On structural interpretation of several distance related topological indices.

We consider the role that individual bonds play in bond-additivities in order to better understand the structural basis of various topological indices. In particular we consider indices closely related to the Wiener index (W) and the distance matrix and search for optimal weights of terminal and interior CC bonds in alkanes for a selection of physicochemical properties. It is interesting to note that different properties are associated with different relative roles of the exterior and the interior CC bonds.     

Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.     

Impersonality of the connectivity index and recomposition of topological indices according to different properties

The connectivity index chi can be regarded as the sum of bond contributions. In this article, boiling point (bp)-oriented contributions for each kind of bond are obtained by decomposing the connectivity indices into ten connectivity character bases and then doing a linear regression between bps and the bases. From the comparison of bp-oriented contributions with the contributions assigned by chi, it can be found that they are very similar in percentage, i.e. the relative importance of each particular kind of bond is nearly the same in the two forms of combinations (one is obtained from the regression with boiling point, and the other is decided by the constructor of the chi index). This coincidence shows an impersonality of chi on bond weighting and may provide us another interpretation of the efficiency of the connectivity index on many quantitative structure-activity/property relationship (QSAR or QSPR) results. However, we also found that chi's weighting formula may not be appropriate for some other properties. In fact, there is no universal weighting formula appropriate for all properties/activities. Recomposition of some topological indices by adjusting the weights upon character bases according to different properties/activities is suggested. This idea of recomposition is applied to the first Zagreb group index M(1) and a large improvement has been achieved.     

Comparison of the Topological and Dynamic Characteristics of Molecules for Calculating Retention Indices of Organic Compounds

This paper reports on a comparative analysis of the linear correlation equations relating the gas-chromatographic retention indices (RI) of several groups of isomeric organic compounds to the topological (Wiener and Hosoya indices) and dynamic (intramolecular energies estimated by molecular dynamic methods and the vibrational components of this energy estimated with simple models) parameters of molecules. It is shown that the choice of one of these parameters does not ensure the best approximation to RI, since the results depend on the chemical nature of substances. Correlations between all of the above quantities have been established. The correlations serve as a basis for physicochemical interpretation of the topological parameters of molecules as quantities proportional to the intramolecular vibrational and rotation energies.     

广义α~N指数及其应用 IV: 广义α~N指数在碳氢化合物中的应用

本文计算了七对顺、反异构体的广义α~N指数, 结果表明, 广义α~N指数不仅能很好地区分碳氢化合物的顺、反异构体, 而且与其物理性质(沸点、密度、折光率、分子折射度等)有良好的单变量线性相关关系, 解决了拓扑指数不能区分顺、反异构体的困难。本文还计算了庚烷、辛烷系列的广义α~N指数, 并比较了多种拓扑指数与辛烷值的相关分析结果。     

Large Nonstatistical Branching Ratio in the Dissociation of Pentane-2,4-dione Radical Cation: An Ab Initio Direct Classical Trajectory Study

The dissociation of pentane-2,4-dione radical cation has been studied by ab initio direct classical trajectory calculations at the MP2/6-31G(d) level of theory. A bond additivity correction has been used to improve the MP2 potential energy surface (BAC-MP2). A microcanonical ensemble was constructed using quasiclassical normal-mode sampling by distributing 10 kcal/mol of excess energy above ZPE for the transition state for the tautomerization of the enol with a terminal double bond, 4-hydroxypent-4-en-2-one radical cation, to the diketo form. A total of 244 trajectories were run starting from this transition state, yielding pentane-2,4-dione radical cation and depositing energy in the terminal CC bond. As a result, the branching ratio for dissociation of the terminal CC bond versus the interior CC bonds is significantly larger than expected from RRKM theory. The branching ratio for the dissociation of the two interior CC bonds is ～20:1, with the one closest to the activated methyl breaking more often. Since the two interior bonds are equivalent and should dissociate with equal probability, this branching ratio represents a very large deviation from statistical behavior. A simple kinetic scheme has been constructed to model the dissociation rates. The nonstatistical behavior is seen because the rate of energy flow within the molecule is comparable to or less than the rates of dissociation for the activated system. In addition to the expected dissociation products, some of the trajectories also lead to the formation of an ester-like product, prop-1-en-2-yl acetate radical cation.     

Topological Estimation of Aromatic Stabilities of Polyacenes and Helicenes: Modeling of Resonance Energy and Benzene Character

The distance-based topological indices viz Wiener (W)-, Szeged (Sz)-, Padmakar-Ivan (PI)- and Sadhana (Sd)-indices have been used for estimating aromatic stabilities as well as % benzene character of polyacenes and helicenes. Excellent models are obtained in combined set and there was no need to split the data set into polyacenes and helicenes. The newly introduced Sd index was found the best index among all the four indices used. In bi-parametric modeling with the combination of the Sd index with the PI index yielded excellent results. The results are discussed critically on the basis of variety of statistical parameters.     

Modification of Wiener Index and its application

A novel topological index based on the Wiener Index is proposed as W* = 1/2 sigma (n)(i,j=1) S(*)(ij), the element S(*)(ij) of the distance matrix is defined either as S(*)(ij) = alpha x square root of I(i)I(j)/R(ij) (atoms i and j are adjacent) or as S(*)(ij) = = alpha x (j-i+1)square root of I(i) x x x x x I(j)/R(ij) (atoms i and j are not adjacent), where I(i) and I(j) represent the electronegativity of vertices i or j, respectively, R(ij)() is the sum of the bond length between the vertices i and j in a molecular graph, and alpha = (Z(i)/Z(j))(0.5), where Z(i) and Z(j) are the atomic numbers of the positive valence atom i and the negative valence atom j, respectively. The properties and the interaction of the vertices in a molecule are taken into account in this definition. That is why the application of the index W to heteroatom-containing and multiple bond organic systems and inorganic systems is possible. Correlation coefficients above 0.97 are achieved in the prediction of the retention index of gas chromatography of the hydrocarbons, the standard formation enthalpy of methyl halides, halogen-silicon, and inorganic compounds containing transition metals.     

Important mathematical structures of the topological index Z for tree graphs

Mathematical importance of the topological index, ZG, or the so-called Hosoya index is stressed by presenting and giving supporting evidence for the proposed conjecture. That is, for a given pair of positive integers (n1or=3), with Z(G1) = n1 and Z(G2) = n2.     

Partitioning of the orbital overlap matrix and the localization criteria

A new molecular orbital localization procedure is proposed. The approach is based on partitioning of the overlap matrix into atomic contributions in accordance with Bader's topological theory of atoms in molecules. The new procedure has several advantages over other schemes. It preserves the 6/7c-separability in planar systems and allows for a straightforward interpretation of the localized orbitals in terms of their localization indices and atomic occupancies. The new procedure is tested on the H2O, LiF, N2, CO, BH₃ · CO and Li₂ molecules.Research partially supported by US DOE.     

Statistical properties of carbon nanostructures

We look at modeling carbon nanostructures from a theoretical graph network view, where a graph has atoms at a vertex and links represent bonds. In this way, we can calculate standard statistical mechanics functions (entropy, enthalpy, and free energy) and matrix indices (Wiener index) of finite structures, such as fullerenes and carbon nanotubes. The Euclidean Wiener index (topographical index) is compared with its topological (standard) counterpart. For many of these parameters, the data have power law behavior, especially when plotted versus the number of bonds or the number of atoms. The number of bonds in a carbon nanotube is linear with the length of the nanotube, thus enabling us to calculate the heat of formation of capped (5,5) and (10,10) nanotubes. These properties are determined from atomic coordinates using MATLAB routines.     

Modeling chromatographic parameters by a novel graph theoretical sub-structural approach.

A novel approach to the study of quantitative relationships between chromatographic parameters and the chemical structure is introduced. It is based on the computation of the spectral moments of the topological bond matrix by using different weights as diagonal entries of this matrix. The main advantage of the present approach is that the quantitative contributions of the structural fragments of molecules to the chromatographic parameters studied can be obtained explicitly. By using this approach we study two data sets: one composed of 156 alkanes and the other of 81 oxygen-containing organic molecules. In both cases excellent quantitative structure-chromatographic retention relationships were obtained. The contributions of the different fragments to the chromatographic retention were generated obtaining tables of additive contributions to the properties studied. The physicochemical interpretation of the results on the basis of the retention mechanisms is also analyzed in light of this new approach.