Sphericities of Double Cosets,Double Coset Representations,and Fujita’s Proligand Method for Combinatorial Enumeration of Stereoisomers

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 and 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.     

Sphericity governs both stereochemistry in a molecule and stereoisomerism among molecules

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, with 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 and permutation group symmetry. Prochirality, stereogenicity, prostereogenicity, and relevant topics are described in terms of the concept of sphericity.     

Group-theoretical Foundations for the Concept of Mandalas as Diagrammatical Expressions for Characterizing Symmetries of Stereoisomers

Group-theoretical foundations for the concept of mandalas have been formulated algebraically and diagrammatically in order to reinforce the spread of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991). Thus, after the introducton of right coset representations (RCR) (H\)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\)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\)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.     

Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _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 theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-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.     

Double coset for symmetry orbitals

In this paper it is shown that, by use of the double coset decompositions of the point symmetry group and permutation group, the related symmetry coefficients and Clebsch–Gordan coefficients can be given in close formulas.     

Comments on group theoretical analysis of NQR spectra of crystals

The use of generating function methods for the number of NQR lines of crystals exhibiting distortion is outlined. The intensity ratios of NQR lines can be obtained using a double coset method.     

Enumerations of chemical structures by subduction of coset representations. Correlation of unit subduced cycle indices to Pólya's cycle indices

Subduction of the coset representations and the related concepts such as unit subduced cycle indices and subduced cycle indices yields the foundation for a new type of generating function for enumerating chemical structures. This method is related to Pó1ya's theorem.     

Symmetry-adapted integral derivatives

A procedure, based on double coset decompositions, is described for reducing formulas for derivatives (with respect to nuclear coordinates) of integrals over symmetry-adapted orbitals to symmetry-distinct integral derivatives over atomic orbitals. The procedure is applicable to any finite point group and to integral derivatives of any order.     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

Global topological approach to highly excited vibration: a case study of H2O, CH2Br2 and CD2Br2

A global topological approach based on the algebraic coset representation to study highly excited molecular vibration is proposed. The algorithm allows us to elucidate ample highly excited vibrational dynamics from a global viewpoint. Topics for highly excited states studied include (1) global topological structures, (2) global symmetric and antisymmetric characters, (3) resonance overlaps and chaotic motion, (4) action transfer coefficients and (5) survival probabilities denoting the decaying of the action stored in a mode, with sample calculations for H₂O, CH₂Br₂ and CD₂Br₂.     

Real irreducible tensorial sets

A simple formalism of real irreducible tensorial sets of real bases is proposed. The definition of the real bases, the coupling of the real bases, and the transformation of the real bases in a group chain including the three-dimensional rotation group and the molecular point groups are studied. The double coset technique is used to derive the close formulas for generating the coupling coefficients and the transformation coefficients. © 1996 John Wiley & Sons, Inc.     

Subduction of coset representations

Summary Enumerations of compounds based on a parent skeleton with and without the influence of obligatory minimum valency (OMV) are reported. The effect of the OMV is formulated by assigning different weights to the respective 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.     

Dynamical similarity in the highly excited vibrations of HCP and DCP: The dynamical potential approach

The fixed points in the dynamical potentials of phosphaethyne (HCP) and deuterated phosphaethyne (DCP) derived in the coset space are identified and shown to govern the various quantal environments in which the vibrational states lie. The state dynamics is interpreted and classified by the classical actions and action integrals. This is closely related to the fixed point structure. Localized modes even at high excitation are identified. Most important is that the dynamical similarity between these two systems is identified which enables us to understand the DCP dynamics simply from that of HCP without repeated elaboration.     

Enumeration of organic reactions by counting substructures of imaginary transition structures. Importance of orbits governed by coset representations

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 Pólya's theorem, in which the transitivity of the positions of the skeletons is explicitly considered. Thus, a permutation representation acting on the positions is reduced into cosec 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 the coset representations. This contains useful concepts such as unit subduced cycle indices and subduced cycle index that afford a new type of generating functions.     

$$PSL\left( 2,7\right) $$ and carbon allotrope D168 Schwarzite

We investigate actions of the modular group \(PSL(2, \mathbb {Z} )\) on the projective line over finite fields \(PL(\mathbf {F}_{7^{n}})\) and find interesting relation between the coset diagram of orbits and the carbon allotrope with negative curvature D168 Shewarzite. We also highlighted some topological aspects of these diagrams.     

Subduction of coset representations. An application to enumeration of chemical structures with achiral and chiral ligands

Molecules derived from a parent skeleton are enumerated where both achiral ligands as well as 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 determines the mode of substitution with chiral and achiral ligands. Several novel concepts such as a unit subduced cycle index and a subduced cycle index are given in general forms.     

Combinatorial properties of graphs and groups of physico-chemical interest

Combinatorial properties of graphs and groups of physico-chemical interest are described. A type of mathematical modeling is applied which involves "translating" algebraic expressions into graphs. The idea is applied to both graph theory and group theory. The former topic includes objects of importance in physics and chemistry such as trees, polyomino graphs, king boards, etc. Our study along these lines emphasizes nonadjacency relations, graph-generation, quasicrystals, continued fractions, fractals, and general ordering schemes of graphs. The second part of the paper considers certain colored graphs as models of several group-theoretical concepts including coset representations of groups, subduction of groups, character tables, and mark tables which are essential to the understanding of recent developments of combinatorial enumeration in chemistry.     

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

Symmetry simplifications are introduced in configuration interaction (CI ) by reducing the number of symmetry-allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group S_n, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.     

Dominant representations and a markaracter table for a group of finite order

Summary The concept of markaracter is proposed to discuss marks and characters for a group of finite order on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominant representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers of fixed points corresponding to each DR are collected to form a row vecter called a dominant markaracter (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 a 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 corresonding to an irreducible representation is equal to zero.     

Stereoisograms for reorganizing the theoretical foundations of stereochemistry and stereoisomerism. Part 3: rational avoidance of misleading standpoints for pro-R/pro-S-descriptors

The stereoisogram approach is introduced to settle the misleading terminology due to ‘prochirality’ in 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 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 and not on prochirality.