Prochirality revisited. An approach for restructuring stereochemistry by novel terminology

The importance of orbits (equivalence classes) has been stressed 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.     

Combinatorial enumeration of three-dimensional trees as stereochemical models of alkanes: an approach based on Fujita’s proligand method and dual recognition as uninuclear and binuclear promolecules

Three-dimensional (3D) trees, which are defined as a 3D extension of trees, are enumerated by Fujita’s proligand method (Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 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 (van’t~Hoff, Arch. Néerlandaises des Sci. Exactes et Nat, 9 (1874) 445–454”) and by LeBel (LeBel, Bull. Soc. Chim. Fr. (2), 22 (1874) 337–347) and the enumeration problems (on the number of trees) by Cayley (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.     

Stereogenicity/Astereogenicity as Global/Local Permutation-Group Symmetry and Relevant Concepts for Restructuring Stereochemistry

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 [S. Fujita, J. Am. Chem. Soc. 112 (1990) 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 local stereogenicity of the present investigation. The terms homotropic, enantiotropic, and 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 CRs. The Cahn–Ingold–Prelog (CIP) system has been discussed 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.     

Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes

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 1–3 of this series [Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously [Fujita in Tetrahedron 47:31–46, 1991], a 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 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 mathematician Cayley [Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

Combinatorial enumeration of nonrigid isomers with given ligand symmetries on the basis of promolecules with a subsymmetry of D infinity h

To enumerate nonrigid isomers with given ligand symmetries on the basis of a D infinity 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 infinity h-group is concealed by adopting the factor group of finite order, D infinity h/C infinity (= 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 infinity h/C infinity (= K3 [symbol: see text] K). Each term of the resulting generating functions has been factorized into a pair of factors to represent ligand constitutions. 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.     

Systematic enumeration of nonrigid isomers with given ligand symmetries

A new method for enumerating nonrigid 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.     

Chemical reactivity concepts in charge sensitivity analysis

Recent reactivity concepts formulated within charge analysis (CSA ) are outlined. The charge stability criteria of equilibrium states in open and closed systems are conveniently characterized in terms of the condensed reactant hardness quantities of reactants; their implications for catatytic systems are examined. A use of characteristics associated with selected collective charge displacement modes, including the populational normal modes and minimum-energy coordinates, as diagnostic tools in the theory of chemical reactivity is proposed. The importance of the mapping relations between modes defined in the electron population and nuclear position spaces, respectively, as the unifying concept linking the conjugate charge and geometry displacements, is commented upon. Recent results for model catalytic clusters are used to illustrate some of the concepts introduced. Finally, the relevant contributions to the quadratic interaction energy between reactants are reexamined and expressed in terms of relevant charge sensitivities. © 1995 John Wiley & Sons, Inc.     

Group-theoretical discussion on the E/Z-nomenclature for ethylene derivatives. Discrimination between RS-stereoisomeric groups and stereoisomeric groups

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 extended stereoisograms and 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.     

Chirality and Stereogenicity of Square‐Planar Complexes

Square‐planar complexes with achiral and chiral ligands have been enumerated exhaustively under the point‐group D _4h and under the symmetry 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.     

Chiral stereochemistry of nanoparticles

The whole array of organic and inorganic nanoparticles, from fullerenes and carbon nanotubes to metals and metal oxides, is discussed in terms of chirality and its stereochemical consequences on the basis of the unified “core-shell” concept.     

Enantiomeric and Diastereomeric Relationships of Ethylene Derivatives. Restructuring Stereochemistry by a Group-Theoretical and Combinatorial Approach

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 ₉ ^[4]), which is a subgroup of the symmetric group of order 4 (S ^[4]) and isomorphic to a point group D _2d . The subgroups of S ₉ ^[4] 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.     

The symmetry of molecular polarization tensors

Polarization tensors are discussed in terms of their intrinsic symmetry group which is a direct product of the point group and the subgroup of the permutation group relevant to the experiment. The study of these latter groups is simplified by use of the isomorphism with certain point groups and permutations of suffixes can be visualized by rotations and reflections of the vertices of various objects in space. The approach unites the previous treatments and provides a means of constructing the bases for the irreducible tensor components. The difficulties introduced by Laplace's equation are explained and the information obtainable from induced birefringence experiments (Kerr and Cotton–Mouton effects) discussed for various systems.     

Electron group functions for the analysis of the electronic structures of molecules

The electronic structure of a vast majority of molecular systems can be understood in terms of electron groups and their wave functions. They serve as a natural basis for bringing intuitive chemical and physical concepts into quantum chemical calculations. This article considers the general electron group functions formalism as well as its simple geminal version. We try to characterize the wave function with the group structure and its capabilities in actual calculations. For this purpose we implement a variational method based on the wave function in the form of an antisymmetrized product of strongly orthogonal group functions and perform a series of electronic structure calculations for small molecules and model systems. The most important point studied is the relation between the choice of electron groups and the results obtained. We consider energetic characteristics as well as optimal geometry parameters. In view of practical importance, the structure of variationally optimized local one-electron states is considered in detail as well as intuitive characteristics of chemical bonds.     

An Alkyl-Aluminium Promoted Stereospecific Addition to Vinyl Olefins

The reaction of tris-crotyl-boron with vinylic olefins in the presence of triethyl-aluminium, results in the addition of an α-methylallyl group and an aluminium atom to the double bond, as indicated by the hydrolysis products. The stereochemical aspect of the reaction is discussed and a mechanism is proposed to explain the preferential formation of one diastereoisomer.     

Biomimetic Assembly Lines Producing Natural Product Analogs: Strategies from a Versatile Manifold to Skeletally Diverse Scaffolds

Biosynthetic assembly lines have evolved in nature, adopting divergent processes to produce a vast number of secondary metabolites. Inspired by these biogenetic processes, this account introduces recent investigations by my research group to formulate a synthetic strategy for establishing a biomimetic assembly line. With the aim not only to construct natural product‐relevant scaffolds within 5–7 steps, but also to systematically diversify skeletal and stereochemical properties and functional groups, divergent synthetic processes exploiting a versatile manifold have been developed. This approach allows for cost‐effective production of skeletally diverse and biologically active natural product analogs inaccessible by other means. Discovery of several lead candidates for a neglected tropical disease is a proof‐of‐concept of this synthetic approach.     

Concept system on ‘quantity’: formation and terminology

The third edition of the International Vocabulary of Metrology occasions continued useful discussion. A Commentator, in four recent papers, presents viewpoints on the generic division of ‘property’, ‘quantity’, and ‘ordinal quantity’, as well as a perceived problem of inheritance. Alternative terms for ‘kind-of-quantity’, ‘kind-of-quantity with chosen sort of component’, and ‘dedicated kind-of-quantity’ are suggested to be formed by prefacing “quantity” with the respective adjectival modifiers “generic”, “subgeneric”, and “specific”. Furthermore, a neoterm for ‘quantity expressed by a measurement unit’ and a redefinition of ‘property’ are offered. The Commentator’s proposals and arguments are discussed with emphasis on ways of constructing concept systems to ensure proper inheritance. Some formal and semantic problems of the proposed alternative terms and definitions are advanced. Terminological concepts are defined in an Appendix.     

Mass spectral study of cyclic alkaneboronates of sugar phosphates. Evidence of interaction between the phosphate group and boron under electron-impact conditions

Cyclic butane- and methaneboronic esters of dimethylphosphates of glucose, galactose, mannose, fructose and methyl gluconate have been examined by mass spectrometry. High resolution information and deuterium labeling of all six carbons of glucose, as well as of the methylphosphate groups, permits elucidation of the origin of various ions, which can be grouped according to their probable mechanism of formation. In each case, the intensity of the [M – R]⁺ ion can be rationalized, in stereochemical terms, by the ease with which incipient positive charge on boron can be stabilized by the phosphate group. This interaction also gives rise to rearrangement ions containing a B? O? P linkage. Other effects of structure on fragmentation pattern are discussed. Retention of specific portions of the glucose skeleton in ions with clearly defined label shift behavior, as well as the preservation of intact sugar moieties in intense [M – R]⁺ ions, suggests applications for stable isotope analyses. These are discussed in terms of their advantages over silicon-containing derivatives.