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.     

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.     

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.     

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.     

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.     

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.     

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.     

Symmetry specified enumeration of substituted derivatives: an easy solution to the complex problem

Several sophisticated methods to solution of symmetry specified enumeration problems are available in the modern literature. In this paper we propose a simple technique that allows one to manually compute the exact numbers of fixed-symmetry derivatives for a given structure either with inclusion or ignoring the substitution patterns. The basic idea of the method suggested consists in the derivation of Pólya-like cycle indices for the automorphism groups of specially constructed orbit partition graphs; the expansion of these indices and subsequent simple calculations result in the desired numbers of substituted derivatives with achiral substituents. Limitations of the new technique (and a method suggested earlier) depend on the relevance of the orbit partitions for particular subgroups of the point symmetry group. For illustration purposes, the results obtained for the prismane (D _3h ) and adamantane (T _d ) structures are discussed. In the former case the numbers of substituted derivatives can be found for all subgroups of the D _3h group, whereas in the latter case these numbers can be determined for eight out of eleven subgroups of the T _d point symmetry group. This work is based on the text of the lecture presented by the authors at the 5th All-Russia Conference on Molecular Modeling (Moscow, April 2007). The paper deals with the methodology and detailed treatment of applied aspects related to solution of enumeration problems for substituted derivatives with prescribed symmetry groups. Unlike the known methods of symmetry specified enumeration, the technique suggested is simple enough and may be regarded as generalization of the Pólya methodology, which is widely used by chemists. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 227–245, February, 2008.     

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.     

A simple algorithm for unique representation of chemical structures-cyclic/acyclic functionalized achiral molecules

An algorithm, based on "vertex priority values" has been proposed to uniquely sequence and represent connectivity matrices of chemical structures of cyclic/acyclic functionalized achiral hydrocarbons and their derivatives. In this method, "vertex priority values" have been assigned in terms of atomic weights, subgraph lengths, loops, and heteroatom contents. Subsequently, the terminal vertices have been considered upon completing the sequencing of the core vertices. This approach provides a multilayered connectivity graph, which can be put to use in comparing two or more structures or parts thereof for any given purpose. Furthermore, the basic vertex connection tables generated here are useful in the computation of characteristic matrices/topological indices and automorphism groups and in storing, sorting, and retrieving chemical structures from databases.     

Analytical calculation of atomic and molecular electrostatic potentials from the Poisson equation

An analytic formulation is given for the total potential in atomic and molecular systems, based on the electrostatic approach from the Hellmann-Feynman theorem. The potential function is obtained from the analytic solution of the Poisson equation using charge densities expressed as a superposition of gaussian functions. The method is independent of the specific LCAO approximation used for the calculation of the charge distribution function. The calculation of the potential and its derivatives to a rapid algorithm form, which can be used for the evaluation of various electronic properties and the treatment of experimental situation, even for large molecular systems.     

Automatic superposition of drug molecules based on their common receptor site

Summary We have prevously developed a new rational method for superposing molecules in terms of submolecular physical and chemical properties, but not in terms of atom positions or chemical structures as has been done in the conventional methods. The program was originally developed for interactive use on a three-dimensional graphic display, providing goodness-of-fit indices on molecular shape, hydrogen bonds, electrostatic interactions and others.Here, we report a new unbiased searching method for the best superposition of molecules, covering all the superposing modes and conformational freedom, as an additional function of the program. The function is based on a novel least-squares method which superposes the expected positions and orientations of hydrogen bonding partners in the receptor that are deduced from both molecules. The method not only gives reliability and reproducibility to the result of the superposition, but also allows us to save labor and time. It is demonstrated that this method is very efficient for finding the correct superposing mode in such systems where hydrogen bonds play important roles.     

Reduced cycle indices and their applications in enumeration of NMR signals and equivalence classes

An efficient technique is formulated based on a polynomial structure which we call the reduced cycle index for the enumeration of equivalence classes and NMR signals under group action. The reduced cycle indices are shown to be the cycle index polynomials of a subset of significantly smaller order of the point group of the molecule. Thus, the reduced cycle indices are much simpler and their use leads to a considerable reduction in tile computation of the generating functions from the cycle indices and irreducible representations contained in a set.Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.     

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.     

Analysis of Mixtures Containing Unknown Components by Gas Chromatography: Determination of Molecular Mass

New methods are proposed for the gas-chromatographic determination of the molecular masses of unknown components in complex mixtures within one cycle of analysis. The methods are based on the use of complex chromatographic data in the form of linear retention indices of compounds for a column with a nonpolar stationary phase and sensitivity indices of a thermal conductivity detector and a flame ionization detector.     

Spectroscopy and the classification of liquid stationary phases in gas-liquid chromatography

Summary A table of data for acidity, basicity, polarity and London potential for stationary phases is given. These provide guidance in choosing a stationary phase for a given task from the 26 substances listed. The indices are calculated from infra-red shifts of 1-chlorobutane and the visible wavelength shifts of l₂ when they are dissolved in the stationary phases, from NMR estimates of acidity and basicity and from relative retention data. They replace the less reliable data of Burns and Hawkes. Methods of determining the indices are suggested that require only a gas chromatograph and can be applied by technicians with no specialized understanding of solution parameters. An attempt to add electron donoracceptor indices to the table was unsuccessful. A further table is given of the slopes of the log plots d(logt)/dC for use in extrapolating from one homologue to another on these stationary phases.     

Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.     

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.