed with respect to the parallelism between stereogenicity and chirality in order to restructure the theoretical foundation of stereochemistry and stereoisomerism. The derivation of the skeletons has been characterized by desymmetrization due to the subduction of CR's. The Cahn-Ingold-Prelog (CIP) system has been disscussed from the permutational point of view to show that it specifies diastereomeric relationships only. The apparent specification of enantiomeric relationships by the CIP system has been shown to stem from the fact that diastereomeric relationships and enantiomeric ones overlap occasionally in case of tetrahedral molecules.
[Abstract] A graphical method of generating one- and (some) two-dimensional characters ($\Gamma$) has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G_{i})] = {h_{1}, ..., h_{d-1},h_{d}} (d = |G|/|G_{i}|) has been generated from a regular body of G so that it has been governed by the coset representation G(/G_{i}). The homomer set has been reduced into a reduced homomer set H^{'}[$\Gamma$] = {h_{1}, ..., h_{d-1}}, where we have placed h_{d} = -(h_{1} + ... + h_{d-1}) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set H^{'}[$\Gamma$] has graphically generated a one- or (some) two-dimensional character ($\Gamma$). The versatility of the graphical method has been tested by using C_{3v}, D_{2h}, C_{2h}, C_{2v}, D_{3h}, and C_{3h} as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., \Gamma = G(/G_{i}) - G(/G).
2004 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The close relationship between stereogenicity and chirality has frequently caused serious confusion in the stereochemistry of organic molecules and inorganic complexes. To clarify the entangled relationship between them, we have proposed the concept of ``holantimer''. In addition, we have newly defined the concept of ``stereoisogram'' in order to correlate a set of stereoiomers based on holantimeric and enantiomeric relationships. These concepts have been applied to tetrahedral organic molecules as well as to square-planar inorganic complexes. The stereogenicity characterized by a stereoisogram has been called ``RS-stereogenicity'', which has been clarified to be a property that should be correlated to chirality. The stereoisograms of tetrahedral molecules have been examined and classified into five types, i.e., Type I (chiral/RS-stereogenic), Type II (chiral/RS-astereogenic), Type III (chiral/RS-stereogenic), Type IV (achiral/RS-astereogenic), and Type V (achiral/RS-stereogenic), where RS descriptors are concluded to be specified in cases of Types I, III, and V. On the other hand, the stereoisograms of square-planar complexes have been classified to two types, i.e., Type II and Type IV. As a result, the confusion on the RS-nomenclature has been concluded to appear within the RS-stereogenic relationships. Such a new viewpoint of stereogenicity and chirality as described in the present paper provides us with a methodology for restructuring stereochemistry.
[Abstract] To discuss the difference between stereogenicity and chirality, we propose the concept of RS-stereoisomeric groups. By starting from this concept, we have further proposed the concepts of holantimers, stereoisograms, and RS-stereogenicity. Thereby, we have clarified that the concept of RS-stereogenicity, but not the conventional stereogenicity, is closely related to chirality. Thus, five RS-stereogenicity types are defined and examined to discuss the difference between stereogenicity and chirality. Combinatorial enumerations have been also studied by considering the RS-stereogenicity.
[Abstract] Coset algebaic theory developed by Fujita (Shinsaku Fujita, {\em Symmetry and Combinatorial Enumeration in Chemistry}, Springer-Verlag, Berlin Heidelbelg (1991)) is studied by using (colored) graphs to model coset representations, marks, characters, and group subductions.
[Abstract] Topicity terms for stereochemical relationships (homotopic, enantiotopic, diastereotopic, etc.) and topicity terms for stereochemical attributes (chirotopic and achiotopic) have been combined to discuss the stereochemistry of tetrahedral molecules. Stereochemical discussions due to such combined usage have exhibited complicated features that would cause misunderstanding or confusion. On the other hand, 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 such stereochemical discussions. Sphericity indices have been defined and applied to examining the existence of derivatives.
[Abstract] The concept of doubly-colored graphs is proposed to model subductions of coset representations, double cosets, and unit subduced cycle indices, which have been mathematically formulated in coset algebaic theory developed by Fujita (Shinsaku Fujita, {\em Symmetry and Combinatorial Enumeration in Chemistry}, Springer-Verlag, Berlin Heidelbelg (1991)).
[Abstract] Chirality, RS-stereogenicity, stereogenicity, and isoskeletal isomerism for allene derivatives have been comprehensively discussed, where we have proposed a new terminology for groups (point groups, RS-permutation groups, RS-stereoisomeric groups, stereoisomeric groups, and isoskeletal groups) as well as a new terminology for isomers (enantiomers, holantimers, RS-diastereomers, diastereomers, and isoskeletal isomers). In the case of allene derivatives, RS-stereoisomeric groups have been clarified to coincide with stereoisomeric groups so that diastereomers are identical with RS-diastereomers. We have proposed the concept of {\em stereoisogram set} in oder to discuss the relationship between (RS-)diastereomers and isoskeletal isomers. As for allene derivatives, each stereoisogram set contains three stereoisograms to represent isoskeletal isomerism. Each of the three stereoisograms has been categorized into five stereogenicity types (Types I--V) so that the combination of such stereogenicity types can be used to characterize the isomerism of allene derivatives. The term ``pseudoasymmetry'' for allene derivatives has been discussed in terms of stereoisograms.
[Abstract] The hierarchy of point groups, RS-stereoisomeric groups, stereoisomeric groups, and isoskeletal groups is discussed to comprehend the chirality, RS-stereogenicity, stereogenicity, and isoskeletal isomerism for ethylene derivatives. The RS-stereoisomeric groups for ethylene derivatives have been clarified not to coincide with their stereoisomeric groups, so that diastereomers (E/Z-isomers) are not identical with RS-diastereomers. To discuss the relationship among RS-diastereomers, m-diastereomers, and isoskeletal isomers, we have proposed the concepts of {\em extended stereoisograms} and {\em extended stereoisogram sets}, where the term ``m-diastereomers'' is coined to show its difference from the traditional term ``diastereomer''. Thereby, ethylene derivatives are classified into Types II-II/II-II/II-II, IV-IV/IV-IV/IV-IV, etc. on the basis of relevant stereoisograms (Types I to V). The stereoisomerism of ethylenes has been concluded to be treated in terms of m-diastereomers characterized by the E/Z-nomenclature, but not to be treated in terms of RS-diastereomers characterized by the RS-nomenclaure.
[Abstract] The concepts of holantimer and stereoisogram are applied to comprehensive discussions on the term ``pseudoasymmetry'', where the concept of RS-stereogenicity is used as a more definite concept than usual stereogenicity. Thereby, three relationships contained in each stereoisogram can be definitely specified: an enantiomeric relationship is related to chiral/achiral, an RS-diastereomeric relationship is related to RS-stereogenic/RS-astereogenic, and a holantimeric relationship is related to scleral/ascleral, which is coined to keep the terminology in a balanced fashion. Such stereoisograms are classified into five types (Types I--V) by virtue of the three relationships. Among them, Type I, III, and V are selected as a set of RS-stereogenic units: chiral/ascleral RS-stereogenic unit (or Type I unit), chiral/scleral RS-stereogenic unit (or Type III unit), and achiral/scleral RS-stereogenic unit (or Type V unit). Thereby, the term ``pseudoasymmetric stereogenic units'' should be replaced by the term ``achiral/scleral RS-stereogenic units'' (or ``Type V units'').
2005 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Polya's theorem has been concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly. In order to take account of chiral ligands along with achiral ones, coset representations (CRs) for cyclic subgroups have been examined to classify permutations of the CRs into proper and improper elements. As a result, a k-cycle contained in each permutation has been classified into an enantiospheric, homospheric, or hemispheric one. Thereby, sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. On the basis of the sphericity indices, cycle indices with chirality fittingness (CI-CFs) have been defined in place of Polya's cycle indices. The CI-CFs have been proved to be capable of enumerating of stereoisomers with chiral and achiral ligands. Their capabilities have been confirmed by using allene derivatives as examples.
[Abstract] In order to enumerate nonrigid isomers, we have proposed the proligand approach, where extended sphericity indices of k-cycles have been defined according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle. Then, cycle indices with chirality fittingness (CI-CFs) have been defined so as to enumerate nonrigid stereoisomers with chiral and achiral ligands. Results of the proligand approach using tetramethylmethane as an example have been compared with those based on Polya's corona. Thereby, Polya's corona is concluded to be concerned with graphs, but not with chemical structures, where it is incapable of treating chiral ligands properly.
[Abstract] The hierarchy of groups for square-planar complexes is determined to be: point groups = $RS$-stereoisomeric groups $\subset$ stereoisomeric groups = isoskeletal groups. The $RS$-nomenclature is not effective to name square-planar complexes, because the $RS$-stereoisomeric groups coincide with the point groups so that all square-planar complexes are determined to be $RS$-astereogenic. There exist no isoskeletal isomers for square-planar complexes, because the isoskeletal groups coincide with the stereoisomeric groups. To discuss the stereogenicity, we propose the concept of {\em extended stereoisogram}, which contains three degenerate stereoisograms. Thereby, square planar-complexes are classified into Types II-II-II, IV-IV-IV, etc. on the basis of relevant stereoisograms (Types I to V). The number 3 of the degenerate stereoisograms in an extended stereoisogram means that square-planar complexes cannot be named by a dichoromus nomenclature such as the $RS$-nomenclature and the $E$/$Z$-nomenclature.
[Abstract] A proof for the existence of five stereogenicity types, which has once been demonstrated intuitively by means of stereoisograms, is mathematically given on the basis of the five factor groups derived from $RS$-stereoisomeric groups. These factor groups are proved to correspond to five types of subgroups appearing in $RS$-stereoisomeric groups. Thereby, the relationship between the concept of chirality and the concept of stereogenicity is clarified after the latter concept is replaced by a more specified concept, i.e., $RS$-stereogenicity. Moreover, the concepts of holantimers and $RS$-diastereomers gain a sound mathematical basis.
[Abstract] The desymmetrization of a regular tetrahedral molecule is discussed by emphasizing the importance of orbits, which appear concurrently in the molecule as equivalence classes of objects such as atoms (vertices), bonds, edges (hypothetical lines containing two atoms), faces (hypothetical planes containing three or more atoms), bond angles, and any other segments. Each orbit is ascribed to a sphericity index, which is determined by the sphericity and the size of the orbit. Concurrent desymmetrizations of such orbits are characterized by unit subduced cycle indices with chirality fittingness (USCI-CF), which are defined as the product of such sphericity indices. Thereby, the validity of models based on the regular tetrahedron is proved systematically.
[Abstract] Molecular symmetries based on a tetrahedral model are discussed in terms of subductions of coset representations, the data of which are obtained manually (non-algebraically) by examining the local symmetries appearing in the orbits of objects, e.g., atoms (vertices), bonds, edges (hypothetical lines containing two atoms), faces (hypothetical planes containing three or more atoms), and bond angles. Because the resulting subduction data are equivalent to those derived algebraically, the former are proved to be also effective in the application of the methods developed previously for the latter algebraically-obtained subduction data. This merit has been examined by tasks for finding molecules matched and mismatched to local symmetries, so that the manual derivation of subduction data is proved to be a novel way for understanding and teaching the desymmetrization of the tetrahedron in introductory courses of organic stereochemistry.
[Abstract] Stereogenicity and chirality of allene derivatives are discussed by means of the concepts of holantimers and stereoisograms. The importance of $RS$-stereogenicity, which is characterized as a kind of stereogenicity by a stereoisogram, is stressed to understand the basis of the $RS$-nomenclature. Thereby, three types of diastereomers, i.e., $RS$-diastereomers, holantimers, and traditional diastereomers are discussed. The traditional definition ``diastereomers are defined as stereoisomers that are not enantiomers'' is concluded to be an oversimplified dichotomy. The stereoisograms of allene derivatives are classified into five types, i.e., Type I (chiral/$RS$-stereogenic), Type II (chiral/$RS$-astereogenic), Type III (chiral/$RS$-stereogenic), Type IV (achiral/$RS$-astereogenic), and Type V (achiral/$RS$-stereo\-genic). Then chirality axes are concerned with Types I and III (chiral/$RS$-stereogenic), while stereogenic axes are concerned with Types I, III, and V (chiral and achiral/$RS$-stereogenic). These types of allene derivatives are compared with those of tetrahedral molecules so as to give a logical framework for discussing $RS$-stereogenicity and chirality.
[Abstract] The XyMTeX2PS system for typesetting chemical documents having structural formulas has been developed to cover both traditional printing and Internet communication. The system is capable of providing chemical documents as PostScript files of high quality. The PostScript files can be converted into PDF files, which serves as a key to cover both of the fields, where more elaborate stereochemical expressions such as wedged bonds are available.
[Abstract] The XyM2Mol system, which consists of the XyM2Mol application and the XyM2Mol applet, is developed to convert XyM-notation codes into connection tables. Thereby, the structural data by XyM Notation become applicable to a wide variety of chemical applications through such connection tables.
[Abstract] Fujita's proligand method is applied to the enumeration of ethane derivatives, where the counting of stereoisomers of tartaric acids is examined in detail as a probe for testing the versatility of the method. The cycle index with chirality fittingness (CI-CF) for enumerating ethane derivatives is obtained by Fujita's proligand method and compared with the CI-CF derived alternatively by the direct calculation of permulations of substitution positions. The two CI-CFs are identical with each other so that the methodology underlying in Fujita's method is demonstrated in a concrete fashion. The enumeration results are compared with those derived by P\'{o}lya's corona. Fujita's proligand method is shown to be capable of enumerating stereoisomers, whereas P\'{o}lya's corona is concluded to enumerate graphs, but not stereoisomers. The conceptually change from graphs to three-dimensional chemical structures is discussed, where the superiority of Fujita's proligand method is demonstrated.
[Abstract]
[Abstract] The present series is devoted to a diagrammatical introduction to the USCI (unit-subduced-cycle-index) approach developed by Fujita (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag, 1991). In Part 1 of this series, intramolecular stereochemistry is discussed by emphasizing orbits as sets of symmetry-equivalent objects. In particular, concurrent appearance of orbits of various kinds in a molecule is discussed diagrammatically, where any orbits are shown to be controlled by three kinds of sphericity indices ($a_{d}$, $c_{d}$, and $b_{d}$) correlated to coset representations (CRs). Derivation of molecules of given symmetries is discussed in terms of concurrent desymmetrization of orbits, where USCI-CFs (unit subduced cycle indices with chirality fittingness) are obtained diagrammatically as products of sphericity indices. The concurrent behaviors of orbits are explained by using a regular body, the positions of which are segmented in terms of segmentation patterns so as to give segmented regular bodies. Such segments are studied as models of ligands or proligands so that segmented regular bodies can be regarded as models of three-dimensional molecules (stereoisomers). The segmented regular bodies are used to generate CRs and to derive subductions of CRs diagrammatically. The generality of the procedure is confirmed so as to be capable of generating the subduction table, the USCI-CF table, the USCI table, and the mark table of D_{2d}-point group, which have been alternatively obtained and used in the Fujita's USCI approach. Diagrammatical correspondence between segments and cosets are examined in detail so that the relationship between subduction of CRs and double cosets is clearly demonstrated. Several terms (e.g., regular bodies, segments, segmentation patterns, tansformulas, and assemblies of transformulas} are introduced in order to give succinct but strict foundations to the present diagramatical approach. It is concluded that any symmetrical properties appear in regular bodies.
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[Abstract] The extended sphericity indices of k-cycles, which were defined in Part 2 of this series (S. Fujita, Theor. Chem. Acc., Online: http:// www.springerlink. com/index/10.1007/s00214-004-0606-z) according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle, are further extended to prove more general theorems for enumerating nonrigid stereoisomers with rotatable ligands. One of the extended points is the use of different sets of sphericity indices to treat one or more orbits contained in skeletons and ligands. Another one of the extended points is to take account of chirality in proligands and sub-proligands, the latter of which are introduced to consider further inner structures of ligands. Two theorems for enumerating nonrigid stereoisomers are proved by adopting two schemes of derivation of them, i.e., the scheme ``positions of a skeleton --- proligands --- ligands (positions of a ligand --- sub-proligands)'' and the scheme ``(positions of a skeleton --- proligands --- ligands (positions of a ligand)) --- sub-proligands''. The theorems are applied to the stereoisomerism of trihydroxyglutaric acids. Thereby, it is demonstrated what Polya's theorem and other previous methods are deficient in, when applied to the enumeration of stereoisomers.
[Abstract] Long-standing confusion on the term ``prochirality'' in stereochemistry has been analyzed by a critical review on the definitions described in the IUPAC recommendations 1996 (Pure Appl. Chem., 1996, 68, 2193). Thereby, the confusion has been clarified to come from the fact that the scope and limitations of the terms ``enantiotopic'', ``diastereotopic'', and ``stereoheterotopic'' are not so fully specified to discriminate between prochirality and prostereogenicity. Entangled situations due to the confusion have been avoided by abstracting the concept of pro-RS-stereogenicity from the conventional ``prostereogenicity'' on the same line as the concept of RS-stereogenicity was separated from the conventional ``stereogenicity'' (S. Fujita, J. Org. Chem., 2004, 69, 3158). This abstraction, which has been based on stereoisograms and RS-stereoisomeric groups, has provided us with a systematic approach for investigating the relationship between the prochirality and the pro-RS-stereogenicity. Thus, the prochirality and the pro-RS-stereogenicity can be discussed in terms of a common theoretical framework, i.e., coset representations of RS-stereoisomeric groups, where the conventional terms on topicities are replaced by unambiguous terms on sphericities (homospheric, enantiospheric, and hemispheric) and RS-tropicities (RS-homotropic, RS-enantiotropic, and RS-hemitropic). Moreover, the pro-RS-stereogenicity defined in the present paper has been correlated to the capability of designating pro-R/pro-S descriptors without any ambiguity. Thereby, the long-standing confusion on the term ``prochirality'' has been settled completely.
[Abstract] The present series is devoted to a diagrammatical introduction to the USCI (unit-subdu-ced-cycle-index) approach developed by Fujita (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag, 1991). In Part 2 of this series, intermolecular stereochemistry (stereoisomerism) is discussed by the concept of assembles of transformulas. Transformulas that are generated from a skeleton of G-symmetry having |G| vertices (i.e., a regular body representing the coset representation (CR) G(/C_{1})) are equivalent under G so as to construct an orbit governed by the CR G(/C_{1}). An assembly of H-symmetry (H\subsetG) is defined as a set of such transformulas as fixed by the action of H. The set of equivalent H-assemblies constructs an orbit of H-assemblies, which is governed by the CR G(/H). Each H-assembly corresponds to an H-molecule derived from the skeleton of G-symmetry. Thus the CR G(/H) is concluded to control the intermolecular stereochemistry (stereoisomerism). Thereby, subduction tables, USCI-CF tables (tables of unit-subduced-cycle-index with chirality fittingness), USCI tables (tables of unit-subduced-cycle-index without chirality fittingness), and mark tables for D_{2d} (as an example of G) are obtained in an alternative way to the method described in Part 1. The parallelism between Part 1 and Part 2 is clearly demonstrated by defining {\em a mandala} as a hypothetical structure (a nested regular body) in which the |G| transformulas generated as above from a regular body by the action of G are placed on the vertices of a regular body. Assembled mandalas and reduced mandalas are defined to integrate intermolecular and intramolecular stereochemistries.
[Abstract] The present series is devoted to a diagrammatical introduction to the USCI (unit-subduced-cycle-index) approach developed by Fujita (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag, 1991). In Part 3 of this series, intramolecular stereochemistry (Part 1) and intermolecular stereochemistry (Part 2) are integrated in terms of {\em reduced mandalas} so as to provide versatile tools for enumerating stereoisomers, i.e., the SCI (subduced-cycle-index) method and the PCI (partial-cycle-index) method. The methodology which these methods depend upon is diagrammatically demonstrated, where two modes of views (i.e., ``column view'' and ``row view'') are applied to the mark table of D_{2d} in stereoisomer enumeration based on an allene skeleton of D_{2d}-symmetry.
[Abstract] Three methods and their extended versions for enumerating stereoisomers, which have been developed by modifying or simplifying Fujita's USCI (unit-subduced-cycle-index) approach based on the concept of {\em sphericities of orbits} in order not to take account of symmetry itemization, are applied to the enumeration problem of ethane and propane derivatives. The proligand method and its extended version based on the concept of {\em sphericities of cycles} are also applied to the same enumeration problems. These results are compared with the results based on P\'{o}lya's theorem (and P\'{o}lya's corona). Thereby, it is shown that P\'{o}lya's theorem enumerates chemical compounds as graphs, not as stereoisomers (3D chemical structures) if all of the permutations corresponding to proper and improper rotations are adopted. Moreover, if the permutations corresponding to proper rotations are adopted, P\'{o}lya's theorem enumerates chemical compounds as chiral ones, where enantiomeric relationship and achiral nature (i.e., self-enantiomeric relationship) are not characterized properly. The two types of applications of P\'{o}lya's theorem do not take account of improper rotations properly. Thereby, what P\'{o}lya's theorem is deficient in is concluded to be the concept of sphericity.
2007 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The concepts of double coset representations and sphericities of double cosets are proposed to characterize stereoisomerism, where double cosets are classified into three types, i.e., homospheric double cosets, enantiospheric double cosets, or hemispheric double cosets. They determine modes of substitutions (i.e., chirality fittingness), where homospheric double cosets permit achiral ligands only; enantiospheric ones permit achiral ligands or enantiomeric pairs; and hemispheric ones permit achiral and chiral ligands. The sphericities of double cosets are linked to the sphericities of cycles which are ascribed to right coset representations. Thus, each cycle is assigned to the corresponding sphericity index (a_{d}, c_{d}, or b_{d}) so as to construct a cycle indices with chirality fittingness (CI-CFs). The resulting CI-CFs are proved to be identical with CI-CFs introduced in Fujita's proligand method (S.\ Fujita, Theor. Chem. Acc., 113 (2005) 73--79 & 80--86). The versatility of the CI-CFs in combinatorial enumeration of stereoisomers is demonstrated by using methane derivatives as examples, where the numbers of achiral plus chiral stereoisomers, those of achiral stereoisomers, and those of chiral stereoisomers are calculated separately by means of respective generating functions.
[Abstract] Fujita's proligand method, which was originally formulated by using the symmetrical properties of cyclic subgroups (Fujita, S. (2005) Theor. Chem. Acc., 113 73--79, 80--86), has been alternatively formulated in terms of the concept of mandalas proposed in Part 2 of this series (Fujita, S. (2006) MATCH Commun. Math. Comput. Chem., 55, 5--38). A set of assemblies of K-symmetry in a mandala of G-symmetry has been characterized by the left coset representation G(/K), where achiral assemblage and chiral assemblage have been discussed in terms of the chirality/achirality of the group K. Each K-assembled mandala has been shown to correspond to one stereoisomer of \bfK-symmetry, i.e., an achiral molecule or a pair of enantiomers. The alternative formulation of Fujita's proligand method has been accomplished by comparing the number of fixed assemblies per stereoisomer with the number of fixed assemblies per permutation. Thereby, stereoisomers can be enumerated in an itemized fashion, i.e., the numbers of achiral plus chiral stereoisomers, of achiral stereoisomers, and of chiral stereoisomers. Deficiency of Polya's theorem in stereoisomer enumeration and merits of Fujita's proligand method have been demonstrated by using allenes and prismanes as examples.
[Abstract] Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita's proligand method formulated in Parts 1 to 3 of this series (Fujita S (2005) Theor. Chem. Acc. 113:73--79, 113:80--86, Fujita S (2006) Theor. Chem. Acc. 115:37--53). By starting from the concepts of proligand and promolecule introduced previously (Fujita S (1991) Tetrahedron, 47:31--46), a {\em planted promolecule} is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands. Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating functions are based on cycle indices with chirality fittingness (CI-CFs), which are composed of three kinds of sphericity indices (SIs), i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles. For the purpose of evaluating c_{d} recursively, the concept of {\em diploid} is proposed, where the nested nature of c_{d} is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x^{2}), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem initiated by a mathematician Cayley (Cayley A (1874) Philos. Mag. 47(4):444--446) has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.
[Abstract] Three-dimensional trees (3D-trees), which are defined as a 3D version of trees, are enumerated by Fujita's proligand method formulated in Part 1 to Part 3 of this series (Fujita S (2005) Theor. Chem. Acc. 113:73--79, 113:80--86, Fujita S (2006) Theor. Chem. Acc. 115:37--53). Such 3D-trees are classified into centroidal and bicentroidal 3D-trees, which correspond to respective promolecules having proligands as substituents. In order to enumerate such centroidal and bicentroidal 3D-trees, cycle indices with chirality fittingness (CI-CFs) are formulated as being composed of three kinds of sphericity indices, i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles. The CI-CFs are capable of giving itemized results with respect to chiral and achiral 3D-trees so that they are applied to derive functional equations (a(x), c(x^{2}), and b(x)). The generating functions of planted 3D-trees, which are formulated and calculated elsewhere, are introduced into such functional equations. Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated and collected up to carbon content 20 in a tabular form. Now, the enumeration problem initiated by a mathematician Cayley (Cayley A (1874) Philos. Mag. 47(4):444--446) has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.
[Abstract] Monosubstituted alkanes as stereoisomers, not as constitutional isomers, are regarded as planted three-dimensional (3D) trees, which are defined as a 3D extension of planted trees (graphs). They are thus recognized as 3D-objects (planted promolecules) and enumerated by Fujita's proligand method (Fujita S (2005) Theor. Chem. Acc. 113:73--79, 113:80--86, Fujita S (2006) Theor. Chem. Acc. 115:37--53). By starting from three kinds of sphericity indices, i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles, cycle indices with chirality fittingness (CI-CFs) are obtained to enumerate planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers. Functional equations (a(x), c(x^{2}), and b(x)) for recursive calculations are derived from the CI-CFs and programmed in three ways by means of the Maple programming language. The three recursive procedures for calculating the numbers of planted 3D-trees are executed to give identical results, which are collected up to 100 carbon content in a tabular form. The results are compared with the enumeration of planted trees (as graphs).
[Abstract] Monosubstituted alkanes are counted as stereoisomers by means of Fujita's proligand method (Fujita S (2005) Theor. Chem. Acc. 113:73--79, 113:80--86, Fujita S (2006) Theor. Chem. Acc. 115:37--53), where the numbers of primary, secondary, and tertiary ones are calculated after deriving respective functional equations. The procedures of counting are programmed by means of the Maple programming language. They are executed and the results collected up to carbon content 100 in a tabular form. By omitting the sphericities of the recursive functions a(x), c(x^{2}), and b(x) so as to give a single dummy variable r(x), such functional equations with sphericity are transformed into Polya's functional equations without sphericity, which are applied to the enumeration of primary, secondary, and tertiary monosubstituted alkanes as graphs (chemically, constitutional isomers). The results of Fujita's proligand method are compared with those based on Polya's theorem in connection with several cases of pseudoasymmetry.
[Abstract] Alkanes are counted as 3D-trees or stereoisomers by means of Fujita's proligand method (S. Fujita, {\em Theor. Chem. Acc.}, {\bf 113}, 73--79, 80--86 (2005); {\bf 115}, 37--53 (2006)), where the 3D-trees are categorized into balanced and unbalanced 3D-trees according to the presence or absence of a balance-edge. Such balanced and unbalanced 3D-trees are enumerated by presuming that they are dually recognized as uninuclear and binuclear 3D-trees, where a tetrahedral skeleton of T_{d}-symmetry is used to generate the uninuclear 3D-trees, while a binuclear skeleton of D_{infty h}-symmetry is examined to generate the binuclear 3D-trees. The values for binuclear 3D-trees are regarded as contaminants in the enumeration of uninuclear 3D-trees so that the subtraction of the contaminants from the latter enumeration leaves unbalanced 3D-trees to be counted. The enumeration of balanced 3D-trees is conducted directly by using the binuclear skeleton of D_{infty h}-symmetry. The enumeration is based on functional equations derived from cycle indices with chirality fittingness (CI-CFs), where the functions a(x^{d}), c(x^{d}), and b(x^{d}) (or their modifications) are substituted for three kinds of sphericity indices (SIs), i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles. Thus, respective functional equations for counting alkanes as well as for itemizing them into achiral and chiral ones are derived by starting from recursive functional equations for counting alkyl ligands as planted 3D-trees. They are programmed by means of the Maple programming language and executed to give respective stereoisomer numbers, which are collected in tabular forms up to carbon content 100.
[Abstract] Alkanes as stereoisomers are categorized according to the dichotomy between centroidal and bicentroidal three-dimensional trees (3D-trees), which are distinctly counted by means of Fujita's proligand method (S. Fujita, {\em Theor. Chem. Acc.}, {\bf 113}, 73--79, 80--86 (2005); S. Fujita, {\em Theor. Chem. Acc.}, {\bf 115}, 37--53 (2006)). The centroidal 3D-trees are enumerated by using a tetrahedral skeleton of T_{d}-symmetry, while the bicentroidal 3D-trees are enumerated by using a binuclear skeleton of D_{infty h}-symmetry. The enumerations based on the two skeletons are conducted by means of respective functional equations derived from cycle indices with chirality fittingness (CI-CFs), where the functions a(x^{d}), c(x^{d}), and b(x^{d}) are substituted for three kinds of sphericity indices (SIs), i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles. The 3D-trees are alternatively counted by using the dichotomy between balanced and unbalanced 3D-trees. The two dichotomies are combined to categorize the 3D-trees into three categories, i.e., centroidal \& unbalanced 3D-trees, bicentroidal & unbalanced 3D-trees, and bicentroidal & balanced 3D-trees, which are counted distinctly by using respective functional equations. These functional equations are programmed by means of the Maple programming language and executed to give respective stereoisomer numbers, which are collected in tabular forms up to carbon content 100. These numbers of stereoisomers obtained by Fujita's proligand method are compared with those of constitutional isomers (graphs) derived by using Polya's theorem.
[Abstract] Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the USCI (unit-subduced-cycle-index) approach (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\backslash)G and left coset representations (LCR) G(/H) for the group G and its subgroup H, a regular body of G-symmetry is defined as a diagrammatical expression for a right regular representation (C_{1}\backslash)G, which is an extreme case of RCRs. The |G| substitution positions of the regular body as a reference are numbered in accord with the numbering of the elements of G and segmented into |G|/|H| of H-segments, which are governed by the RCR (H\backslash)G. By regarding each H-segment as a substitution position, the H-segmented regular body is reduced into a reduced regular body, which can be regarded as a secondary skeleton for generating a molecule. The reference regular body (or H-segmented one) is operated by every symmetry operations of G to generate regular bodies (or H-segmented ones), which are placed on the vertices of a hypothetical regular body of G-symmetry. The resulting diagram (a nested regular body) is called a mandala (or a reduced mandala), which is a diagrammatical expression for specifying the G-symmetry of a molecule. The effect of a K-subduction on the regular bodies of a mandala (or a reduced mandala) results in the K-assemblage of the mandala (or the reduced mandala), where the resulting K-assemblies governed by the LCR G(/K) construct a |G|/|K|-membered orbit, which corresponds to a molecule of K-symmetry. The sphericity of the RCR (or the LCR) is used to characterize symmetrical properties of substitution positions and those of stereoisomers. The fixed-point vector for each mandala (or reduced mandala) in terms of row view and the number of fixed points of K-assembled mandalas (or K-assembled reduced mandalas) in terms of column view are compared to accomplish combinatorial enumeration of stereoisomers. The relationship between a mandala and a reordered multiplication table is discussed.
[Abstract] Alkanes are counted as stereoisomers or three-dimensional trees (3D-trees) by means of Fujita's PCI (partial-cycle-index) method (Fujita, S., {\em Chem. Inf. Comput. Sci.}, {\bf 2000}, {\it 40}, 135--146; Fujita, S., {\em Bull. Chem. Soc. Jpn.}, {\bf 2000}, {\it 73}, 329--339) after they are categorized according to the dichotomy between centroidal and bicentroidal 3D-trees. The centroidal alkanes are enumerated by using a tetrahedral skeleton of $\bfT_{d}$-symmetry under the criterion of defining such centroidal 3D-trees, where they are itemized in terms of the eleven subgroups of the $\bfT_{d}$-symmetry. On the other hand, the bicentroidal alkanes are enumerated by using a two-nodal skeleton belonging to the $\bfK{}$-symmetry, where they are itemized in terms of the five subgroups of the factor group $\bfK$ = $\bfD_{\infty h}/\bfC_{\infty}$. Both the enumerations are based on functional equations derived from partial cycle indices with chirality fittingness, where the component functions $a(x^{d})$, $c(x^{d})$, and $b(x^{d})$ (or their modifications) are substituted for three kinds of sphericity indices, i.e., $a_{d}$ for homospheric orbits, $c_{d}$ for enantiospheric orbits, and $b_{d}$ for hemispheric orbits. Respective functional equations based on the itemization by subgroups are programmed by means of the Maple programming language. The resulting programs are executed to give respective stereoisomer numbers up to carbon content 100, which are collected in tabular forms with subgroup itemization.
[Abstract] Promolecules are derived from a given skeleton by putting proligands (ligands with chirality/achirality only) on its substitution positions. They are regarded as RS-stereoisomers, which are characterized by the concepts of chirality, RS-stereogenicity, and sclerality. They are categorized into five types by means of RS-stereoisomeric groups: Type I (chiral/RS-stereogenic/ascleral), Type II (chiral/RS-astereogenic/scleral), Type III (chiral/RS-stereogenic/scleral), Type IV (achiral/RS-astereogenic/ascleral), and Type V (achiral/RS-stereogenic/scleral). They are counted under the action of the maximum point subgroup, the maximum RS-permutation subgroup, and the maximum ligand-inversion subgroup, which are subgroups of an RS-stereoisomeric group. After respective cycle indices with chirality fittingness are derived for Type I to Type V, the itemized numbers of promolecules are obtained as generating functions. The general method is applied to the enumerations of allene derivatives and of methane derivatives. The results are verified in comparison with manual enumerations reported previously.
2008 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Three-dimensional (3D) trees, which are defined as a 3D extension of trees, are enumerated by Fujita's proligand method (S. Fujita, Theor. Chem. Acc., 113 (2005) 73--79 & 80--86; 115 (2006) 37--53). Such 3D-trees are dually recognized as uninuclear promolecules and as binuclear ones. The 3D-trees regarded as uninuclear promolecules are enumerated to give the gross number of 3D-trees, which suffers from redundancy due to contaminants. To evaluate the number of such contaminants, the 3D-trees are alternatively enumerated as binuclear promolecules. Cycle indices with chirality fittingness (CI-CFs) composed of three kinds of sphericity indices (SIs), i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles are obtained for evaluating promolecules of the two kinds. The CI-CFs for the uninuclear promolecules and those for the related binuclear promolecules are compared in terms of the dichotomy between balanced 3D-trees and unbalanced 3D-trees. Thereby, the redundancy due to such contaminants is deleted effectively so as to give the net number of 3D-trees. The validity of this procedure is proved in three ways, all of which are based on the respective modes of the correspondence between uninuclear promolecules and binuclear ones. In order to enumerate 3D-trees by following this procedure, the CI-CFs are converted into functional equations by substituting the SIs for a(x^{d}), c(x^{d}), and b(x^{d}). Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated under various conditions and collected up to 20 carbon content in a tabular form. Now, the stereochemical problems (on the number of stereoisomers) by van't Hoff (J. H. van't Hoff, Archives Neerlandaises des Sciences Exactes et Naturelles, 9 (1874) 445--454) and by LeBel (J. A. LeBel, Bull. Soc. Chim. Fr. (2), 22 (1874) 337--347) and the enumeration problems (on the number of trees) by Cayley (A. Cayley, Philos. Mag., 47 (1874) 444--446), both initiated in the 1870s, have been solved in a common theoretical framework, which satisfies both chemical and mathematical requirements.
[Abstract] Monosubstituted alkanes as stereoisomers have been counted combinatorially by regarding them as nested planted promolecules, where the resulting numbers have been itemized with respect to carbon content (k), to the number of asymmetric carbons (l), as well as to the number of pseudoasymmetric carbons (m). To accomplish such itemization, the definitions of RS-stereogenic centers, asymmetric centers, and pseudoasymmetric centers have been discussed in detail. Each itemized number has been obtained as the coefficient of the term x^{k}y^{l}z^{m} appearing in a respective generating function, which has been derived by following Fujita's proligand method (S. Fujita, Theor. Chem. Acc. 2005, 113, 73--79, 113, 80--86; S. Fujita, Theor. Chem. Acc. 2006, 115, 37--53). The itemized values up to carbon content 30 have been listed in tabular forms, which are distinctively concerned with achiral stereoisomers, chiral stereoisomers, and constitutional isomers (graphs).
[Abstract] Asymmetric and pseudoasymmetric centers in alkanes (as tree-dimensional trees of degree 4) have been specified by means of newly-defined criteria based on three kinds of attributes of RS-stereoisomers (chirality, RS-stereogenicity, and sclerality), where the classification into five RS-stereoisomeric types (Types I to V) according to S. Fujita, J. Org. Chem., 69, 3158--3165 (2004), S. Fujita, MATCH Commun. Math. Comput. Chem., 54, 39--52 (2005), and S. Fujita, MATCH Commun. Math. Comput. Chem., 58, 611--634 (2007) plays an important role. Among three types of RS-stereogenic promolecules (Types I, III, and V), the central atom of each alkane as a promolecule of Types I and III (chiral/RS-stereogenic) is regarded as an asymmetric center, while the central atom of each alkane as a promolecule of Types V (achiral/RS-stereogenic) is regarded as a pseudoasymmetric center. The data of alkyl ligands, which have been recursively calculated by a personal computer and stored as the coefficients of the term x^{k}y^{l}z^{m} of generating functions involving carbon content (k), the number of asymmetric carbons (l), as well as the number of pseudoasymmetric carbons (m), have been used to count centroidal and bicentroidal alkanes on the basis of a tetrahedral skeleton. Each itemized number has been obtained as the coefficient of the term x^{k}y^{l}z^{m} appearing in a respective generating function, which has been derived by following Fujita's proligand method. The itemized values up to carbon content 30 have been listed in tabular forms, which are distinctively concerned with Type I, ..., or Type V as well as with achiral stereoisomers and chiral stereoisomers.
[Abstract] Itemized numbers of achiral and/or chiral monosubstituted alkanes (MSAs) as three-dimensio\-nal structures have been obtained by means of a newly-developed method based on Fujita's proligand method (S. Fujita, Theor. Chem. Acc. 2005, 113, 73--79, 113, 80--86; S. Fujita, Theor. Chem. Acc. 2006, 115, 37--53), where the itemization with respect to internal branching has been embodied by employing branching indicators (BIs), i.e., q for quaternary carbons, t for tertiary carbons, s for secondary carbons, and p for primary carbons. Each of the numbers appears as the coefficient of a branching monomial q^{nq} t^{nt} s^{ns} p^{np} which is contained in a generating function to characterize MSAs having n_{q} quaternary carbons, n_{t} tertiary carbons, n_{s} secondary carbons, and n_{p} primary carbons (k = n_{q} + n_{t} + n_{s} + n_{p}). By following Fujita's proligand method, functional equations for recursive calculations have been obtained to treat respective cases, where previous approaches without considering BIs have been extended so as to incorporate BIs with no violation of consistency. The resulting functional equations have been used to obtain generating functions which give the numbers of achiral MSAs, chiral MSAs, total (achiral and chiral) MSAs. They have been compared with the corresponding numbers of MSAs as graphs. Thereby, difference between stereoisomers (3D structures) and constitutional isomers (graphs) has been discussed on a quantitative basis.
[Abstract] Combinatorial enumeration of alkanes as three-dimensional structures has been investigated, where the degrees of internal branching have been taken into consideration by introducing branching indicators (BIs, i.e, q, t, s, and p). After generating functions for counting preformed alkyl moieties (PAMs) were calculated by following Fujita's proligand method (S. Fujita, Theor. Chem. Acc. 2005, 113, 73--79, 113, 80--86; S. Fujita, Theor. Chem. Acc. 2006, 115, 37--53), they have been introduced into functional equations for counting alkanes so as to give the corresponding generating functions, where such alkanes of carbon content k have been categorized into four cases in terms of centroidal/bicentroidal nature and achirality/chirality. Thereby, the generating functions of the respective cases have given the numbers of alkanes, where each number appears as the coefficient of the term called a branching monomial (BM) q^{nq} t^{nt} s^{ns} p^{np}, when such an alkane contains n_{q} quaternary carbons, n_{t} tertiary carbons, n_{s} secondary carbons, and n_{p} primary carbons (k = n_{q} + n_{t} + n_{s} + n_{p}). The results of the enumeration have been verified by drawing alkanes of several representative cases, where a mode of divergence, i.e., [n_{q}, n_{t}, n_{s}, n_{p}], has been used for the purpose of qualitative discussions. Previous approaches without considering BIs have been derived by disregarding the effect of internal branching. Moreover, the functional equations for counting 3D structures have been systematically reduced into those for counting graphs, where graph-reduction conditions have been formulated to rationalize the reduction processes. Thereby, Polya's theorem for counting graphs has been shown to be a special case of Fujita's proligand method for counting 3D structures. As a result, difference between stereoisomers (3D structures) and constitutional isomers (graphs) has been discussed by the common criterion due to BMs (or modes of divergence), which provides more detailed enumerations than previous enumerations due to carbon contents (or constitutions, or strictly speaking molecular formulas).
2009 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The versatility of stereoisograms has been discussed in order to comprehend terms related to stereoisomerism. Each stereoisogram characterizes three kinds of relationships (an enantiomeric, an RS-diastereomeric, and a holantimeric relationship), which collectively formulate RS-stereoisomerism. In contrast to the inclusion relationship of conventional usage, i.e.,
[Abstract] In contrast to the mathematical formulation reported in Fujita, S. Tetrahedron, 2006, 62, 691--705, the terms prochirality and pro-RS-stereogenicity have been alternatively formulated by emphasizing relational terms which specify relationships between two objects (ligands etc.), i.e., enantiotopic and RS-diastereotopic. By developing the substitution criteria for enantiotopicity and for RS-diastereotopicity on the basis of stereoisograms, the relational terms are linked to the corresponding terms for specifying RS-stereoisomerism: i.e., enantiomeric relationships for chirality, RS-diastereomeric relationships for RS-stereogenicity, and holantimeric relationships for sclerality. Then, enantiotopic, RS-diastereotopic, and holantitopic relationships are employed to specify prochirality, pro-RS-stereogenicity, and prosclerality. The concept of prochirality characterized by the term enantiotopic gives a sound basis to the capability of deriving chiral compounds selectively from achiral precursors. On the other hand, the concept of pro-RS-stereogenicity characterized by the term RS-diastereotopic gives a sound basis to the capability of giving pro-R/pro-S-descriptors. The substitution criteria are correlated to the previous mathematical formulation through the membership criteria which are defined on the basis of coset representations. The difference between ``stereoheterotopic'' of conventional usage and RS-stereoheterotopic of the present approach is discussed in connection with the differences between ``prostereoisomerism'' and the present term pro-RS-stereoisomerism as well as between ``diastereotopicity'' and the present term RS-diastereotopicity. Addition criteria for prochirality and for pro-RS-stereogenicity are also developed to discuss two faces of a carbonyl ligand.
[Abstract] A quadruplet of RS-stereoisomers appearing in a stereoisogram is categorized into either one of five types by means of chirality, RS-stereogenicity, and sclerality, i.e., Type I (chiral/RS-stereogenic/ascleral), Type II (chiral/RS-astereogenic/scleral), Type III (chiral/RS-stereogenic/scleral), Type IV (achiral/RS-astereogenic/ascleral), and Type V (achiral/RS-stereogenic/scleral). Each quadruplet is considered to be an entity to be counted just once, where the entity can be regarded as an equivalence class under the corresponding RS-stereoisomeric group G (e.g., T_{d\widetilde{\sigma}\widehat{I}} for a quadruplet of tetrahedral promolecules). To accomplish itemized enumeration of such quadruplets, three modes of action of G are discussed by considering three subgroups of index 2, i.e., the maximum point subgroup G_{C\sigma}, the maximum RS-permutation group G_{C\widetilde{\sigma}}, and the maximum ligand-inversion group G_{C\widehat{I}} (e.g., T_{d}, T_{\widetilde{\sigma}}, or T_{\widehat{I}} for a quadruplet of tetrahedral promolecules). Such a quadruplet consists of two E-pairs, each of which is defined as a pair of enantiomers (chiral promolecules) or a pair of self-enantiomers (an achiral promolecule); it consists of two D-pairs, each of which is defined as a pair of RS-diastereomers (RS-stereogenic promolecules) or a pair of self-RS-diastereomers (an RS-astereogenic promolecule); and it consists of two H-pairs, each of which is defined as a pair of holantimers (scleral promolecules) or a pair of self-holantimers (an ascleral promolecule). The two E-pairs, D-pairs, or H-pairs contained in each quadruplet are considered to construct an equivalence class of G and also an equivalence class of G_{C\sigma}, G_{C\widetilde{\sigma}}, or G_{C\widehat{I}}. Thereby, such equivalence classes are enumerated under G and G_{C\sigma} (or G_{C\widetilde{\sigma}} or G_{C\widehat{I}}) to give partially itemized generating functions, where the number of the equivalence classes can be regarded as the number of quadruplets to be counted. The results of the enumerations by means of E-pairs, D-pairs, and H-pairs are combined to accomplish itemized enumeration with respect to Types I--V. Polya's theorem is discussed as a special case of the present approach.
[Abstract] The concept of RS-stereoisomers is proposed as an intermediate concept between enantiomers and stereoisomers. Stereoisograms developed for examining relationships between RS-stereoisomers contain three kinds of relationships (enantiomeric, RS-diastereomeric, and holantimeric relationships). Because these relationships correspond to three attributes (chirality, RS-stereogenicity, and sclerality), a quadruplet of RS-stereoisomers can be regarded as equivalence classes. The intermediacy of such RS-stereoisomeric relationships provides a paradigm shift from the conventional terminology of stereochemistry to a new terminology based on the the concept of RS-stereoisomers. In addition, the recognition of a quadruplet of RS-stereoisomers as equivalence classes provides another paradigm shift on the basis of such equivalence classes. For example, the basis of the Cahn-Ingold-Prelog system for generating RS-descriptors is changed from the stereogenicity of conventional stereochemistry to the RS-stereogenicity specified by stereoisograms.
[Abstract] Primary, secondary, and tertiary monosubstituted alkanes are counted as three-dimen\-sional (3D) structures, where branching indicators (q for quaternary carbons, t for tertiary carbons, s for secondary carbons, and p for primary carbons) are introduced to evaluate the effect of internal branching. Each monosubstituted alkane of carbon content k, which is composed of n_{q} quaternary carbons, n_{t} tertiary carbons, n_{s} secondary carbons, and n_{p} primary carbons (k=n_{q} + n_{t} + n_{s} + n_{p}), is characterized by the product of branching indicators (named a branching monomial), q^{nq}t^{nt}s^{ns} r^{nr}x^{k}. The number of primary, secondary, or tertiary monosubstituted alkanes as 3D structures is calculated as the coefficient of the term q^{nq}t^{nt}s^{ns} r^{nr}x^{k} in a generating function, which is obtained by starting from functional equations involving branching indicators. Three kinds of sphericity indices, i.e., a_{d} for homospheric cycles, c_{d} for enantiospheric cycles, and b_{d} for hemispheric cycles, which have been developed by Fujita's USCI (unit-subduced-cycle-index) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, 1991) and modified by Fujita's proligand method (S. Fujita, Theor. Chem. Acc. 2005, 113, 73--79, 113, 80--86; (S. Fujita, Theor. Chem. Acc. 2006, 115, 37--53), are replaced by a(x^{d}, q^{d}, t^{d}, s^{d}, p^{d}), c(x^{d}, q^{d}, t^{d}, s^{d}, p^{d}), and b(x^{d}, q^{d}, t^{d}, s^{d}, p^{d}) so as to produce functional equations for recursive calculation. Respective functional equations for counting primary, secondary, and tertiary monosubstituted alkanes are derived and used to produce such generating functions. The respective results are collected in tabular forms up to carbon content 15, where the numbers of monosubstituted alkanes as 3D structures are further itemized into achiral and chiral ones. By omitting the sphericity concept, the functional equations for 3D structures are transformed into the counterparts for graphs, which are applied to the enumeration of primary, secondary, and tertiary monosubstituted alkanes as graphs (chemically, constitutional isomers) or as planted trees, more specifically speaking.
[Abstract] The present paper is devoted to clarify how asymmetric and pseudoasymmetric centers in monosubstituted alkanes participate in processes during which their three-dimensional (3D) structures are reduced into graphs. After categorized into five types (S. Fujita, J. Org. Chem., 2004, 69, 3158--3165), monosubstituted alkanes have been enumerated recursively by considering the numbers of asymmetric and pseudoasymmetric carbons as well as carbon content, where the recursive process adopted in the original program of Fujita's method (S. Fujita, Bull. Chem. Soc. Jpn., 2008, 81, 193--219; S. Fujita, MATCH Commun. Math. Comput. Chem., 2008, 59, 509--554) has been improved so as to rely on a more simplified structure of recursiveness. To compare between 3D structures and graphs, a graph-reduction condition has been employed. Thereby, the non-recursive functional equations have been reduced into such equations as relevant to graphs. The functional equations for 3D structures have been discussed from the present viewpoint of "3D structures with asymmetries and pseudoasymmetries", which has been compared with the viewpoint of "graphs with asymmetries". In particular, Polya's special questions described in Section 60 of his famous article (G. Polya, Acta Math., 1937, 68, 145--254) have been revisited and discussed critically by comparing the two viewpoints. A stereochemical convention that the number of stereoisomers with l asymmetric carbons is equal to 2^{l} has been examined also by comparing the two viewpoints.
[Abstract] Alkanes have been enumerated from two viewpoints, i.e., the first viewpoint of ``3D structures with asymmetries and pseudoasymmetries'' and the second viewpoint of ``graphs with asymmetries''. After alkanes are categorized into Types I--V by means of stereoisograms (S. Fujita, J. Org. Chem., 2004, 69, 3158--3165), non-recursive functional equations for counting alkanes of Types I--V (as 3D structures) are derived by using three functional equations i.e., a(x,y,z) for enumerating achiral ligands, c(x^{2},y^{2},z^{2})$ for enumerating diploids, and b(x,y,z) for enumerating achiral and chiral ligands as steric isomers. The resulting generating functions give isomer numbers from the first viewpoint, where each coefficient of the term x^{k}y^{l}z^{m} represents the number of respective objects with k carbons, l asymmetric carbons, and $m$ pseudoasymmetric carbons. On the other hand, the non-recursive functional equations for counting Types I--V are reduced into the corresponding ones for counting alkanes as graphs, where a graph-reduction condition represented by a(x,y,z) = c(x,y,z) = b(x,y,z) = r(x,y) is employed. During this reduction, the non-recursive functional equations for counting Types II, III, and V vanish to zero. In particular, the variable $z$ for characterizing pseudoasymmetric carbons disappear during the graph-reduction process, while the variable $y$ for characterizing asymmetric carbons remains. The resulting non-recursive functional equations give isomer numbers from the second viewpoint of ``graphs with asymmetries''. The data from the two viewpoints are systematically compared with each other so that the fate of asymmetries and pseudoasymmetries are demonstrated. A stereochemical convention that the number of stereoisomers with l asymmetric carbons is equal to $2^{l}$ has been examined by comparing the two viewpoints.
[Abstract] The foundation of the Cahn-Ingold-Prelog (CIP) system has been redefined by employing the concept of RS-stereogenicity derived from stereoisograms for analyzing stereoisomerism, where R- and S-stereodescriptors are pairwise given to two molecules in an RS-diastereomeric relationship. The conventional definitions of RS-stereodescriptors which are based on the original term “chirality” (along with enantiomeric and diastereomeric relationships) and on the revised term “stereogenicity'” are altogether abandoned. Thereby, the present foundation is shown to be distinct from and to be more consistent than the conventional one which assigns R- and S-stereodescriptors to two molecules in an enantiomeric relationship as well as in a diastereomeric relationship. In order to harmonize the present foundation with the conventional one and to leave conventional results unchanged as far as possible, RS-stereodescriptors given originally to a pair of RS-diastereomers are considered to be translated into RS-stereodescriptors for a pair of enantiomers, where the concept of chirality-faithfulness of priority sequences has been proposed to testify whether the translation process of RS-stereodescriptors is faithful or not. As a result, the specification of configurations is rationalized on a more succinct basis than the conventional foundation which emphasizes the dichotomy between enantiomers and diastereomers. A paradigm shift from enantiomers to RS-stereoisomers has been pointed out so that over-simplified features of the conventional dichotomy between enantiomers and diastereomers have been discussed by means of stereoisograms.
[Abstract] The pro-R/pro-S system has been redefined by using the concept of pro-RS-stereogenicity, where two ligands in an RS-diastereotopic relationship are pairwise characterized by pro-R- and pro-S-descriptors on the basis of stereoisograms. As for a practical criterion for determining RS-diastereotopic relationships, a symmetry criterion has been developed as a new matter by extending the concept of stereoisograms to testify RS-diastereotopic relationships. The conventional definitions of pro-R/pro-S-descriptors which are based on the original term "prochirality" (along with enantiotopic and diastereotopic relationships) and on the revised term "prostereogenicity" (along with stereoheterotopic relationships) are altogether abandoned. The revised concept of prochirality has a purely geometric meaning, which stems from the term enantiotopic relationship or equivalently enantiospheric equivalence class (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg (1991)). The two concepts for describing intramolecular environments, i.e., the pro-RS-stereogenicity and the prochirality, are clarified to correspond to the concepts of RS-stereogenicity and chirality, which have been formulated on the basis of stereoisograms in order to bring about a harmonized viewpoint to stereoisomeric and geometric intermolecular features of stereochemistry (S. Fujita, J. Org. Chem., 69, 3158-3165 (2004); J. Comput. Aided Chem., 10, 16-29 (2009)). Chirality-faithfulness of pro-R/pro-S-descriptors specified by RS-diastereotopic relationships is discussed to harmonize pro-RS-stereogenicity with prochirality.
[Abstract] Preparation methods of chemical documents containing chemical structural formulas have been surveyed by referring to the author's experiences of publishing books, emphasizing differences before and after the adoption of TeX/LaTeX-typesetting as well as before and after the development of XyMTeX. The recognition of XyMTeX commands as linear notations has led to the concept of the XyM notation, which has further grown into XyMML (XyM Markup Language) as a markup language for characterizing chemical structural formulas. XML (Extensible Markup Language) documents with XyMML are converted into HTML (Hypertext Markup Language) documents with XyM notations, which are able to display chemical structural formulas in the Internet by means of the XyMJava system developed as a Java applet of an internet browser. On the other hand, the same XML documents with \XyM{}ML are converted into LaTeX documents with XyMTeX commands (the same as XyM notations), which are able to print out chemical structural formulas of high quality. Functions added by the latest version (4.04) of XyMTeX have enhanced abilities of drawing complicated structures such as steroids. LaTeX documents with XyMTeX formulas can be converted into PDF (Portable Document Format) documents directly or via PostScript document. Applications of such PDF documents in online or semi-online submission to scientific journals have been discussed.
2010 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Correlation diagrams of stereoisomers are developed as a versatile device for discussing the stereoisomerism of cyclobutane derivatives. A cyclobutane skeleton belongs to an RS-stereoisomeric group, which is constructed by starting from the point group D_{4h} in order to represent a global symmetry. Stereoisomers derived from the cyclobutane skeleton are treated by a main correlation diagram of stereoisograms under the action of the RS-stereoisomeric group. In order to discuss the local symmetry of each RS-stereogenic center, on the other hand, a promolecule is generated at each RS-stereogenic center, where the RS-stereoisomeric group for specifying the promolecule is constructed by starting from the point group T_{h}. Such promolecules derived from respective stereoisomers are correlated to each other by using stereoisograms, which are further correlated to give a correlation diagram of stereoisograms. RS-stereodescriptors are discussed on the basis of such correlation diagrams of stereoisograms.
[Abstract] The concept of correlation diagrams of stereoisograms has been developed to characterize a set of stereoisomers. Each of the stereoisomers is regarded as a tetrahedral uninuclear promolecule at an RS-stereogenic center or a binuclear promolecule at a bond selected to be considered. Stereoisograms, each of which is constructed from a quadruplet of such promolecules according to Fujita's formulation (Fujita,~S. J. Org. Chem. 2004, 69, 3158--3165), are collected to give a correlation diagram at the center or bond. When all of the RS-stereogenic centers (along with the bond) are taken into consideration, a set of correlation diagrams are generated to characterize the set of stereoisomers. Because each stereoisogram represents the local chirality/achirality and the local RS-stereogenicity/RS-astereogenicity of the RS-stereogenic center (or the bond), the corresponding correlation diagram indicates total features of such local symmetries. Non-degenerate and degenerate cases having two or four RS-stereogenic centers along with a central bond to be considered have been investigated by using correlation diagrams of stereoisograms. Thereby, over-simplified features of the conventional dichotomy between enantiomers and diastereomers have been discussed in detail.
[Abstract] Correlation diagrams of stereoisograms for characterizing stereoisomers have been developed so as to provide more information on both geometric and stereoisomeric features than a separate use of a stereoisogram. They are capable of solving most problems which have been left unsolved within the traditional terminology of stereochemistry and related chemoinformatics practices, e.g., over-simplified features of the conventional dichotomy between enantiomers and diastereomers, incomplete separation of RS-stereogenicity from chirality, unconscious disregard of local $RS$-stereogenicity and confusion of it with local chirality, implications of reflection-invariant cases of the CIP priority system, and others.
2011 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] A new theory of stereoisomerism in harmony with molecular symmetry has been developed by starting from RS-stereoisomeric groups correlated to stereoisograms and their correlation diagrams. The substitution positions of a stereoskeleton are permuted by a set of epimerizations at the RS-stereogenic centers of the stereoskeleton. The product of epimerizations and the mirror-image transformation of the skeleton characterize the total feature of isomerization, which is based on the axiom of organic stereoisomerism. Then, stereoisomeric groups are formulated to develop the theory of stereoisomerism after several related groups are defined, e.g., stereoisogram groups, epimerization groups, local symmetry groups, epimeric stereoisogram groups, epimeric RS-stereoisomeric groups, and multiple epimerization groups. On the basis of the stereoisomeric groups, stereoisomeric representations are derived and employed to discuss correlation diagrams of stereoisograms. On the other hand, molecular-symmetry representations are derived from the stereoisomeric groups and employed to discuss molecular symmetries. Typical topics of stereochemistry, e.g., the CIP system for giving RS-stereodescriptors and the Fischer-Rosanoff convention for naming {\sc dl}-series of sugars, are discussed on the basis of the present theory.
[Abstract] Cubane derivatives with chiral and achiral proligands are counted as three-dimensional (3D) structural isomers by the fixed-point matrix method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag (1991)). The numbers of such 3D structural isomers are itemized with respect to their point-group symmetries, which are subgroups of the point group O_{h} for a cubane skeleton. For this purpose, the full list of unit subduced cycle indices with chirality fittingness (USCI-CFs) of O_{h} is constructed in a tabular form. Fixed-point vectors or fixed-point matrices, which are calculated by starting from USCI-CFs, are used to count 3D structural isomers in the form of isomer-counting matrices. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation.
2012 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Cubane derivatives with chiral and achiral proligands are counted as 3D structural isomers and as steric isomers in the light of the proligand method developed by us (S. Fujita, Theor. Chem. Acc., 113, 73--79, (2005); 113, 80--86 (2005); and 115, 37--53 (2006)). The results are further applied to count achiral derivatives as well as enantiomeric pairs of chiral derivatives. By taking account of the sphericities of cycles, the proligand method is capable of counting cubane derivatives with chiral and achiral proligands, where the chirality of each proligand is judged in isolation. Polya's theorem is concluded to lack such sphericities of cycles, so that it is restricted to counting cubane derivatives with achiral proligands only. A Maple program source for counting cubane derivatives as 3D structural isomers etc. is given as an example of practical calculation.
[Abstract] Cubane derivatives with chiral and achiral proligands are counted as 3D structural isomers and as steric isomers in the light of the markaracter method developed by us (S. Fujita, Theor. Chim. Acta, 91, 291--314 (1995), 91, 315--332 (1995)). The results are further applied to count achiral derivatives as well as enantiomeric pairs of chiral derivatives. The markaracter tables of the point groups O_{h} and O, their inverse tables, the dominant USCI-CF (unit subduced cycle indices with chirality fittingness) tables, and the non-dominant USCI-CF tables are prepared for the purpose of further applications of combinatorial enumeration. A Maple program source for calculating non-dominant USCI-CF tables is given as an example of practical calculation.
[Abstract] The CM (characteristic monomial) method developed by us (S. Fujita, Theor. Chem. Acc., 99, 224--230 (1998), S. Fujita, J. Chem. Inf. Comput. Sci., 40, 1101--1112 (2000)) is applied to enumeration of cubane derivatives with chiral and achiral proligands. For this purpose, CM-CFs (characteristic monomials with chirality fittingness) are calculated by emphasizing the interconvertivity between Q-conjugacy characters and markaracters. Thereby, a dominant CM-CF table is constructed so as to correspond to an irreducible Q-conjugacy character table in the present article (Part III of this series), just as a dominant USCI-CF (unit subduced-cycle index with chirality fittingness) table corresponds to a dominant markaracter table as discussed in Part II of this series. The present set of tables is used to prepare dominant and non-dominant USCI-CFs, which are in turn used to prepare SCI-CFs (subduced cycle indices with chirality fittingness) and CI-CFs (cycle indices with chirality fittingness). The CI-CFs of O_{h} and O for enumerating cubane derivatives as three-dimensional structural isomers and as steric isomers as well as for enumerating achiral and chiral cubane derivatives are clarified to be equivalent to those prepared by the proligand method (Part I of this series) as well as to those prepared by the markaracter method (Part II of this series). A Maple program source for calculating USCI-CFs from CM-CFs is given as an example of practical calculation.
[Abstract] The extended superposition method, which has been developed by us as an extension of the concept of elementary superposition (S. Fujita, Theor. Chim. Acta, 82, 473--498 (1992)), is applied to enumeration of cubane derivatives with chiral and achiral proligands. This method provides us with a tool for evaluating the respective contribution of each USCI-CF (unit subduced-cycle index with chirality fittingness) to the corresponding CI-CF (cycle index with chirality fittingness), which is in turn calculated by means of the proligand method, the markaracter method, the characteristic-monomial method or others. The extended superposition method does not require generating functions but requires cycle indices (CI) for regular and irregular cases, which depend upon permutational features of chiral and/or achiral (pro)ligands. Calculated values by the extended superposition method are clarified to be identical with those obtained in terms of generating functions. Effects of chiral proligands (as three-dimensional structures) on the numbers of cubane derivatives are detailedly compared with those of the corresponding graphs (as two-dimensional constitutions).
[Abstract] The double coset representation (DCR) method for combinatorial enumeration is reformulated on the basis of unit subduced cycle indices (USCI-CFs) defined in terms of sphericities of double cosets for cyclic subgroups. Its theoretical framework is in sharp contrast to the original version of the DCR method (S. Fujita, J. Math. Chem., 42, 215--263 (2007)), in which products of sphericity indices assigned to respective operations are used in place of USCI-CFs. From the viewpoint of methodology, the present version of the DCR method succeeds to that of the markaracter method (S. Fujita, Theor. Chim. Acta, 91, 291--314, 315--332 (1995)), while the original version succeeds to that of the proligand method (S. Fujita, Theor. Chem. Acc., 113, 73--79 (2005)). Thereby, cycle indices with chirality fittingness (CI-CFs) defined by starting from the newly-defined USCI-CFs are used to conduct gross enumeration of 3D-structural isomers. Cubane derivatives with chiral and achiral proligands are counted as 3D structural isomers by relying on a CI-CF based on the new definition.
[Abstract] The restricted-subduced-cycle-index (RSCI) method for symmetry-itemized enumeration of Kekule structures has been developed as an extension of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)). In the RSCI method, such Kekule structures are considered to be generated by edge substitution of a skeleton under a restricted condition that no selected edges have a common vertex. The restricted condition is formulated by means of the concept of territory indicators and territory discriminants, by which subduced cycle indices (SCIs) derived from unit subduced cycle indices (USCIs) are converted into restricted subduced cycle indices (RSCIs). The resulting RSCIs are used to evaluate marks (numbers of fixed points) in place of the SCIs used in the USCI approach, so that the numbers of Kekule structures are calculated in a symmetry-itemized fashion. Moreover, the resulting data of enumeration are used to draw respective Kekule structures by means of orbit indicators and edge indicators. The application of the RSCI method to fullerene C_{60} of I_{h}-symmetry is discussed in detail.
[Abstract] The restricted-subduced-cycle-index (RSCI) method for generating Z-counting polynomials and the Hosoya indices (Z-indices) as well as matching polynomials has been developed by starting from subduced cycle indices (SCI) defined in the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)). In the RSCI method, k-matchings of a given skeleton (or graph) for deriving these matters are regarded as restricted structures in which vertex substitution and edge substitution occur concurrently under a restricted condition that occupation of a common vertex does not occur. For the purpose of counting such restricted structures, the concepts of territory indicators and territory discriminants are introduced. Thereby, a restricted subduced cycle index (RSCI) for the skeleton (or graph) is derived from an SCI by the subduction to C_{1} (non-symmetry). The RSCI gives the generating function for counting the numbers of restricted structures, which is further converted into a Z-counting polynomial, the Hosoya indices, as well as a matching polynomial. The versatility of the RSCI method is illustrated by applying to benzene, naphthalene, dodecahedron, and fullerene C_{60}.
[Abstract] By placing chiral and achiral proligands on the eight positions of a cubane skeleton of O_{h}, resulting cubane derivatives are counted as three-dimensional (3D) structural isomers by the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991); S. Fujita, "Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers", University of Kragujevac, Kragujevac (2007)), where the numbers of 3D structural isomers with given constitutional formulas are itemized with the subsymmetries of O_{h}. For this purpose, partial cycle indices with chirality fittingness (PCI-CFs) for respective subgroups of O_{h} are calculated by using unit subduced cycle indices with chirality fittingness (USCI-CFs) and an inverse matrix of the mark table of O_{h}. After introducing ligand-inventory functions into the PCI-CFs, expansions of the resulting equations provide us with generating functions for giving the numbers of 3D structural isomers. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation for listing the resulting data in tabular forms.
[Abstract] Cubane derivatives with a given chemical formula and a given symmetry are enumerated by applying the elementary-superposition method (S. Fujita, Theor. Chim. Acta, 82, 473--498 (1992)) of the unit-subduced-cycle-index (USCI) approach to a cubane skeleton of the point group O_{h}. A cycle index (CI) for a regular or an irregular case of chiral and achiral ligands is calculated in accord with such a given chemical formula. The CI is superposed elementarily onto respective subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs). Thereby, the numbers of fixed points (derivatives) are obtained with respect to the respective SCI-CFs to give a fixed-point vector (FPV). The resulting FPV is multiplied by an inverse matrix of the mark table of O_{h} to give an isomer-counting vector (ICV), which contains the numbers of 3D-structural isomers in an itemized fashion with respect to point-group symmetries. The concept of prochirality is examined by using cubane derivatives enumerated by the ICV.
[Abstract] Pseudoasymmetry is discussed in an extended fashion by means of the stereoisogram approach (Fujita, S. J. Org. Chem. 2004, 69, 3158--3165; Fujita, S. Tetrahedron 2004, 60, 11629--11638), where an adamantane skeleton and an 8-azabicyclo[3.2.1]octane skeleton are used as polycyclic skeletons for constructing stereoisograms of five types (Types I--V). The term pseudoasymmetry is characterized as the concurrent participation of achirality and RS-stereogenicity in terms of a stereoisogram of Type V, while the term asymmetry is characterized as the concurrent participation of chirality and RS-stereogenicity in terms of a stereoisogram of Type I. Prochirality as a purely geometric concept is also discussed from the viewpoint of the stereoisogram approach. The stereoisogram approach has brought about a paradigm shift from enantiomers to RS-stereoisomers as equivalence classes, so that the categorization of compounds by the stereoisogram approach is shown to be conceptually different from the conventional categorization described in the Nobel Prize lecture of Prelog (1975). Thereby, the stereoisogram approach is concluded to serve as an integrated methodology for organic and inorganic stereochemistry.
[Abstract] The stereoisogram approach, which has originally been developed to rationalize organic stereochemistry (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004); S. Fujita, Tetrahedron, 62, 691--705 (2006); 65, 1581--1592 (2009)), is extended and applied to inorganic stereochemistry by using trigonal bipyramidal compounds as examples. The point group D_{3h} of a trigonal bipyramidal skeleton is extended into the RS-stereoisomeric group of order 24, which is considered to control a stereoisogram of the trigonal bipyramidal skeleton. Stereoisograms of trigonal bipyramidal compounds derived from the skeleton correspond to subgroups of the RS-stereoisomeric group. Thereby, they are discussed in terms of attributive terms (chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality) or equivalently in terms of relational terms (enantiomeric/self-enantiomeric, RS-diastereomeric/self-RS-diastereomeric, and holantimeric/self-holantimeric), where the stereoisograms are categorized into five types (Types I--V). Among them, stereoisograms of Types I, III, and V are shown to be capable of giving C/A-descriptors because of their RS-stereogenicity (or RS-diastereomeric relationships).
[Abstract] The stereoisogram approach (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004); S. Fujita, Tetrahedron, 62, 691--705 (2006); 65, 1581--1592 (2009)) is applied to trigonal bipyramidal compounds, where chiral and achiral proligands are taken into consideration. After configurations of trigonal bipyramidal compounds are enumerated by using the partial-cycle-index (PCI) method (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag, 1991), they are categorized into Type I--V cases according to the stereoisogram approach. The enumerated configurations are specified by configuration indices and C/A-descriptors, which are assigned in terms of RS-diastereomeric relationships, (not of enantiomeric relationships). The concept of a multiplet of stereoisograms is proposed to formulate the concept of ortho-stereogenicity, which is concerned with ortho-diastereomeric relationships between stereoisograms. On the other hand, the concept of stereogenicity (which has been used in the conventional stereochemistry) is redefined by starting from RS-stereogenicity and by comparing with the ortho-stereogenicity, where the stereogenicity is concerned with diastereomeric relationships between pairs of enantiomers. Berry's pseudorotation for isomerization of trigonal bipyramidal compounds is reinterpreted in order to cover more general cases in which chiral moieties along with achiral moieties (i.e., all of Type I--V cases) are taken into consideration. A modified Desargues-Levi graph is proposed to cover Type I--V cases. In addition, an adamantane-like graph is proposed to formulate Berry's pseudorotation on the basis of a multiplet of stereoisograms, where quadruplets of RS-stereoisomers occupy the nodes of the graph. Thereby, a multiplet of stereoisograms is shown to be a versatile tool to characterize stereoisomerization processes of inorganic stereochemistry in addition to those of organic stereochemistry.
[Abstract] Among the four methods of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)), the fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method have been applied to the combinatorial enumeration of prismane derivatives. These enumeration processes are based on the proligand-promolecule model, which enables us to take account of achiral and chiral proligands. Prochirality in a geometric meaning has been discussed in general by emphasizing the presence of enantiospheric orbits in enumerated prismane derivatives. An enantiospheric orbit accommodating chiral proligands (along with achiral ones) has been shown to exhibit prochirality by using various prismane derivatives as examples. On the other hand, the scope of pseudoasymmetry has been extended to cover such a rigid skeleton as prismane in addition to a usual pseudoasymmetric enter as a single atom, where the proligand-promolecule model plays an essential role.
[Abstract] The stereoisogram approach (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004), Tetrahedron, 60, 11629--11638 (2004)) has been applied to comprehensive discussions on geometric aspects and stereoisomeric aspects of stereochemistry, where a prismane skeleton has been selected as a rigid skeleton for the underlying proligand-promolecule model. The existence of five types of stereoisograms (Types I--V) has been demonstrated by using prismane derivatives as illustrative examples in a consistent way with a general proof using the group theory (S. Fujita, MATCH Commun. Math. Comput. Chem., 54, 39--52 (2005)). After a C/A-convention for characterizing absolute configurations was proposed on the basis of the stereoisogram approach, such geometric and stereoisomeric aspects of stereochemistry as chirality, RS-stereogenicity, and sclerality have been discussed by putting emphasis on the independence between chirality and RS-stereogenicity, on extended features of pseudoasymmetry, and on the assignability of C/A-descriptors. By following a general rationalization (S. Fujita, Tetrahedron, 62, 691--705 (2006)), prochirality, pro-RS-stereogenicity, and prosclerality have been discussed on the basis of such attributive terms as sphericities, RS-tropicities, and cercalities, where illustrative examples are selected from prismane derivatives. Thereby, the stereoisogram approach has been clarified to be a versatile device for integrating geometric and stereoisomeric aspects of stereochemistry.
[Abstract] On the basis of restricted subduced cycle indices with chirality fittingness (RSCI-CFs), the restricted-fixed-point-matrix (RFPM) method has been developed as a new method for the combinatorial enumeration of sterically hindered derivatives of a given skeleton. Such RSCI-CFs are derived from subduced cycle indices with chirality fittingness (SCI-CFs) defined in the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)). Thus, the SCI-CFs are combined with territory indicators to give territory discriminants, which are treated by considering a restriction condition that rejects the occupation of any adjacent sites, so as to generate RSCI-CFs. The resulting RSCI-CFs are used to evaluate marks (the numbers of fixed points) in place of the original SCI-CFs and combined with the fixed-point-matrix (FPM) method of the USCI approach to develop the RFPM method. The RFPV method based on RSCI-CFs is applied to enumeration of sterically hindered derivatives of dodecahedrane.
[Abstract] The restricted-partial-cycle-index (RPCI) method for combinatorial enumeration under the restriction of no adjacency of ligands has been developed as a restricted version of the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)). To take account of the restriction condition, (unrestricted) subduced cycle indices with chirality fittingness (SCI-CFs) of the USCI approach are converted into restricted subduced cycle indices with chirality fittingness (RSCI-CFs). Then, restricted partial cycle indices with chirality fittingness (RPCI-CFs) are derived from the RSCI-CFs, just as partial cycle indices with chirality fittingness (PCI-CFs) are derived from the SCI-CFs in the USCI approach. The resulting RPCI-CFs provide generating functions for restricted enumerations. The RPCI method using such RPCI-CFs is applied to enumeration of dodecahedrane derivatives under the restriction of no adjacency of ligands. Several enumerated derivatives are depicted and their symmetries are discussed to comprehend stereochemical properties such as pseudoasymmetry, sphericity, prochirality, and so on.
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[Abstract] In order to treat steric hindrance due to monodentate and bidentate ligands in isomer enumeration, the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach is extended to enumerate derivatives by taking restricted modes of vertex and/or edge substitutions into consideration. Factorization of subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs) assigned to respective subgroups of the group of a given skeleton, is discussed to generate restricted SCI-CFs. The restricted SCI-CF for each subgroup is effective to evaluate the numbers of fixed points (promolecules) on the action of the subgroup under the restricted conditions of enumeration. The set of the restricted SCI-CFs is multiplied by the inverse mark table to generate partial cycle indices with chirality fittingness (PCI-CFs), by which itemized enumerations for every subgroups are conducted under the restricted conditions. Several results starting from an icosahedral skeleton are discussed to examine differences between unrestricted and restricted enumerations.
[Abstract] The restricted-fixed-point-matrix (RFPM) method is developed as an extension of the fixed-point-matrix (FPM) method, which is one of the four methods of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)). The RFPM method is capable of combinatorial enumerations of 3D structures or graphs under a restricted condition, where orbits of vertices and edges interact each other. Subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated for an unrestricted condition by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs), are converted into restricted SCI-CFs by means of newly-defined territory indicators (TIs) of vertices and edges. Such restricted SCI-CFs as calculated for respective subgroups are effective to evaluate the numbers of fixed points (promolecules) on the action of the subgroups under the restricted condition, where the occupation of a common vertex or the occupation of adjacent edges is avoided.
[Abstract] The restricted partial-cycle-index (RPCI) method has been developed by starting from the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)), where enumerated derivatives are generated by means of vertex substitution (monodentate ligands) and/or edge substitution (bidentate ligands) under a restriction condition that occupation of a common vertex (or occupation of adjacent edges) is avoided. Thus, restricted partial cycle indices with chirality fittingness (PCI-CFs) are derived from unit subduced cycle indices with chirality fittingness (USCI-CFs) via restricted subduced cycle indices with chirality fittingness (SCI-CFs). The resulting restricted PCI-CFs enable us to enumerate derivatives under the restricted condition in a symmetry-itemized fashion. The restricted PCI-CFs are further transformed into restricted cycle indices with chirality fittingness (restricted CI-CFs) for gross enumerations of total, achiral, chiral derivatives. A maple program for the RPCI method is reported as an appendix.
[Abstract] Restricted enumerations based on the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag (1991)) are discussed from a viewpoint of subductions into the C_{1}-group, where only edges contained in a given skeleton are taken into consideration under a restriction condition that occupation of a common vertex (or occupation of adjacent edges) is avoided. The restriction condition for edge occupation is formulated by introducing territory indicators and territory discriminants so as to give a restricted subduced cycle index (RSCI) for the C_{1}-group. Thereby, the restricted-subduced-cycle-index (RSCI) method for enumerating Kekule structures (or equivalently perfect matchings of a graph) is developed as a specialized application of the USCI approach. To show the versatility of the RSCI method, the numbers of Kekule structures for a dodecahedral skeleton, coronene, dibenzo[b,n]picene, dibenzo[bc,mn]ovalene, dibenzo[fg,op]anthanthrene, dibenzo[hi,sl]ovalene, naphtho[8,1,2-efg]anthanthrene, and fullerene C_{60} are calculated by the RSCI method. Maple programs for the RSCI method are reported as examples of practical calculations.
[Abstract] As clarified by the publication of the interdisciplinary chemistry/mathematics books, the XyMTeX system coupled with the LaTeX system has been proven to be a reliable tool for publishing books of high printing quality which contain structural formulas along with mathematical equations.
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[Abstract] The stereoisogram approach, which has originally been developed to rationalize organic stereochemistry (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004); S. Fujita, Tetrahedron, 62, 691--705 (2006); 65, 1581--1592 (2009)), is extended and applied to inorganic stereochemistry by using octahedral complexes as examples. Stereoisograms of octahedral complexes are constructed and discussed in terms of attributive terms (chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality) or equivalently in terms of relational terms (enantiomeric/self-enantiomeric, RS-diastereomeric/self-RS-diastereomeric, and holantimeric/self-holantimeric). After they and categorized into five types (Types I--V), stereoisograms of Type I, III, and V are shown to be characterized by A/C-descriptors, where the capability of giving A/C-descriptors is ascribed to RS-stereogenicity (or RS-diastereomeric relationships), which is determined to be a common nature to Types I, III, and V. Several textbook errors are pointed out and corrected in terms of the stereoisogram approach. For example, although the symbols A and C are originally called ``chirality symbols'' in the IUPAC recommendations 2005 (so-called The Red Book), the naming is inadequate and they should be called ``RS-stereogenicity symbols".
[Abstract] Stereoisomeric groups are defined to describe the stereoisomerism of octahedral complexes by starting from RS-stereoisomeric groups for characterizing stereoisograms (cf. Part I of this series) and the symmetric group of degree 6. On the basis of the stereoisomeric groups, multiplets of stereoisograms are defined so as to clarify the difference between RS-stereoisomerism and stereoisomerism. Stereogenic groups are also defined as subgroups of the stereoisomeric groups, so that the difference between RS-stereogenicity and stereogenicity is determined decisively after group-theoretical consideration. Thus, stereogenicity is shown to have nothing to do with the capability of giving C/A-descriptors, which is, in turn, ascribed to RS-stereogenicity (or RS-diastereomeric relationships), but not to chirality. Both the concept of stereogenicity and the concept of RS-stereogenicity are concluded to be independent to the concept of chirality, although RS-stereogenicity interacts with chirality as formulated by a stereoisogram. Thereby, the stereoisogram approach, which has originally been developed to rationalize organic stereochemistry (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004); S. Fujita, Tetrahedron, 62, 691--705 (2006); 65, 1581--1592 (2009)), is clarified to be effective to inorganic stereochemistry.
[Abstract] The concept of prochirality is defined to discuss inorganic stereochemistry on the basis of enantiosphericity, which has been developed as a purely geometric formulation for organic stereochemistry (S. Fujita, J. Am. Chem. Soc., 112, 3390--3397 (1990); S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer-Verlag, Berlin-Heidelberg (1991)). In addition, the concept of pro-RS-stereogenicity is defined to discuss inorganic stereochemistry on the basis of RS-enantiotropicity (S. Fujita, J. Math. Chem., 33, 113--143 (2003); S. Fujita, Tetrahedron, 62, 691--705 (2006)). These concepts are applied to the discussion on intramolecular features of octahedral complexes. Alternatively, the stereoisogram approach for organic stereochemistry (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004); S. Fujita, Tetrahedron, 60, 11629--11638 (2004)) is extended to discuss prochirality and pro-RS-stereogenicity of octahedral complexes in inorganic stereochemistry. After octahedral complexes are categorized into five types (Types I--V) by means of stereoisograms, stereoisograms for testifying prochirality and pro-RS-stereogenicity are proposed as alternative devices, which are an extension of the counterparts developed originally for the symmetry criterion in organic stereochemistry (S. Fujita, J. Comput. Aided Chem., 10, 76--95 (2009)). The relational terms enantiotopic and RS-diastereotopic are introduced to make the symmetry criterion effective. Thereby, pro-A/pro-C-descriptors are concluded to be based on an RS-diastereotopic relationship. The further concept of pro-ortho-stereogenicity is proposed to rationalize intramolecular (in)equivalency of proligands, which cannot be characterized by pro-A/pro-C-descriptors.
[Abstract] The RS-stereoisomeric group T_{d\widetilde{\sigma}\widehat{I}} is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group O_{h} of order 48. The non-redundant set of subgroups (SSG) of T_{d\widetilde{\sigma}\widehat{I}} is obtained by referring to the non-redundant set of subgroups of O_{h}. The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be T_{d\widetilde{\sigma}\widehat{I}}(/C_{3v\widetilde{\sigma}\widehat{I}}) which is closely related to the O_{h}(/D_{3d}). According to the unit-subduced-cycle-index (USCI) approach (S. Fujita, ``Symmetry and Combinatorial Enumeration of Chemistry'', Springer, 1991), the subdution of T_{d\widetilde{\sigma}\widehat{I}}(/C_{3v\widetilde{\sigma}\widehat{I}}) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix (FPM) method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of T_{d\widetilde{\sigma}\widehat{I}}. After the subgroups of T_{d\widetilde{\sigma}\widehat{I}} are categorized into types I to V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.
[Abstract] The symmetry-itemized enumeration of quadruplets of stereoisograms is discussed by starting from a tetrahedral skeleton, where the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, ``Symmtetry and Combinatorial Enumeration of Chemistry'', Springer, 1991) is combined with the stereoisogram approach (S. Fujita, J. Org. Chem., \textbf{69}, 3158--3165 (2004), Tetrahedron, \textbf{60}, 11629--11638 (2004)). Such a tetrahedral skeleton as contained in the quadruplet of a stereoisogram belongs to an RS-stereoisomeric group denoted by T_{d\widetilde{\sigma}\widehat{I}}, where the four positions of the tetrahedral skeleton accommodate achiral and chiral proligands to give quadruplets belonging to subgroups of T_{d\widetilde{\sigma}\widehat{I}} according to the stereoisogram approach. The numbers of quadruplets are calculated as generating functions by applying the PCI method. They are itemized in terms of subgroups of T_{d\widetilde{\sigma}\widehat{I}}, which are further categorized into five types. Quadruples for stereoisograms of types I--V are ascribed to subgroups of T_{d\widetilde{\sigma}\widehat{I}}, where their features are discussed in comparison between RS-stereoisomeric groups and point groups.
[Abstract] A tetrahedral skeleton is considered to belong to the RS-stereoisomeric group denoted by T_{d sigma I}. By placing proligands on the four positions of the tetrahedral skeleton, the resulting promolecule is considered to belong to a subgroup of T_{d sigma I}, where its RS-stereoisomeric properties are illustrated by the corresponding stereoisogram. Three aspects of an absolute configuration, i.e., a chiral aspect, an RS-stereogenic aspect, and a scleral aspect, are formulated on the basis of three attributes of a stereoisogram, i.e., chirality, RS-stereogenicity, and sclerality. The RS-stereodescriptors of the Cahn-Ingold-Prelog (CIP) system are clarified to specify the RS-stereogenic aspect, so that they are assigned to a pair of RS-diastereomers contained in a type-I, type-III, or type-V stereoisogram. To apply the RS-stereodescriptors to the chiral aspect of an absolute configuration, the concept of chirality faithfulness is redefined by proposing odd and even priority permutations.
[Abstract]
After the RS-stereoisomeric group D_{2d sigma I}
of order 16 has been defined by starting point group D_{2d} of order 8,
the isomorphism between D_{2d sigma I} and
the point group D_{4h} of order 16 is throughly discussed.
The non-redundant set of subgroups (SSG) of D_{2d sigma I} is obtained
by referring to the non-redundant set of subgroups of D_{4h}.
The coset representation for characterizing the orbit of the four positions of an allene skeleton
is clarified to be D_{2d sigma I}(/C_{s sigma I})
which is closely related to the D_{4h}(C'''_{2v}).
According to the unit-subduced-cycle-index (USCI) approach
(S. Fujita, ``Symmetry and Combinatorial Enumeration of Chemistry'', Springer, 1991),
the subduction of D_{2d sigma I}(/C_{s sigma I})
is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs).
Then, the fixed-point matrix (FPM) method of the USCI approach is applied to the USCI-CFs.
Thereby, the numbers of quadruplets are calculated in an itemized fashion
with respect to the subgroups of D_{2d sigma I}.
After the subgroups of D_{2d sigma I} are categorized into
types I to V, type-itemized enumeration of quadruplets is conducted to
illustrate the versatility of the stereoisogram approach.
(Correction) The numbering 1 and 3 of the diagram 2 in Figure 1 should be exchanged.
The numbering of the diagram 1 in Figure 1 is correct.
[Abstract] The partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, ``Symmetry and Combinatorial Enumeration of Chemistry'', Springer, 1991) is extended to meet the stereoisogram approach (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004), Tetrahedron, 60, 11629--11638 (2004)). Then, the PCI method is applied to the symmetry-itemized enumeration of quadruplets of stereoisograms, where an allene skeleton is considered to belong to the RS-stereoisomeric group D_{2d sigma I} derived from the point group D_{2d}. The resulting numbers of quadruplets are itemized in terms of subgroups of D_{2d sigma I}, which are further categorized into five types (types I--V). The term `absolute configuration', which has been correlated solely to chirality in the conventional stereochemistry, is extended to have three aspects (i.e., a chiral aspect, an RS-stereogenic aspect, and a scleral aspect) according to the three attributes of a stereoisogram (i.e., chirality, RS-stereogenicity, and sclerality). Thereby, the RS-stereodescriptors of the Cahn-Ingold-Prelog (CIP) system are clarified to specify the RS-stereogenic aspect, not to specify the chiral aspect. They are assigned to a pair of RS-diastereomers contained in a type-I, type-III, or type-V stereoisogram, but not to a pair of enantiomers. To judge whether or not the RS-stereodescriptors (assigned originally on the basis of the RS-stereogenic aspect of an absolute configuration) are permitted to be applied to the chiral aspect, the concept of chirality faithfulness (S. Fujita, J. Comput. Aided Chem., 10, 16--29 (2009)) is used after redefined by proposing odd and even priority permutations.
[Abstract] Three kinds of groups, i.e., point groups, RS-permutation groups, and ligand-reflection groups, are integrated in a consistent way. Thereby, RS-stereoisomeric groups are generated so as to be capable of constructing a strict and succinct mathematical framework for reorganizing the modern stereochemistry. A stereoisogram is proposed as a diagrammatic expression of an RS-stereoisomeric group, which contains a quadruplet of promolecules derived from a given skeleton in terms of the proligand-promolecule model. The four promolecules of the quadruplet are correlated to one another by means of enantiomeric, RS-diastereomeric, and holantimeric relationships according to the three kinds of groups. These relationships are respectively correlated to three pairs of attributes, i.e., chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality. The mathematical rationalization of the three pairs of attributes is in sharp contrast to the presumption of a single pair of attributes (i.e., chirality/achirality) supporting the modern stereochemistry. By simple mathematical treatments, it is proven that there are only five types of stereoisograms (type I--V). The group-subgroup relationship concerned with an RS-stereoisomeric group, e.g., T_{d sigma I} for a tetrahedral skeleton, permits us to accomplish qualitative and quantitative discussions for reorganizing the modern stereochemistry. As Part I of this series, this article is devoted to a theoretical formulation of the stereoisogram approach.
[Abstract] The stereoisogram approach presupposes three distinct pairs of attributes, i.e., chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality, which are integrated into RS-stereoisomerism represented by stereoisograms. The mathematical rationalization of the three pairs of attributes is in sharp contrast to the presumption of a single pair of attributes (i.e., chirality/achirality) supporting the modern stereochemistry. Thereby, a pair of R/S-stereodescriptors of the Cahn-Ingold-Prelog (CIP) system is clarified to be assigned to a pair of RS-diastereomers, not to a pair of enantiomers. In particular, the `units' of the CIP system (e.g., `chirality units', `stereogenic units', and `pseudoasymmetric units') are replaced by promolecules having type-I, type-III, or type-V stereoisograms; the `chirality rule' of the CIP system is replaced by the RS-stereogenicity rule of the stereoisogram approach; as well as the use of `reflection invariance' for `pseudoasymmetric units' is avoided by developing the concept of chirality faithfulness/unfaithfulness. Global and local symmetries are discussed in terms of correlation diagrams of stereoisograms.
[Abstract] The stereoisogram approach is introduced to settle the misleading terminology due to `prochirality' in the modern stereochemistry. After the term prochirality is redefined as having a purely geometric meaning, a method based on probe stereoisograms and another method based on equivalence classes (orbits) are introduced to determine prochirality and/or pro-RS-stereogenicity. Enantiotopic and RS-diastereotopic relationships appearing in probe stereoisograms are respectively used to determine prochirality and pro-RS-stereogenicity, where `stereoheterotopic' relationships used in the modern stereochemistry are abandoned. Alternatively, an enantiospheric orbit for specifying prochirality and an RS-enantiotropic orbit for specifying pro-RS-stereogenicity are emphasized by using coset representations and Young tableaux. The pro-R/pro-S-system is clarified to be based on pro-RS-stereogenicity, not on prochirality.
[Abstract] The problematic terms `diastereomer' and `pseudoasymmetry' in the terminology of modern stereochemistry are discussed and redefined according to the stereoisogram approach. After the term enantiomeric is redefined as an equivalence relationship, the term diastereomeric is redefined as a relational term for specifying inequivalent pairs of enantiomers. After a brief introduction of the stereoisogram approach, stereoisograms are categorized into five types (Types I--V). The term pseudoasymmetric is correlated to a stereoisogram of type V, which is characterized to be achiral, RS-stereogenic, and scleral. Three aspects of absolute configuration are discussed according to the examination of type-I, type-III, and type-V stereoisograms. Among them, the RS-stereogenic aspect of absolute configuration is specified by R/S-descriptors. Thus, a pair of R/S-descriptors is clarified to be assigned to a pair of RS-diastereomers, not to a pair of enantiomers. The concept of chirality (un)faithfulness is introduced to explain lowercase labels of R/S-descriptors.
[Abstract]
[Abstract]
[Abstract] The conventional schemes for classifying isomers are clarified to suffer from serious confusion, because they are misleadingly based on the pairing of stereoisomers and `constitutional isomers' and the pairing of enantiomers and `diastereomers'. The misleading features are demonstrated in detail by critically examining isomers of dihalobenzenes, cycloalkanes, 2,3,4-trihydroxyglutaric acids, and pentanols as representative examples. The serious confusion is concluded to stem from disregard for the concepts of equivalence relationships and equivalence classes. The conventional definitions of an isomeric relationship, a stereoisomeric relationship, and an enantiomeric relationship are revised respectively to connote a self-isomeric relationship, a self-stereoisomeric relationship, and a self-enantiomeric relationship. Thereby, these relationships are capable of working up to equivalence relationships, which generate equivalence classes of isomers, stereoisomers, and enantiomers. An isoskeletomeric relationship is proposed as an additional equivalence relationship, which generates an equivalence class of isoskeletomers. A new flowchart of classifying isomers is devised on the basis of these equivalence relationships. On the other hand, the term `constitutionally-isomeric' is replaced by the term constitutionally-anisomeric to emphasize the nature of an inequivalence relationship, i.e., the difference nature at which the relationship aims. The term `constitutionally-isomeric' and the plural form `constitutional isomers' are permitted only for the purpose of characterizing 2D structural formulas. The terms skeletally-anisomeric and diastereomeric are also used as inequivalence relationships. The plural form `diastereomers' is permitted if a reference molecular entity is given to be fixed, but its easy usage should be avoided because they do not mean equivalence classes.
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[Abstract] For the purpose of characterizing the stereochemistry and stereoisomerism of oxirane derivatives, the RS-stereoisomeric group C_{2v sigma I} of order 8 has been defined by starting point group C_{2v} of order 4, which specifies the geometric features of an oxirane skeleton. The isomorphism between C_{2v sigma I} and the point group D_{2h} of order 8 is discussed algebraically and diagrammatically. The data necessary to combinatorial enumeration under C_{2v sigma I} e.g., the non-redundant set of subgroups, the subduction of coset representations, and the inverse of the mark table, are prepared by referring to the data of D_{2h}. The fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method, which have been originally developed to accomplish combinatorial enumeration under point groups in the unit-subduced-cycle-index approach (S. Fujita, ``Symmetry and Combinatorial Enumeration of Chemistry'', Springer, 1991), are extended and applied to the combinatorial enumeration of oxirane derivatives under the RS-stereoisomeric group C_{2v sigma I}. Thereby, the numbers of inequivalent quadruplets are calculated in an itemized fashion with respect to the subgroups of C_{2v sigma I} where each quadruplet contained in a stereoisogram is counted once. Such quadruplets are further categorized into five types (type I to V). The enumeration of oxiranes under the point group C_{2v} as well as under the RS-permutation group C_{2 sigma} is also conducted and the results are compared with those of the RS-stereoisomeric group C_{2v sigma I} .
[Abstract] The stereoisogram approach is applied to promolecules derived from an oxirane skeleton. First, the four substitution positions of the oxirane skeleton are examined under the action of the RS-stereoisomeric group C_{2v sigma I}, where point groups for chirality (or enantiomeric relationships), RS-permutation groups for RS-stereogenicity (or RS-diastereomeric relationships), and ligand-reflection groups for sclerality (or holantimeric relationships) are integrated in a consistent way. A notation system for giving R_{a}/S_{a}-descriptors is proposed to specify the absolute configurations of oxirane derivatives on the basis of RS-stereogenicity (or RS-diastereomeric relationships) inherent in type-I, -III, or -V stereoisograms. The concept of chirality faithfulness is revised to give a rational judgement on whether R_{a}/S_{a}-descriptors are labelled in uppercase or lowercase letters. Pseudoasymmetry and extended pseudoasymmetry are discussed on the basis of type-V stereoisograms. Second, the stereoisomeric group C~_{2v sigma I}, which is a supergroup of the RS-stereoisomeric group C_{2v sigma I} is used to characterize the cis/trans- or Z/E-isomerism, where multiple stereoisograms are introduced as graphic representations of stereoisomeric groups. The notation system of specifying Z/E-descriptors is modified to be applicable to oxirane derivatives by adopting ortho-stereogenicity (or ortho-diastereomeric relationships). Finally, the isoskeletal group C~~_{2v sigma I}, which is a supergroup of the stereoisomeric group C~_{2v sigma I}, is used to characterize total features of isomerism based on an oxirane skeleton. Multiple stereoisogram sets are introduced as graphic representations of such isoskeletal groups, where flowcharts for determining types of multiple stereoisogram sets are proposed.
[Abstract] The stereoisogram approach presumes the presence of three pairs of attributes, i.e., chirality/achirality, RS-stereogenicity/RS-astereogenicity, and sclerality/asclerality, in contrast to the modern stereochemistry which presumes the presence of a single pair of chirality/achirality. From this viewpoint of the stereoisogram approach, the scope and limitations of R/S-stereodescriptors of the Cahn-Ingold-Prelog (CIP) system are discussed, where oxirane derivatives are used as probes. In particular, correlation diagrams of stereoisograms are used to comprehend the global and local symmetries of oxirane derivatives. Thereby, R/S-stereodescriptors of the CIP system are found to specify local symmetries at each positions of an oxirane derivative, so that the global symmetry of the oxirane derivative requires the other types of descriptors proposed in Part II of this series R_{a}/S_{a}-descriptors and Z/E-descriptors). A correlation diagram of each position clarifies that R/S-stereodescriptors characterize the RS-stereogenic aspect of absolute configuration in the specification of local symmetries, but by no means the chiral aspect of absolute configuration. In other words, a pair of R/S-stereodescriptors is assigned to a pair of RS-diastereomers, not to a pair of enantiomers. This fact stems from the misleading foundation of the modern stereochemistry that it presumes a pair of chirality/achirality as a single pair of attributes. The pair of R/S-stereodescriptors originally assigned to a pair of RS-diastereomers should be interpreted subsidiarily to characterize the corresponding pair of enantiomers, where the concept of chirality faithfulness is necessary. Chirality-unfaithful cases are discussed by means of stereoisograms. Exceptional cases in the practices of the modern stereochemistry, e.g., pseudoasymmetry and `geometric enantiomers', are discussed in a rational fashion according to the stereoisogram approach.
[Abstract] Fujita's proligand method developed originally for combinatorial enumeration under point groups (S. Fujita, Theor. Chem. Acc., 113, 73--79 (2005)) is extended to meet the group hierarchy, which stems from the stereoisogram approach for integrating geometric aspects and stereoisomerism in stereochemistry (S. Fujita, J. Org. Chem., 69, 3158--3165 (2004)). Thereby, it becomes applicable to enumeration under respective levels of the group hierarchy. Combinatorial enumerations are 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. In these enumerations, stereoskeletons of ligancy 4 are used as examples, i.e., a tetrahedral skeleton, an allene skeleton, an ethylene skeleton, an oxirane skeleton, a square planar skeleton, and a square pyramidal skeleton. Two kinds of compositions are used for the purpose of representing molecular formulas in an abstract fashion, that is to say, the compositions for differentiating proligands having opposite chirality senses and the compositions for equalizing proligands having opposite chirality senses. Thereby, the classifications of isomers are accomplished in a systematic fashion.
[Abstract] Stereoisograms of five types are devised as graphical expressions of the action of an RS-stereoisomeric group on a trigonal pyramidal skeleton. A quadruplet of promolecules (i.e., a reference promolecule, its enantiomer, its RS-diastereomer, and its holantimer) contained in a stereoisogram is counted once as an equivalence class under the action of an RS-stereoisomeric group. An RS-stereoisomeric group C_{3v sigma I} is constructed by starting from the point group C_{3v} of a trigonal pyramidal skeleton, where a point group C_{3v}, an RS-permutation group C_{3 sigma} a ligand-reflection group C_{3 I} are integrated. Fujita's unit-subduced-cycle-index (USCI) approach for enumeration under point groups is extended to cover enumeration under RS-stereoisomeric groups. The fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method of the USCI approach are applied to symmetry-itemized and type-itemized enumeration of quadruplets of promolecules under the action of the RS-stereoisomeric group C_{3v sigma I}. Theoretical foundations of stereochemical nomenclature are discussed on the basis of Fujita's stereoisogram approach, where three aspects of absolute configuration are emphasized. Prochirality and pro-RS-stereogenicity are clarified to be conceptually distinct, just as chirality and RS-stereogenicity are conceptually distinct. Young's tableaux for (pro)chirality and those for (pro-)RS-stereogenicity are compared to demonstrate such conceptual distinctions. It follows that a pair of R/S-stereodescriptors is assigned on the basis of RS-stereogenicity, not of chirality; and a pair of pro-R/pro-S-descriptors is assigned on the basis of pro-RS-stereogenicity, not of prochirality.
[Abstract] For the purpose of characterizing cubane derivatives, the RS-stereoisomeric group O_{h i I} and the corresponding stereoisograms are formulated by the extension of the point group O_{h}, where RS-permutations and ligand-reflections are defined on the basis of the rotations and reflections of O_{h}. According to the hierarchy concerning stereoisomers (i.e., pairs of enantiomers ⊆ quadruplets of RS-stereoisomers ⊆ multiplets of stereoisomers), respective enumerations under the point group O_{h}, under the RS-stereoisomeric group O_{h i I}, and under the corresponding stereoisomeric group are conducted. The results of these enumerations are discussed by categorizing cubane derivatives into those with achiral proligands only and those with achiral and chiral proligands. The importance of the intermediate concept of RS-stereoisomers is emphasized by proposing a flowchart for classifying isomers and stereoisomers. C/A-Descriptors are proposed as a stereochemical notation for specifying global symmetries of cubane derivatives. The assignability of C/A-descriptors to type-I, type-III, and type-V stereoisograms is discussed in comparison with the unassignability of C/A-descriptors to type-II and type-IV stereoisograms. R/S-Descriptors are applied to the specification of local symmetries of cubane derivatives and compared with C/A-descriptors.
[Abstract] The edge strategy of Fujita's unit-subduced-cycle-index (USCI) approach (S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer, 1991) is shown to be effective to the derivation based on a cubane skeleton, where twelve edges accommodate a set of methano-bridges and/or ethano-bridges.
2016 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The conventional flowchart for the classification of isomers into constitutional isomers and stereoisomers and for the further classification of stereoisomers into enantiomers and diastereomers is concluded to be misleading from a viewpoint of equivalence relationships and equivalence classes. A new flowchart based on equivalence relationships and equivalence classes is proposed, where a set of isomers a set of stereoisomers, and a pair of enantiomers are recognized as equivalence classes under the respective equivalence relationships, i.e., an isomeric relationship, a stereoisomeric relationship, and an enantiomeric relationship. A constitutionally-anisomeric relationship (a revisal for a constitutionally-isomeric relationship) and a diastereomeric relationship are redefined as non-equivalence relationships. After the introduction of stereoisograms, a more comprehensive flowchart is proposed, where an RS-stereoisomeric relationship is incorporated as an intermediate concept for mediating between a stereoisomeric relationship and an enantiomeric relationship. The comprehensive flowchart detects RS-stereoisomers, which are classified into five types in accord with type-I to type-V stereoisograms. The comprehensive flowchart is applied to tetrahedral derivatives, allene derivatives, ethylene derivatives, square planar complexes, and oxirane derivatives. The results are summarized in the form of isomer-classification diagrams, which are correlated to correlation diagrams of stereoisograms. The different ways taken by van't Hoff (`asymmetry', `stereogenicity') and Le Bel (`dissymmetry', `chirality') at the beginning of stereochemistry are now integrated by the comprehensive flowchart.
[Abstract] Stereoisograms of octahedral complexes are classified into five types (type I--type V) under the action of the corresponding RS-stereoisomeric group. Their enumeration is accomplished in a type-itemized fashion, where Fujita's proligand method developed originally for combinatorial enumeration under point groups (S. Fujita, Theor. Chem. Acc., 113, 73--79 (2005)) is extended to meet the requirement of Fujita's stereoisogram approach. The cycle index with chirality fittingness (CI-CF) of the point group O_{h} is modulated by taking account of the CI-CF for calculating type-V quadruplets contained in stereoisograms. The modulated CI-CF is combined with a CI-CF of the maximum chiral point group (O), a CI-CF of the maximum RS-permutation group, a CI-CF of the maximum ligand-reflection group, and a CI-CF of the RS-stereoisomeric group, so as to generate CI-CFs for evaluating type-I to type-V quadruplets. By introducing ligand-inventory functions into the CI-CFs, the numbers of quadruplets of octahedral complexes are obtained and shown in tabular forms. Several stereoisograms for typical complexes are depicted. Their configuration indices and C/A-descriptors are discussed on the basis of Fujita's stereoisogram approach.
[Abstract] Programs for calculating cycle indices with chirality fittingness (CI-CFs) have been developed as functions of the GAP (Groups, Algorithms, Programming) system in order to reinforce the practical usage of Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, Mathematical Chemistry Monographs Series, Vol. 15, Kragujevac, 2013). After a mirror-permutation representation of a given point group is defined to differentiate between rotations and (roto)reflections, a combined-permutation representation is newly defined as a computer-oriented representation of the point group. Because such a combined-permutation representation can be regarded as a permutation group, it is generated from an appropriate generators by using the Group function of the GAP system. Thereby, the program (CalcCICF) for generating CI-CFs is developed as a function of the GAP system, which is used to calculate generating functions for combinatorial enumeration of 3D structures under point groups. The program (CalcCICF_A) for calculating the number of achiral 3D structures and the program (CalcCICF_E) for calculating the number of enantiomeric pairs of chiral 3D structures are also developed. A practical procedure for combinatorial enumeration of 3D structures is described on the basis of the GAP system. The source codes of these programs are stored in a file attached as an appendix.
[Abstract] Three kinds of symmetries are derived from stereoisograms, i.e., point-group symmetry, RS-permutation-group symmetry, and ligand-reflection-group symmetry. 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 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-stereogenicity should be used in place of van't Hoff's ``asymmetric carbon atom'', which is discussed by referring to the asymmetry^{(RSP)} restricted to tetrahedral derivatives.
[Abstract] By using allene derivatives as probes, asymmetry under point-group symmetry (denoted as asymmetry^{(P)}) and asymmetry under RS-permutation-group symmetry (denoted as asymmetry^{(RSP)}) are compared with chirality and RS-stereogenicity based on stereoisograms. Van't Hoff's asymmetric carbon atom is extended to asymmetry^{(H)} for supporting allene and other skeletons, which is compared with stereogenicity. The term asymmetry^{(P)} should be adopted to discuss geometric features. To discuss stereoisomeric features, the term asymmetry^{(H)} should be replaced by the term stereogenicity, which supports R/S-stereodescriptors, Z/E-descriptors, and so on. The use of asymmetry^{(RSP)} is desirable to be restricted to analytical purpose. Instead, the term RS-stereogenicity should be used for the purpose of supporting R/S-stereodescriptors.
2017 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] The recognition of chirality as a single kind of handedness is a conceptual defect of modern stereochemistry, which has caused serious confusion in its theoretical foundations and stereochemical nomenclature. To remedy this defect, RS-stereogenicity is developed as another kind of handedness. These two kinds of handedness are integrated to give RS-stereoisomerism, which is formulated diagrammatically by stereoisograms. During the process of the remedy, the lack of the concept of ligand reflections has been revealed as another conceptual defect. The lack of the concept of orbits (equivalence classes) has been found to be one more defect, which has caused a misleading classification of isomers. By adopting the concept of orbits, the revised hierarchy of isomerism is developed, i.e., isomers --- isoskeletomers --- stereoisomers --- RS-stereoisomers --- enantiomers --- 3D-structures. Thereby, the theoretical foundations of modern stereochemistry are restructured rationally.
[Abstract] A new method for the α,β-itemized enumeration has been developed to count inositol derivatives, where each substitution position of a hexagonal skeleton accommodates an α-ligand or a β-ligand exclusively under the D6- or D6h-symmetry. The concept of hedralities of cycles (homohedral cycles, enantiohedral cycles, and hemihedral cycles) has been proposed to treat α,β-handedness for characterizing such exclusive α- or β-substitution. During the α,β-itemized enumeration, the concept of hedralities is found to cooperate with the concept of sphericities of cycles (homospheric cycles, enantiospheric cycles, and hemispheric cyles), which was proposed to treat chirality in Fujita’s proligand method. The effect of hedralities is evaluated by developing the cycle index with α,β-handedness fittingness (CI-HF), which enables us to accomplish the α,β-itemized enumeration under D6. The cooperative effect of hedralities and sphericities can be evaluated by extending Fujita’s proligand method, where the cycle index with pairwise fittingness (CI-PF) is developed to support the α,β-itemized enumeration under D6h. The GAP codes for the enumeration using CI-HFs and CI-PFs are programmed on the basis of the GAP system. The enumeration method based on CI-HFs and CIPFs is generalized to support the α,β-itemized enumeration of m-gonal derivatives under Dm- or Dmh-symmetry. The method for gross enumeration without α,β-itemization and without ligand discrimination is developed as a special case of the extended proligand method described above. General formulas are obtained to give gross numbers as integer sequences.
[Abstract] In order to enhance the applicability of Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, Mathematical Chemistry Monographs Series, Vol. 15, Kragujevac, 2013), functions for calculating cycle indices with chirality fittingness (CI-CFs) have been developed on the basis of the GAP (Groups, Algorithms, Programming) system. After a mirror-permutation representations is defined to characterize a (roto)reflection, a combined-permutation representation is defined as a computer-oriented representation of a point group. Such a computer-oriented representation has been used to develop the GAP functions for calculating CI-CFs, where conjugacy classes are taken into consideration for the purpose of simplifying calculation processes. The source program containing these GAP functions is attached as an appendix. The resulting CI-CFs have been applied to combinatorial enumeration of promolecules derived from Oh-skeletons (an octahedron, a cube, a cuboctahedron, a truncated octahedron, and a truncated hexahedron). A GAP program for calculating the numbers of cubane derivatives is attached as another appendix in order to illustrate a straightforward procedure of Fujita's proligand method.
[Abstract] A point group for specifying chirality and an RS-permutation group for specifying RS-stereogenicity are differentiated by means of mirror-coset representations. They are integrated to generate an RS-stereoisomeric group, after the computer-oriented representation of the RS-stereoisomeric group has been developed by starting from mirror-coset representations. Thereby, the processes of combinatorial enumeration are computerized on the basis of the GAP system. The construction of an RS-stereoisomeric group stems from an appropriate generators by using the Group function of the GAP system. The calculation of the corresponding cycle index with chirality ttingness (CI-CF) is based on the function CalcConjClassCICF developed as a function of the GAP system. The calculation of the corresponding generating function and the evaluation of the coecient of each term are conducted by using newly-developed GAP functions. The source lists of practical procedures for combinatorial enumeration of 3D structures are attached as appendices.
[Abstract] The combined-permutation representations, which have been originally developed for point groups and defined as the combination of permutation representations of groups and a mirror-coset representation, are extended to cover RS-stereoisomeric groups (RS-SIGs) and stereoisomeric groups (SIGs). They are applied to the combinatorial enumeration based on an allene skeleton, where they are used to calculate the cycle indices with chirality fittingness (CI-CFs) according to Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, University of Kragujevac, Faculty of Science, Kragujevac, 2013). They are also applied to the combinatorial enumeration of allene derivatives under the corresponding isoskeletal group (ISG). The CI-CFs for the RS-SIG, the SIG, and the ISG are calculated by using the GAP function CalcConjClassCICF developed recently (S. Fujita, MATCH Commun. Math. Comput. Chem. 76 (2017) 443{478). Although the RS-SIG and the SIG for the allene skeleton are formally identical with each other, they are dierentiated by using different sets of ligand-inventory functions. Then, the calculated data are reported in tabular forms, which are itemized in terms of the compositions (or partitions). A source list for type-itemized enumeration is attached as an appendix.
[Abstract] Combined-permutation representations, which have been originally developed for point groups and defined as the combination of permutation representations of groups and a mirror-coset representation (S. Fujita, MATCH Commun. Math. Comput. Chem. 76 (2016) 379--400), are applied to group hierarchy for characterizing an oxirane skeleton. According to the group hierarchy, the point group C2v (for enantiomerism), the RS-stereoisomeric group C2vsI (for RS-stereoisomerism), the stereoisomeric group eC2sI (for stereoisomerism), and the isoskeletal group eeC2vsI (for isoskeletomerism) are successively defined as combined-permutation representations by starting from respective sets of generators. Then, hierarchical enumeration of oxirane derivatives is conducted by calculating cycle indices with chirality fittingness (CI-CFs) according to Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, University of Kragujevac, Faculty of Science, Kragujevac, 2013). A set of ligand-inventory functions for 3D enumerations and a single ligand-inventory function for 2D (graph) enumeration are defined to discuss interfacial behaviors between 3D structures and 2D structures (graphs). The introduction of these ligand-inventory functions into the CI-CFs provide generating functions for giving enumeration data of oxirane derivatives. These enumeration data are examined by drawing respective isomer-classification diagrams.
[Abstract] Mathematical stereochemistry is discussed by surveying books written on Fujita's USCI (unit-subduced-cycle-index) approach, Fujita's proligand method, Fujita's stereoisogram approach, and related matters from a viewpoint of developing an interdisciplinary chemistry/mathematics field.
2018 [Top of List of Articles] [Bottom of List of Articles]
[Abstract] Group hierarchy for characterizing a cyclopropane skeleton with six substitution positions has been discussed by defining the point group (PG), the RS-stereoisomeric group (RS-SIG), the stereoisomeric group (SIG), and the isoskeletal group (ISG) successively as follows: PG D_{3h} (order 12) < RS-SIG D_{3hσ˜ I-} (order 24) < SIG D^{˜}_{3hσ ˜I-} (order 96) < ISG D^{˜-}_{3hσ ˜I-}(order 1440). Combined-permutation representations, which have been originally developed for point groups and defined as the combination of permutation representations of groups and a mirror-permutation representation (S. Fujita, MATCH Commun. Math. Comput. Chem. 76 (2016) 379--400), are applied to the above group hierarchy. Then, according to Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, University of Kragujevac, Faculty of Science, Kragujevac, 2013), enumerations of cyclopropane derivatives (as 3D structures) under PG and under RS-SIG are conducted by using cycle indices with chirality fittingness (CI-CFs) and a set of three ligand-inventory functions. On the other hand, enumerations of cyclopropane derivatives (as 2D structures or graphs) under SIG and under ISG are conducted by using a single ligand-inventory function, which implies the degeneration of CI-CFs into cycle indices without chirality fittingness (CIs). The enumeration results are discussed systematically in terms of isomer-classification diagrams.
[Abstract] Group hierarchy for characterizing a prismane skeleton with six substitution positions has been discussed by defining the point group (PG), the RS-stereoisomeric group (RS-SIG), the stereoisomeric group (SIG), and the isoskeletal group (ISG) successively as follows: PG D_{3h} (order 12) < RS-SIG D_{3hσ˜ I-} (order 24) = SIG (the same as RS-SIG D_{3hσ˜ I-}) (order 24) < ISG S^{[6]}_{σ I-}(order 1440). On the basis of combined-permutation representations (S. Fujita, MATCH Commun. Math. Comput. Chem. 76 (2016) 379--400), Fujita's proligand method (S. Fujita, Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds, University of Kragujevac, Faculty of Science, Kragujevac, 2013) is modulated and applied to the enumeration of prismane derivatives under the group hierarchy. A set of three ligand-inventory functions is introduced into cycle indices with chirality fittingness (CI-CFs) to give generating functions for the enumeration of 3D structures under PG and RS-SIG. On the other hand, a single ligand-inventory function is introduced into CI-CFs to give generating functions for the enumeration of 2D structures under SIG and ISG. The enumeration results are discussed systematically in terms of isomer-classification diagrams.
[Abstract] The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG G_{σ}) ⊆ RS-stereoisomeric group (RS-SIG G_{σσ ˜I-}) ⊆ stereoisomeric group (SIG G^{˜}_{σσ ˜I-}) ⊆ isoskeletomeric group (ISG G^{˜˜}_{σσ ˜I-} = S^{[6]}_{σ I-}), where we start from the PG G_{σ} = D_{6h} for the Kekule benzene skeleton, from the PG G_{σ} = D_{3h} for the Ladenburg benzene skeleton, from the PG G_{σ} = C_{2v} for the Dewar benzene skeleton, or from the PG G_{σ} = C_{2v} for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms ``skeletal isomerism'', ``positional isomerism'', and ``constitutional isomerism'' are discussed, where the effects of the concept of isoskeletomerism are emphasized.
[Abstract] Hierarchical enumerations of octahedral derivatives are conducted in accord with the hierarchy of groups for characterizing an octahedral skeleton, i.e., point groups (O and O_{h}); orders 24 and 48) ⊂ RS-stereoisomeric group (O_{hσ ˜I-}; order 96) ⊂ stereoisomeric group (O^{˜}_{σ ˜I-} = S^{[6]}_{σ I-}; order 1440) $=$ isoskeletal group (O^{˜}_{σ ˜I-} = S^{[6]}_{σ I-}; order 1440). The corresponding cycle indices with chirality fittingness (CI-CFs) are calculated by using combined-permutation representations (S. Fujita, MATCH Commun. Math. Comput. Chem. 76 (2016) 379--400). Then, a set of three ligand-inventory functions for 3D enumeration is introduced into the CI-CFs for the enumerations under the point groups and the RS-stereoisomeric group, while a single ligand-inventory function for 2D (graph) enumeration is introduced into the CI-CFs for the enumerations under the stereoisomeric group and the isoskeletal group. The expansion of the resulting equations gives generating functions, in which the coefficients of respective terms show the numbers of octahedral derivatives. They are discussed by drawing isomer-classification diagrams after they are categorized into octahedral derivatives with achiral proligands, those with achiral and chiral proligands, and those with chiral proligands. Type-I and type-V stereoisograms are drawn to demonstrate the conceptual distinction between RS-stereogenicity and stereogenicity, where the combination of configuration indices and \textit{C/A}-descriptors is discussed.
[Abstract] Stereoisomerism of cyclopropanes is systematically discussed by means of stereoisograms and their correlation diagrams. A cyclopropane skeleton is governed by a main RS-stereoisomeric group constructed from its point group D_{3h}, where the main RS-stereoisomeric group generates a set of main stereoisograms. An epimerization process at each of carbon centers in the cyclopropane skeleton is characterized by an epimeric RS-stereoisomeric group, which generates a set of epimeric stereoisograms. The main RS-stereoisomeric group is combined with the epimeric RS-stereoisomeric groups, generating a stereoisomeric group for characterizing the cyclopropane skeleton. A set of epimeric stereoisograms at each RS-stereogenic center along with a set of main stereoisograms constructs a correlation diagram of stereoisograms. Such correlation diagrams are used to discuss local chirality and local RS-stereogenicity, where the RS-stereogenicity is correlated to the capability of giving RS-stereodescriptors.
[Abstract]
The stereoisomerism of propane derivatives with bond rotations is investigated
by means of correlation diagrams of stereoisograms, where the diagrams are
governed by epimeric RS-stereoisomeric groups formulated as subgroups
of stereoisomeric groups. A molecular-symmetry group
G_{Cσ}
for specifying the eight positions of a propane skeleton is obtained
as a wreath product with chirality fittingness
(C
_{2v}[C_{3v},C_{∞}]),
where bond rotations are taken into consideration.
The group
G_{Cσ}
of order 36 is extended to give the corresponding RS-stereoisomeric group
G_{Cσσ
˜I-}
of order 72, which characterizes the global symmetry of the propane skeleton.
Then the group
G_{Cσσ
˜I-}
is further extended to give a stereoisomeric group
H
of order 288 for treating chiral and achiral proligands
(or
[Abstract] The conceptual difference between RS-stereoisomerism and stereoisomerism as well as between RS-stereogenicity and stereogenicity is discussed according to Fujita's stereoisogram approach (S. Fujita, Mathematical Stereochemistry; De Gruyter: Berlin, 2015). Enumeration of [2.2]paracyclophanes is conducted to clarify hierarchy of stereoisomerism, i.e., under the point group D_{2} (individual derivatives), the point group D_{2h} (enantiomerism), the RS-stereoisomeric group D_{2hσ ˜I-} (RS-stereoisomerism), the stereoisomeric group sD_{2hσ ˜I-} (stereoisomerism). Stereoisograms for characterizing RS-stereoisomerism are discussed briefly for the purpose of examining nomenclature of [2.2]paracyclophane derivatives. Global and local RS-stereogenicities as well as global and local RS-stereoisomerism are discussed on the basis of two kinds of handedness. Nomenclature for global RS-stereogenicity and for local RS-stereogenicity are discussed after two modes of numbering of skeletal carbons are determined explicitly.
[Fujita's Home Page] [Top of List of Articles]
1969-1979 [Top of List of Reviews and Accounts] [Bottom of List of Reviews and Accounts]
[Abstract] Reactions of active carbon species, carbenes and benzynes, are reviewed (in Japanese).
[Abstract] Chapter 5 (Reactions of Nitrenes) in the monograph of 10 chapters (in Japanese).
[Abstract] A procedure of preparing the title compound from cyclododeca-1,5,9-triene is described (in Japanese).
[Abstract] A procedure of preparing the title compound from 9b-Boraperhydrophenalene is described (in Japanese).
[Abstract] Syntheses and structures of heterophanes are reviewed. In particular, conformational changes of pyridinophanes, pyrazolophanes, furanophanes, and thiophenophanes are discussed in detail. (in Japanese)
[Abstract] A procedure of preparing the title compound from Butyroin is described (in Japanese).
[Abstract] A procedure of preparing the title compound by the oxidative ring-opening of N-Ethoxycaronyl-7-azabicyclo[4.1.0]heptane is described (in Japanese).
[Abstract] N,N-Diethylhydroxylamine is a convenient and versatile reagent for reduction of quinones. Its chemoselectivity and scope and limitaions are discussed in comparison with other reducing agents (in Japanese).
1980-1989 [Top of List of Reviews and Accounts] [Bottom of List of Reviews and Accounts]
[Abstract] This review deals with various image-forming compounds for instant (diffusion transfer) color photography. The processes of the dye-image fromation are discussed in terms of dye-releasing or dye-stopping mechanisms. Dyes containing sufonamide group, temporarily-shifted dye moieties, and after-chelating dye moieties are described. Other organic components such as electron transfer agents, nucleating agents, oxidized developer scavengers, and indicator dyes having high pK_{a} value are also discussed (in Japanese).
[Abstract] Image-providing compounds for instant color photography are classified as positive- or negative-working according to their photographic functions, and as dye-releasing or -stopping to their image-providing mechanisms. Their characteristics and properties as functionalized dyes are discussed briefly. Several synthetic approaches to the compounds selected, e.g., dye developers, p-sulfonamidonaphthol and o-sulfonamidophenol dye-releasers, are reviewed. (in Japanese)
[Abstract] Preparative methods (P), reactions (R), and synthetic applications (S) of quinone bisacetals (I) and monoacetals (II) are reviewed. (P)-I: (1) electrochemical methods (anodic oxidations) and (2) organochemical methods; (R)-I: (1) hydrolysis, (2) hydrogenations, (3) reactions with nucleophiles, and (4) reactions of lithio derivatives of I; and (S)-I: (1) isoprenoid quinones and (2) anthracyclinones. (P)-II: (1) electrochemical methods, (2) organochemical oxidation of p-alkylphenols, (3) partial hydrolysis of quinone bisacetals, and (4) hydrolysis of the monoacetals of p-quinone monosulfonimides or nucleophilic reaction of o-quinone monosulfonimdes; (R)-II: (1) nucleophilic additions, (2) 1,4-additions of carbanions, and (3) reductions; and (S)-II: (1) bishomoquinone and triasteranetrione, (2) tropolones, (3) neolignans, (4) cherylline, (5) anthracyclinones, (6) gymnomitrol, (7) asatone, (8) ryanodol, and (9) alpha-tocopherol (in Japanese).
[Abstract] Mechanism for the dye-releasing process of instant color photography is discussed in terms of the functions of respective layers in multi-layer instant color films.
[Abstract] Research and development of o-sulfonamidophenols for instant color film are described in detail. This work is awarded the 1982 Synthetic Organic Chemistry Award. (in Japanese).
[Abstract] Properties and applications of o-sulfonamidophenol dye-releasers are reviewed in terms of functionalized dyes. (in Japanese)
[Abstract] Rapid access of instant color photography is discussed on the basis of (1) separation of dye images from silver images by diffusion transfer and (2) formation of positive dye images by direct reversal processes. The first point is clarifies by comparing with conventional color photography. With respect of second point, image-providing compounds for instant color photography are classified into positive-working dye-stopping compounds (e.g. dye developers), negative-working dye-releasing compounds (e.g. cye-forming couplers, dye-releasing couplers, amidorazones, thiazolidines, p-sulfonamidonaththol and o-sulfonamidophenol dye-releasers), and positive-working dye-releasing compounds (e.g. dye-releasers by intramolecular cyclizations or by quinone methide formation, BEND compounds, and carquins). And then, direct reversal processes using these compounds are reviewed. The importance of precursors in instant color photography (e.g. temporarily shifted dyes, precursors of silver halide solvents, and of reducing agents) is also outlined. (in Japanese)
[Abstract] Functionalized dyes for color photography (color-forming couplers, dye develpers, dye releasers, and azo dyes for silver dye bleaching process) are reviewed with respect to thier properties and applications. (in Japanese).
[Abstract] Organosulfur compounds for photography are reviewed: (1) thiols and unsaturated thiols for stabilizers or development inhibitors (DI), (2) sulfides used as timing-precursors of DI and as imagewise-precursors of DI, (3) esters of thiols used as timing- or imageqise-presursors of DI, (4) sulfones used as silver halide solvents (and their precursors), as quinone-methide-type dye-releasers and as hardners, and (5) thiazoline dye-releasers. (in Japanese)
[Abstract] Organosulfur compounds for photography, especially those used in timing-mechanisms for instant color photograhy, are reviewed. (in Japanese)
[Abstract] Syntheses, structures, and reactions (reductions, 1,4-additions, hydrolyses, Deils-Alder additions, reactions with aromatic compounds, Grignard reagents, amines, sulfonylchloramides, phosphorus compounds, active methylene compounds, enamines, diazoalkanes, etc.) of the title compounds are reviewed. Their applications to color photography are also described. (in Japanese)
[Abstract] Features of conventional color photography and those of instant color photgraphy are compared. (in Japanese)
[Abstract] Molecules derived from a methane and an adamantane skeleton of T_{d}-point group are classified and their stereochemistries are discussed systematically. (in English)
[Abstract] I discuss various methods of describing organic reactions in order to construct such an integrated system as will support both retriveval and synthetic design: keywords, codes (such as Thilheimer's, Cohen's, GREMAS, Hendrickson's and Letter's one), graphical expressions by Roberts and Hendrickson, mehtods based on connection tables of substrates and products (Corey, Matsuura, and Dubois), and matrix representations (Ugi and Arens). I reveal ``Structure-Reaction type'' paradigm which has been hidden in these conventional methods. (in Japanese)
[Abstract] Several items on dyes for photography are contributed. (in Japanese)
[Abstract] An imaginary transition structure (ITS) is defined as a kind of structure that has par-, out- and in-bonds in accord with structural changes during an aorganic reacation. Various subgraphs (or substructures) of an ITS afford effective pieces of information on reacation types. Thus, imaginary rings in an ITS are the descriptors of ring-opening, ring-closure, and rearrangement reacations. Three- and four-nodal subgraphs correspond to reaction-site changes. Other useful concepts such as reaction-center graphs, reaction graphs, and reaction strings are also discussed. An algorithm for selecting an essential set of essential rings is proposed. The first method for giving a canonical name to ITS is presented. Enumeration of reacation types is also discussed (in Japanese).
1990-1999 [Top of List of Reviews and Accounts] [Bottom of List of Reviews and Accounts]
[Abstract] Molecular and syntheitic design of o-sulfonamidophenol dye-releasers for instant color photography is described. These dye-releasers are oxidized to the corresponding o-quinone monosulfonamides, which are in turn hydrolyzed to release diffusible dyes. The efficiency of the dye-releasing process is influenced by substituents on the phenyl group of the o-sulfonamidophenyl moiety. Side reactions of the intermediate o-quinone-monosulfonamide are reduced by steric hindrance of t-alkyl substituents. The effects of substituents on the heat-fastness of released dyes are discussed. Several unit reactions are developed: e.g., bezoxazole synthesis, convenient methods of preparing sulfonyl chlorides, a procedure of neucleophilic substitution of an aromatic ring, and a new reducing agent of quinoes (in Japanese).
[Abstract] The multilayer structure of instant color film is discussed in relation to the dye-releasing mechanism of o-sulfonamidophenol dye releasers. This discussion reveals several requirements necessary for such dye releasers. In order to satisfy these equirements, we design appropriate targets, which is the frist phse of R&D (molecular design). For developing the o-sulfonamidophenol moiety of a dye releaser, the dye-releasing efficiency of o-sulfonamidophenols with various substituents is sexamined in terms of steric hindrance, elecronic effect, and so on. As for the dye moiety of a dye releaser, we deal with the dark stability of magenta dyes and temporary shift of visible absorption. The second phase of R&D is the synthetic design of dye releasers, which involves the selection of synthetic routes and investigation of unit reacations. For the selection of synthetic routes, we have developed several new rouutes to alkyl-substituted o-aminophenols. Of the unit reaction developed, we discuss benzoxazole synthesis, aromatic nucleophilic substitution for preparing 2-methoxyethoxy-containing moieties, preparation of sulfonyl chlorides, and reduction of quinone derivatives. (in English)
[Abstract] The importance of molecular design and synthetic design is emphasized for the R&D of functionalized organic compounds. The cooperation of design, evaluation, and synthesis in the development of dye releasers for instant color photography is discussed as a typical case. (in English)
[Abstract] FORTUNITS (Fuji Organic Rection Treating Unity based on Imaginary Transition Structures) is a computer system for the registration and retrieval of orgnaic ractions, which consists of three time-sharaing subsystems (the subsystems of registration, of registration of recipe and bibliography, and of retrival) and two batch subsystems (the subsystems of analyzing imaginary transition structures and of analyzing compound data). The developement and the functions of FORTUNITS are described. (in English)
[Abstract] FORTUNITS based on imaginary transition structures (ITSs) is developed as an organic-reaction database, in which the retrieval of reacation types is carried out in terms of substructures of ITSs as queries. (in Japanese)
[Abstract] Merits and demerits in the preparation of manuscripts with LaTeX are discussed in order to aim at electronic publication. LaTeX as a markup language is compared with word processors as a layout-oriented method. (in Japanese)
[Abstract] Applications of the group theory to chemistry are discussed with emphasis on the importance of coset representations. (in English)
[Abstract] The basic ideas for the USCI (unit-subduced-cycle-index) appraoch are discussed. The importance of coset representations and their subductions is emphasized. (in Japanese)
[Abstract] An in-house database system of organic compounds on local area network is developed. It has advantages over conventional database systems of time-sharing stragegy, especially in user-interphase. (in Japanese)
[Abstract] SPHINCS Light, an in-house chemical substance database system in the LAN system, has been constructed by employing ORACLE as a server database engine, where a searching system based on tree-structured data is adopted. (in Japanese) On line
[Abstract] The polyoxa- or polyaza-derivatives of superphanes are enumerated combinatorially by using the partial-cycle-index method of the USCI (unit-subduced-cycle-index) approach. Subsequently, the SCR (set-of-coset-representation) notation is discussed for designating the stereochemistry of each superphane derivative. (in English)
[Abstract] An introductory guide to the Laboratory of Information Materials Technology, Kyoto Institute of Technology (in Japanese).
2000-2009 [Top of List of Reviews and Accounts] [Bottom of List of Reviews and Accounts]
[Abstract] XyM Notation proposed for printing chemical structural formulas has been extended to be applied to WWW communiction. XyMTeX and XyMJava systems based on XyM Notation have been developed. (in Japanese)
[Abstract] The concept of sphericity and relevant fundamental concepts that we have proposed have produced a systematized format for comprehending stereochemical phenomena. Permutability of ligands in conventional approaches is discussed from a stereochemical point of view. After the introduction of orbits governed by coset representations, the concepts of subduction and sphericity are proposed to characterize desymmetrization processes by using a tetrahedral skeleton as an example. The stereochemistry and stereoisomerism of the resulting promolecules (molecules formulated abstractly) are discussed in terms of the concept of sphericity as a common mathematical and logical framework. Thus, these promolecules are characterized by point-group symmetry and permutation-group symmetry. Prochirality, stereogenicity, prostereogenicity, and relevant topics are described in terms of the concept of sphericity. (in English)
[Abstract] Fundamental concepts proposed for comprehending stereochemical phenomena are described in order to bring out a more systematized format to conventional concepts and terminologies. The concept of coset representation (CR), the symbol of which is G(/G_{i}), is intuitively introduced by considering the global symmetry G and the local symmetry G_{i} for an orbit of stereochemically equivalent objects (e.g. atoms in a molecule). The SCR (set-of-coset-representations) notation for classifying molecular symmetries is proposed to remedy the usual insufficient classification based on point groups. According to the chirality/achirality of G and G_{i}, the sphericity concept (homospheric, enantiospheric, and hemispheric) and the relevant concept of chirality fittingness are proposed to specify the stereochemical phenomena of the G(/G_{i})-orbit. The sphericity terms are shown to be superior to the well-known topicity terms (homotopic, enantiotopic, and heterotopic; chirotopic and achirotopic) especially in comprehending complicated stereochemical phenomena. In fact, the topicity terms as well as the terms ``stereogenicity and prostereogenicity'' and ``prochirality'' can be derived subsidiarily from the sphericity terms. The concept of subduction of CRs, for which the symbol G(/G_{i})|G_{j} has been coined, is proposed to characterize the desymmetrization of molecules. Thereby, the design of high-symmetry molecules is discussed in term of desymmetrization by atom replacement and by bond replacement. In order to characterize the stereochemistry of non-rigid molecules with rotatable ligands, the concept of proligand/promolecule is proposed; here biphenyl derivatives, methane derivatives, ethane derivatives, and ferrocene derivatives are examined in terms of matched and mismatched molecules. Applications to combinatorial enumeration, symmetry adapted functions, flexible six-membered cyclic compounds, and symmetry numbers are also described. (in English)
[Abstract] The XyM Notation has been proposed and developed as a linear notation that is applicable to printing and internet distribution of chemical documents containing structural formulas. It has been compared with other representations of chemical structural formulas (e.g., connection tables). (In Japanese).
[Abstract] Stereoisograms proposed for reexamining the relationships between stereoisomers (RS-stereoisomers) contain three relationships (enantiomeric, RS-diastereomeric, and holantimeric relationships), which correspond to three attributes (chirality, RS-stereogenicity, and sclerality). The three relationships are correlated to three relationships for describing internal structures: enantiotopic, RS-diastereotopic, and holantitopic relationships. Among them, the term enantiotopic is used to specify the term {\em prochirality}, while the term RS-diastereotopic is used to specify the term pro-RS-stereogenicity. The term pro-RS-stereogenicity is clarified to correspond to the pro-R/pro-S-nomenclature, just as the term RS-stereogenicity corresponds to the RS-nomenclature. Substitution criteria based on stereoisograms are introduced to determine pro-RS-stereogenicity as well as prochirality (In Japanese).
[Abstract] Global situations when I published "Organic Chemistry of Photography" from Springer-Verlag (2004) are discussed from the viewpoint of the rise and fall of technology: the global trend shifted from color photography based on silver-salt chemistry to digital photography, the convenience of patent databases, the utility of the XyMTeX system for drawing structural formulas, etc.
[Abstract] The drawing of chemical structural formulas by the XyMTeX system has been introduced with referring to the IUPAC nomenclature of organic compounds.
2010-2019 [Top of List of Reviews and Accounts] [Bottom of List of Reviews and Accounts]
[Abstract] Enumeration of alkanes and monosubstituted alkanes with given carbon contents has been investigated by chemists and mathematicians over 130 years and solved recently by the present author in agreement with stereochemical and mathematical requirements. In the present account, the advances of the methodologies for solving the problem are discussed from an interdisciplinary point of view between chemistry and mathematics. Historical backgrounds of the interdisciplinary problem are introduced by emphasizing three epochs, i.e., the first epoch marked by Cayley, a mathematician (the 1870s), the second epoch by Polya, a mathematician (the 1930s), and the third epoch by Fujita, an organic chemist (the first decade of this century). Among them, the accomplishments of the second epoch and those of the third epoch are compared in detail, where graphs (trees, rooted trees, or planted trees) and three-dimensional (3D) objects as mathematical terms are correlated to constitutions (two-dimensional structures) and 3D structures as chemical terms. After an introduction to terminology on isomerism and stereoisomerism, alkanes and monosubstituted alkanes are enumerated as graphs or constitutional isomers by Polya's theorem, while they are alternatively enumerated as 3D objects or 3D-structural isomers by Fujita's proligand method. The present account of the long-standing problem would provide readers with a hint or a motivation for pursuing a concrete route to "the Heavens of Fujita", which have caricatured stereochemical and mathematical barriers lying in wait for them.
[Abstract] The USCI (unit subduced cycle index) approach proposed by us is explained by using proligands, chirality fittingness as keywords for underestaning.
[Abstract] The markaracter method proposed by us is applied to the combinatorial enumeratin of fullerene C_{60} derivatives. The features of this method is compared with those of the USCI approach.
[Abstract] The characteristic-monomial method and the proligand method proposed by us are applied to the combinatorial enumeratin of fullerene C_{60} derivatives. Their features of these method are compared.
[Abstract] The USCI approach is extended to count the Kekule structures of of fullerene C_{60}, where the substitution of edges in fullerene C_{60} is counted under the restriction condition of no occupation of a common vertex by proposing territory variables and territory indicators.
[Abstract] The stereoisogram approach concludes that chirality and RS-stereogenicity are independent concepts and that prochirality and pro-RS-stereogenicity are independent concepts. Accordingly, the conventional approach should be reorganized, where such confused terms as ``chirality'', ``stereogenicity'', ``prochirality'', and ``prostereogenicity'' should be replaced by newly-defined attributive terms based on the stereoisogram approach, e.g., chirality (as a purely geometric term), RS-stereogenicity, prochirality (as a purely geometric term), and pro-RS-stereogenicity. Such relational terms as ``enantiotopic'', ``diastereotopic'', and ``stereoheterotopic'' in the conventional approach should be replaced by newly-defined relational terms based on the stereoisogram approach, e.g., enantiotopic (as a purely geometric term), RS-diastereotopic, and RS-stereoheterotopic.
[Abstract] After the conventional stereochemistry was demonstrated to be based on enantiomeric and stereoisomeric relationships as relationships which correspond to equivalence classes, its theoretical basis has been discussed to have the following weaknesses to be overcome: (1) oversimplified dichotomy between enantiomers and diastereomers, (2) no single operation for judging a diastereomeric relationship, (3) no sufficient rationalization on the difference between chirality and stereogenicity, (4) no sufficient theoretical basis on RS-stereodescriptors, (5) different, sometimes contradictory definitions of prochirality, (6) no sufficient theoretical basis on pro-R/pro-S-descriptors, and (7) no integrated theory with sound mathematical foundations. On the other hand, the present approach based on stereoisograms has been clarified to have merits capable of overcoming these weaknesses, because enantiomeric relationships, RS-diastereomeric relationships, holantimeric relationships, RS-stereoisomeric relationships, and stereoisomeric relationships are introduced as relationships which are capable of generating equivalence classes. Further discussions have been developed so that a key point to construct a consistent system of stereochemistry is found to be whether relationships selected as foundations correspond to equivalence classes.
[Abstract] Written for Lounge (Letter from Yugo-kai). The review published in Shinsaku Fujita Bull. Chem. Soc. Jpn，83, 1--18 (2010) is summarized in Japanese.
[Abstract] Written for Lounge (Letter from Yugo-kai).
[Abstract] My half-century journey started from synthetic organic chemistry. During the first stage of my journey, my interest in stereochemistry was initiated through the investigation on the participation of steric effects in reactive intermediates, cylophanes, strained heterocycles, and organic compounds for photography. In chemoinformatics as the next stage of the journey, I proposed the concept of imaginary transition structures (ITSs) as computer-oriented representation of organic reactions. My interest was stimulated to attack combinatorial enumeration through the investigation on enumeration of subgraphs of ITSs. Stereochemistry and combinatorial enumeration was combined in my interest, so that I reached mathematical stereochemistry as the final stage of my journey. Fujita's unit-subduced-cycle-index (USCI) approach, Fujita's proligand method, and Fujita's stereoisogram approach were developed, so as to integrate van't Hoff's way (asymmetry, stereogenicity) and Le Bel's way (dissymmetry, chirality), which caused continuous confusion in the history of stereochemistry.
[Abstract] The feasibilities of Fujita's unit-subduced-cycle-index (USCI) approach (Monographs: S. Fujita, ``Symmetry and Combinatorial Enumeration in Chemistry'', Springer, 1991; and S. Fujita, ``Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers'', University of Kragujevac, 2007), Fujita's proligand method (Monograph: S. Fujita, ``Combinatorial Enumeration of Graphs, Three-Dimensional Structures, and Chemical Compounds'', University of Kragujevac, 2007), and Fujita's stereoisogram approach (Monograph: S. Fujita, ``Mathematical Stereochemistry'', De Gruyter, 2015) have been demonstrated by applying them to cubane derivatives as probes. They provide us with a new set of theoretical foundations for comprehensive investigation on geometric features and stereoisomeric features of stereochemistry. The new set of theoretical foundations is based on mathematical formulations so as to explore mathematical stereochemistry as a new interdisciplinary field of stereochemistry.
[Abstract] Chirality under point-group symmetry and RS-stereogenicity under RS-permutation-group symmetry are discussed from a viewpoint of two kinds of handedness, which are proposed on the basis of whether or not ligand reflections are taken into consideration. After the additional formulation of sclerality under ligand-reflection-group symmetry, the three groups are integrated to generate RS-stereoisomeric groups, which are represented by stereoisograms as diagrammatic expressions. The vertical direction of a stereoisogram is concerned with (self-)enantiomeric relationships or chirality/achirality for supporting Le Bel's way, while the horizontal direction is concerned with (self-)RS-diastereomeric relationships or RS-stereogenicity/RS-astereogenicity for supporting van't Hoff's way. By taking account of the diagonal direction for characterizing (self-)holantimeric relationships or sclerality/asclerality, the two kinds of handedness (chirality and RS-stereogenicity) are integrated to develop the concept of RS-stereoisomerism, which is an intermediate concept for mediating between enantiomerism and stereoisomerism. The creation of RS-stereoisomerism as an intermediate concept means the Aufheben of van't Hoff's way and Le Bel's way. Thereby, Fujita's stereoisogram approach has brought about a paradigm shift, so that modern stereochemistry has been restructured substantially on the basis of mathematical formulations, where true remedies for the misleading terminology of modern stereochemistry have been developed in a rational fashion. In particular, the hierarchy of isomers and stereoisomers has been thoroughly revised by adding RS-stereoisomerism, so as to develop a new flowchart based on equivalence relationships and equivalence classes, as exemplified by using various skeletons as probes.
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[Abstract] The development of instant color photography based on o-sulfonamidophenol dye-releasers is described: (1) development of o-sulphonamidophenol moeities, (2) development of dye moeties, and (3) syntheses.
[Abstract] The concept of imaginary transition structures (ITSs) is proposed as a new representation of organic reactions. Since ITSs are regarded as extended structural formulas, the retrieval of reaction types are considered to be the substructure search of ITSs.
[Abstract] The concept of imaginary transition structures (ITSs) is proposed for the purpose of contributing to chemical informatics. This lecture attracted much attention so as to be cited in Chem. Eng. News, 64, 75--76 (1986) as one of the topics of the ACS Meeting.
[Abstract] The concept of imaginary transition structures and its application to the comuper-oriented taxonomy of organic reactions were presented as one of the lectures invited to the Colloqium on Computer Chemistry in the IUPAC Symposium.
[Abstract] Combinatorial enumeration of organic reactions are ascribed to the problem of counting basic reaction graphs with an appropriate set par-bonds. The problem is solved by means of the USCI approach.
[Abstract] The construction of an organic-reaction database based on imaginary transition structure was presented.
[Abstract] The construction of an organic-reaction database based on imaginary transition structure was presented.
[Abstract] By starting from imaginary transition structures, the necessity of combinatorial enumeration, especially the USCI approach, in information chemistry was discussed.
[Abstract] A methodology of Japanese vertical typesetting (tagegumi) was discussed in detail.
[Abstract] A new method based on characteristic monomials (CMs) is proposed for combinatorial enumeration of isomers. This method is compared with the one based on Polya's theorem so that Q-conjugacy representations adopted for deriving CMs are shown to be a key to discuss linear and permutation representations on a common basis. These methods are compared with the methods of the unit-subduced-cycle-index approach proposed earlier by us (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry", Springer-Verlag, Berlin-Heidelberg, 1991).
[Abstract] The concept of maginary transition structures and its application to the taxonomy of organic reactions and the combinatorial enumerations of reaction types are discussed in detail.
[Abstract]
The USCI approach based on
the concept of sphericity has been discussed.
Thirty-First Year of the Topological Index Z, Tokyo (2002).
[Abstract] The USCI approach based on the concept of sphericity has been discussed.
[Abstract] How to use the XyMTeX system in the publication of books containing organic structural formulas in addition to mathematical formulas. Slides for TUG2013
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