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.     

Subduction of dominant representations for combinatorial enumeration

Summary 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 representations corresponding to the dominant subgroups. By starting from the markaracter table, we propose an essential set of subdominant markaracter tables and a magnification set for the group G; the latter concept clarifies the relationship between each subdominant markaracter table and the markaracter table of a dominant subgroup. The subduction of DRs is obtained by the markaracter table to produce a dominant subduction table and a dominant USCI (unit-subduced 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 couterpart of our previous approach concerning both cyclic and non-cyclic subgroups. The latter, in turn, has been proved to be equivaltent to the cycle index obtained by the Redfield-Pólya theorem.     

Enumeration of digraphs with a given automorphism group

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 roup is obtained.Dedicated to Professor Frank Harary.     

Reactivity indices: Remarks on present state and prospects

Summary After arguing that reactivity indices are still a very important tool of molecular design, the general perturbation treatment of reactivity indices, originally due to Coulson and to Fukui, is revisited on the basis of its most general (but little familiar) form, when all electrons are taken into account, and the AO basis is nonorthogonal. The difficulties in the physical interpretation of the generalized "perturbabilities" are discussed. An example of how a new reactivity index can be derived starting with a supermolecule model in the reduced coupling approximation is proposed not so much with the intent to add one more individual to the bestiary of those indices as to give a concrete illustration of the path to be followed to get an index of actual significance and sufficiently general scope. A number of comments on the situation and prospects of work in the field are proposed.     

On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.     

The USCI approach and elementary superposition for combinatorial enumeration

Summary The elementary superposition theorems are presented for enumerating chemical compounds that contain achiral and chiral ligands. Subduced cycle indices (SCI-CF), partial cycle indices (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 methodology 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 operation is defined on the basis of the elementary superposition. Thereby, we derive superposition theorems concerning the PCI-CFs and the CI-CFs. These are applicable to combinatorial enumeration.     

Computer generation of nuclear spin species and nuclear spin statistical weights

This article develops computer programs for computer generation of nuclear spin species and nuclear spin statistical weights of rovibronic levels. The programs developed here generate nuclear spin species and statistical weights from the group structures known as generalized character cycle indices (GCCI s) which are computed easily from the character table of the PI group of the molecule under consideration. Procedures are illustrated with examples.     

Structural interpretation of the topological index. 2. The molecular connectivity index, the Kappa index, and the atom-type E-State index

The structural interpretation is extended to the topological indices describing cyclic structures. Three representatives of the topological index, such as the molecular connectivity index, the Kappa index, and the atom-type E-State index, are interpreted by mining out, through projection pursuit combining with a number theory method generating uniformly distributed directions on unit sphere, the structural features hidden in the spaces spanned by the three series of indices individually. Some interesting results, which can hardly be found by individual index, are obtained from the multidimensional spaces by several topological indices. The results support quantitatively the former studies on the topological indices, and some new insights are obtained during the analysis. The combinations of several molecular connectivity indices describe mainly three general categories of molecular structure information, which include degree of branching, size, and degree of cyclicity. The cyclicity can also be coded by the combination of chi cluster and path/cluster indices. The Kappa shape indices encode, in combination, significant information on size, the degree of cyclicity, and the degree of centralization/separation in branching. The size, branch number, and cyclicity information has also been mined out to interpret atom-type E-State indices. The structural feature such as the number of quaternary atoms is searched out to be an important factor. The results indicate that the collinearity might be a serious problem in the applications of the topological indices.     

Refractive Index of Liquid Mixtures: Theory and Experiment

An innovative approach is presented to interpret the refractive index of binary liquid mixtures. The concept of refractive index “before mixing” is introduced and shown to be given by the volume‐fraction mixing rule of the pure‐component refractive indices (Arago–Biot formula). The refractive index of thermodynamically ideal liquid mixtures is demonstrated to be given by the volume‐fraction mixing rule of the pure‐component squared refractive indices (Newton formula). This theoretical formulation entails a positive change of refractive index upon ideal mixing, which is interpreted in terms of dissimilar London dispersion forces centred in the dissimilar molecules making up the mixture. For real liquid mixtures, the refractive index of mixing and the excess refractive index are introduced in a thermodynamic manner. Examples of mixtures are cited for which excess refractive indices and excess molar volumes show all of the four possible sign combinations, a fact that jeopardises the finding of a general equation linking these two excess properties. Refractive indices of 69 mixtures of water with the amphiphile (R,S)‐1‐propoxypropan‐2‐ol are reported at five temperatures in the range 283–303 K. The ideal and real refractive properties of this binary system are discussed. Pear‐shaped plots of excess refractive indices against excess molar volumes show that extreme positive values of excess refractive index occur at a substantially lower mole fraction of the amphiphile than extreme negative values of excess molar volume. Analysis of these plots provides insights into the mixing schemes that occur in different composition segments. A nearly linear variation is found when Balankina’s ratios between excess and ideal values of refractive indices are plotted against ratios between excess and ideal values of molar volumes. It is concluded that, when coupled with volumetric properties, the new thermodynamic functions defined for the analysis of refractive indices of liquid mixtures give important complementary information on the mixing process over the whole composition range.     

An algorithm for the generation of nuclear spin species and nuclear spin statistical weights

This article develops a set of algorithms for the computer generation of nuclear spin species and nuclear spin statistical weights potentially useful in molecular spectroscopy. These algorithms generate the nuclear spin species from group structures known as generalized character cycle indices (GCCI s). Thus the required input for these algorithms is just the set of all GCCI s for the symmetry group of the molecule which can be computed easily from the character table. The algorithms are executed and illustrated with examples.     

Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems

On the great success of bond-additive topological indices such as Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a quantitative refinement of the distance nonbalancedness and also a peripherality measure in molecular graphs and networks. In this direction, we introduce other variants of bond-additive indices, such as edge-Mostar and total-Mostar indices. The present article explores a computational technique for Mostar, edge-Mostar, and total-Mostar indices with respect to the strength-weighted parameters. As an application, these techniques are applied to compute the three indices for the family of coronoid and carbon nanocone structures.     

Measurements of the density and refractive index for 1-n-butyl-3-methylimidazolium-based ionic liquids

Correlations between density and refractive index of pure systems of ionic liquids were examined in this work. To this end, the density and refractive index of four 1-n-butyl-3-imidazolium-based ionic liquids were measured at atmospheric pressure and temperature up to 353.2 K. Densities and refractive indices of the ionic liquids investigated are presented as a function of temperature. A group contribution-based equation was modified to calculate the density as a function of temperature. An empirical equation was used to study the temperature-dependence of refractive index. The Lorentz–Lorenz, Dale–Gladstone, Eykman, Oster, Arago–Biot, and Newton equations, as well as a modified Eykman were used to correlate the relation between the densities and refractive indices of the different ionic liquid systems. The correlations give satisfactory results. The results of this study can add to the newly organized database for ionic liquids and can also be used for various process design calculations.     

Group electronegativity and Fukui function studies of the substituent effects in aromatic and inorganic systems

The group electronegativities (GE ) of molecular fragments, including the environmental contributions due to both the electrostatic interactions and electron distribution relaxation, and the Fukui function (FF ) indices of the charge sensitivity analysis (CSA ) are correlated with the known substituent effects in molecular systems. The semiempirical CSA in the atoms-in-molecules (AIM ) resolution has been applied to substituted benzenes and square platinum complexes treated as illustrative examples. The calculated FF indices and GE are both shown to constitute adequate reactivity criteria that qualitatively reproduce the known substituent effects. The FF index (second-order property) is found to be a more sensitive detector of the substituent influence than is the corresponding GE parameter (first-order property). © 1992 John Wiley & Sons, Inc.     

Methods of studies on quantitative structure–activity relationships for chiral compounds

Chiral compounds are very important in drug development, organic synthesis, materials science, toxicology, or environmental chemistry. Therefore, for creating new drugs, several methods have been suggested in recent years. In several laboratories in the world, some new methods for the derivations of the parameters were constructed and used for studies on quantitative structure–activity/property relationships of chiral molecules. The algorithms reviewed in this paper involve Zargeb group chiral indices, chiral molecular connectivity index, chiral topological charge index, chiral A_m index, chiral indices based on the matrixes, chiral indices based on chiral product, conformation‐independent chirality code, conformation‐dependent chirality code, quantitative two‐dimensional chirality degrees of benzenoids, and so on. Copyright © 2012 John Wiley & Sons, Ltd.     

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.