Modification of the Wiener Index. 2

A novel topological index based on the Wiener Index is proposed as W = 1/2, summation(n/ij)=S*(ij) the element S*(ij) of the distance matrix is defined either as S*(ij)= square root (E(i)E(j)/R(ij) (atoms i and j are adjacent) or as S*(ij)=(j-i+1)square root (E(i)...xE /R(ij) (atoms i and j are not adjacent), where E(i) and E(j) represent the total energy of the valence electrons of vertexes i and j, respectively, R(ij) is the sum of the distance between the vertexes i and j in a molecular graph. The distance and the energy of the vertexes in a molecule are taken into account in this definition. Hence the application of the index W to multiple bond and heteroatom-containing organic systems and inorganic systems is possible. Good correlation coefficients are achieved not only in the standard formation enthalpy of methyl halides, halogen-silicon, but also in the retention index of gas chromatography of the hydrocarbons.     

Graphs of unbranched hexagonal systems with equal values of the wiener index and different numbers of rings

Graphs of unbranched hexagonal systems consist of hexagonal rings connected with each other. Molecular graphs of unbranched polycyclic aromatic hydrocarbons serve as an example of graphs of this class. The Wiener index (or the Wiener number) of a graph is defined as the sum of distances between all pairs of its vertices. Necessary conditions for the existence of graphs with different numbers of hexagonal rings and equal values of the Wiener index are formulated, and examples of such graphs are presented.     

Three methods for calculation of the hyper-Wiener index of molecular graphs

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.     

Novel topological index F based on incidence matrix

Based on incidence matrix W, the novel topological index F is defined by the matrices L, W, X as F = LWX. The properties, the chemical environments, and the interaction of the vertexes in a molecule are taken into account in this definition. Several good QSPR models in the hetero-atom-containing organic compounds and inorganic compounds are obtained. Moreover, based on the definition of F and the values C(i) of the vertexes or the values (m)F(ij) of the chemical bonds, we have obtained serial indices, (m)F(v), (m)F(b), and F(w). They are successfully applied to QSPR models and good correlation results have been obtained as well.     

How to compute the Wiener index of a graph

The Wiener index of a graphG is equal to the sum of distances between all pairs of vertices ofG. It is known that the Wiener index of a molecular graph correlates with certain physical and chemical properties of a molecule. In the mathematical literature, many good algorithms can be found to compute the distances in a graph, and these can easily be adapted for the calculation of the Wiener index. An algorithm that calculates the Wiener index of a tree in linear time is given. It improves an algorithm of Canfield, Robinson and Rouvray. The question remains: is there an algorithm for general graphs that would calculate the Wiener index without calculating the distance matrix? Another algorithm that calculates this index for an arbitrary graph is given.     

Weak distance dependence of through-bond interactions in tetrahydro-4H-thiopyran-4-ylidene end-capped oligo(cyclohexylidenes); a computational survey

Calculations on members of the oligo(cyclohexylidene) series [(n), n = 1-5)] and related tetrahydro-4H-thiopyran end-capped analogues [(n), n = 1-4)] show a strong through-bond coupling between their pi bonds and sulfur lone pairs (Lp(pi)S). This coupling is mediated by an interaction between the H(ax)-C-C-H(ax) structural sub-units and the pi bonds connecting the cyclohexyl moieties. A comparison of the length dependency of the through-bond coupling via an oligo(cyclohexylidene) and an alkane bridge [divinyl alkanes (n)] shows that oligo(cyclohexylidenes) are more efficient in mediating through-bond couplings over large distances. Oligo(cyclohexylidene) bridges exhibit molecular wire characteristics.     

Harary index of Eulerian graphs

Journal of Mathematical Chemistry - The Harary index of a connected (molecular) graph is defined as the sum of reciprocals of distances between all its pairs of vertices. In this paper, we...     

Conformations of poly{G}-poly{C} pi stacks with high hole mobility

Charge transfer properties of DNA depend strongly on the pi stack conformation. In the present paper, we identify conformations of homogeneous poly-{G}-poly-{C} stacks that should exhibit high charge mobility. Two different computational approaches were applied. First, we calculated the electronic coupling squared, V(2), between adjacent base pairs for all 1 ps snapshots extracted from 15 ns molecular dynamics trajectory of the duplex G(15). The average value of the coupling squared is found to be 0.0065 eV(2). Then we analyze the base-pair and step parameters of the configurations in which V(2) is at least an order of magnitude larger than . To obtain more consistent data, approximately 65,000 configurations of the (G:C)(2) stack were built using systematic screening of the step parameters shift, slide, and twist. We show that undertwisted structures (twist<20 degrees) are of special interest, because the pi stack conformations with strong electronic couplings are found for a wide range of slide and shift. Although effective hole transfer can also occur in configurations with twist=30 degrees and 35 degrees, large mutual displacements of neighboring base pairs are required for that. Overtwisted conformation (twist> or =38 degrees) seems to be of limited interest in the context of effective hole transfer. The results may be helpful in the search for DNA based elements for nanoelectronics.     

Bounds to average interelectronic angles in Hartree-Fock theory of atoms

The average interelectronic is the expectation value of the angle thetaij (0 < or = thetaij < or = pi) subtended by the position vectors ri and rj of a pair of electrons i and j. In the Hartree-Fock theory of atoms, we point out that the angle and its subshell-pair components nl,n'l' are bounded from above and below, where n and l are the principal and azimuthal quantum numbers. The upper bounds for nl,n'l' with 0 < or = l, l' < or = 3 are 9pi/16 (=101.25 degrees), 135pi/256 (approximately 94.922 degrees), 265pi/512 (approximately 93.164 degrees), and 129pi/256 (approximately 90.703 degrees) for sp, pd, df, and sf pairs, respectively, while they are pi/2 (=90 degrees ) for the other ll' pairs, independent of n and n'. A weighted sum of these subshell-pair bounds gives an upper bound to . The lower bounds are pi/2 in all the cases.     

Manifestation of stereoelectronic effects on the calculated carbon-hydrogen bond lengths and one-bond 1J(C-H) NMR coupling constants. Relative acceptor ability of the carbonyl (C=O), thiocarbonyl (C=S), and methylidene (C=CH2) groups toward C-H donor bonds

Theoretical examination [B3LYP/6-31G(d,p), PP/IGLO-III//B3LYP/6-31G(d,p), and NBO methods] of six-membered cyclohexane 1 and carbonyl-, thiocarbonyl-, or methylidene-containing derivatives 2-27 afforded precise structural (in particular, C-H bond distances) and spectroscopic (specifically, one-bond (1)J(C)(-)(H) NMR coupling constants) data that show the consequences of stereoelectronic hyperconjugative effects in these systems. Major observations include the following. (1) sigma(C)(-)(H)(ax)() -->(C)(=)(Y) and pi(C)(=)(Y) --> sigma(C)(-)(H)(ax)() (Y = O, S, or CH(2)) hyperconjugation leads to a shortening (strengthening) of the equatorial C-H bonds adjacent to the pi group. This effect is reflected in smaller (1)J(C)(-)(H)(ax)() coupling constants relative to (1)J(C)(-)(H)(eq)(). (2) Comparison of the structural and spectroscopic consequences of sigma(C)(-)(H)(ax)() --> pi(C)(=)(Y) hyperconjugation in cyclohexanone 2, thiocyclohexanone 3, and methylenecyclohexane 4 suggests a relative order of acceptor orbital ability C=S > C=O > C=CH(2), which is in line with available pK(a) data. (3) Analysis of the structural and spectroscopic data gathered for heterocyclic derivatives 5-12 reveals some additivity of sigma(C)(-)(H)(ax)() --> pi(C)(=)(Y), pi(C)(=)(Y) --> sigma(C)(-)(H)(ax)(), n(X) --> sigma(C)(-)(H)(ax)(), n(beta)(O) --> sigma(C)(-)(H)(eq)(), and sigma(S)(-)(C) --> sigma(C)(-)(H)(eq)() stereoelectronic effects that is, nevertheless, attenuated by saturation effects. (4) Modulation of the C=Y acceptor character of the exocyclic pigroup by conjugation with alpha-heteroatoms O, N, and S in lactones, lactams, and methylidenic analogues 13-24 results in decreased sigma(C)(-)(H)(ax)() --> pi(C)(=)(Y) and pi(C)(=)(Y) --> sigma(C)(-)(H)(ax)() hyperconjugation. (5) Additivity of sigma(C)(-)(H)(ax)() --> pi(C)(=)(Y) and pi(C)(=)(Y) --> sigma(C)(-)(H)(ax)() hyperconjugative effects is also apparent in 1,3-dicarbonyl derivative 25 (C=Y equal to C=O), 1,3-dithiocarbonyl derivative 26 (C=Y equal to C=S), and 1,3-dimethylidenic analogue 27 (C=Y equal to C=CH(2)).     

Structural paradigms in macropolyhedral boranes

Macropolyhedral borane clusters are concave polyhedra constituting fused convex simple polyhedra. They are formally obtained by condensation of simple polyhedral boranes under elimination of between one and four BH(3) or isoelectronic units. The number of eliminated vertexes from simple polyhedra equals the number of shared vertexes in macropolyhedral boranes. For each of the eight classes with general formulae ranging from B(n)H(n-4) to B(n)H(n+10), more than one structure type is possible, differing in the number of shared vertexes and in the types of the two combined cluster fragments. However, only one type of "potential structures" is represented by experimentally known examples and is found to be favored by theoretical calculations. A sophisticated system exists among the favored macropolyhedral borane structures. For each class of macropolyhedral boranes, the number of skeletal electron pairs is directly related to the general formula, the number of shared vertexes and the type of fused cluster fragments. In order to predict the distribution of vertexes among the fused fragments, we propose the concept of preferred fragments. Preferred fragments are those usually present in the thermodynamically most stable structure of a given class of macropolyhedral boranes and are also frequently observed in the experimentally known structures. This allows us to completely predict the cluster framework of the thermodynamically most stable macropolyhedral borane isomers.     

Determination of the Wiener molecular branching index for the general tree

The many applications of the distance matrix, D(G), and the Wiener branching index, W(G), in chemistry are briefly outlined. W(G) is defined as one half the sum of all the entries in D(G). A recursion formula is developed enabling W(G) to be evaluated for any molecule whose graph G exists in the form of a tree. This formula, which represents the first general recursion formula for trees of any kind, is valid irrespective of the valence of the vertices of G or of the degree of branching in G. Several closed expressions giving W(G) for special classes of tree molecules are derived from the general formula. One illustrative worked example is also presented. Finally, it is shown how the presence of an arbitrary number of heteroatoms in tree-like molecules can readily be accommodated within our general formula by appropriately weighting the vertices and edges of G.     

A novel definition of the overall hyper-wiener index for unsaturated hydrocarbons

By replacing the distances between pairs of vertices with the relative distances, we define a novel overall hyper-Wiener index (NOR); the novel overall hyper-Wiener index extends the usefulness of the hyper-Wiener index and the overall hyper-Wiener index to unsaturated hydrocarbons.     

The valence overall wiener index for unsaturated hydrocarbons

By replacing the distances between pairs of vertices with the relative distances, we define a novel valence overall Wiener index (VOW); the valence overall Wiener index extends the usefulness of the Wiener index and the overall Wiener index to unsaturated hydrocarbons.     

Simultaneous analysis of ultrafast fluorescence decays of FMN binding protein and its mutated proteins by molecular dynamic simulation and electron transfer theory

Ultrafast fluorescence decays of FMN binding proteins (FBP) from Desulfovibrio vulgaris (Miyazaki F) were analyzed with an electron transfer (ET) theory by Kakitani and Mataga (KM theory). Time-dependent distances among isoalloxazine (Iso) and Trp-32, Tyr-35, and Trp-106 in wild-type FBP (WT), among Iso and Tyr-32, Tyr-35, and Trp-106 in W32Y (Trp-32 was replaced by Tyr-32), and among Iso and Tyr-35 and Trp-106 in W32A (Trp-32 was replaced by Ala-32) were determined by molecular dynamic simulation (MD). Electrostatic energies between Iso anion and all other ionic groups, between Trp-32 cation and all other ionic groups, and between Tyr-32 cation and all other ionic groups were calculated in WT, W32Y, and W32A, from the MD coordinates. ET parameters contained in KM theory, such as frequency (nu 0), a coefficient of the ET process (beta), a critical distance of the ET process ( R 0), standard free energy related to the electron affinity of the excited Iso ( G Iso (0)), and the static dielectric constant in FBP species (epsilon 0), were determined with and without inclusion of the electrostatic energy, so as to fit the calculated fluorescence decays with the observed decays of all FBP species, by a nonlinear least-squares method according to the Marquardt algorithm. In the analyses the parameters, nu 0, beta, and R 0 were determined separately between Trp residues and Tyr residues among all FBP species. Calculated fluorescence intensities with the inclusion of the electrostatic energy fit quite well with the observed ones of all WT, W32Y, and W32A.     

Modern valence-bond description of chemical reaction mechanisms: the 1,3-dipolar addition of diazomethane to ethene

The electronic mechanism for the gas-phase concerted 1,3-dipolar cycloaddition of diazomethane (CH2N2) to ethene (C2H4) is described through spin-coupled (SC) calculations at a sequence of geometries along the intrinsic reaction coordinate obtained at the MP2/6-31G(d) level of theory. It is shown that the bonding rearrangements occurring during the course of this reaction follow a heterolytic pattern, characterized by the movement of three well-identifiable orbital pairs, which are initially responsible for the pi bond in ethene and the C-N pi bond and one of the N-N pi bonds in diazomethane and are retained throughout the entire reaction path from reactants to product. Taken together with our previous SC study of the electronic mechanism of the 1,3-dipolar cycloaddition of fulminic acid (HCNO) to ethyne (C2H2) (Theor. Chim. Acc. 1998, 100, 222), the results of the present work suggest strongly that most gas-phase concerted 1,3-dipolar cycloaddition reactions can be expected to follow a heterolytic mechanism of this type, which does not involve an aromatic transition state. The more conventional aspects of the gas-phase concerted 1,3-dipolar cycloaddition of diazomethane to ethene, including optimized transition structure geometry, electronic activation energy, activation barrier corrected for zero-point energies, standard enthalpy, entropy and Gibbs free energy of activation, have been calculated at the HF/6-31G(d), B3LYP/6-31G(d), MP2/6-31G(d), MP2/6-31G(d,p), QCISD/6-31G(d) and CCD/6-31G(d) levels of theory. We also report the CCD/6-311++G(2d, 2p)//CCD/6-31G(d), MP4(SDTQ)/6-311++G(2d,2p)//CCD/6-31G(d) and CCSD(T)/6-311++G(2d, 2p)//CCD/6-31G(d) electronic activation energies.