On topological and geometrical distance matrices

Differences between topological and geometrical distance matrices are examined. Some examples of geometrical distances when graphs are embedded in spaces of different dimensions are given. Relations of topological distance matrices to other graph matrices are shown. The topological distance matrices are defined in the Hilbert space and their elements are distances through the graph lattices.     

Cluster self-organization theory for crystal-forming systems: The geometrical and topological model of formation, selection, and evolution of precursor nanoclusters of molecular and framework compounds

This review concerns the contemporary state of the problem of self-organization in crystal-forming systems, where a long-range order appears spontaneously in the arrangement of nanoscale structural units of any nature (atomic clusters and molecules), which initially existed as a chaotic mixture. Examples are provided where combinatorial topological analysis algorithms are used to restore, from structure data, the convergent matrix self-assembly code of crystal structures in the form of the sequence of significant elementary events e _i . For a cyclic six-node cluster S ₃ ⁰ , the geometrical and topological modeling of various self-organization levels of hierarchic structures was carried out for six types of primary chains S ₃ ¹ , fifteen types of networks S ₃ ² , and thirty types of frameworks S ₃ ³ . The model is universal and has been used to model the self-assembly of the following crystal structures: monomolecular compound S₆ and bimolecular compound S₆ + S₁₀, ozone O₃, benzene C₆H₆, cubane C₈H₈, Zn₄O₄ (NaCl structure type), carbon oxides C₆-GRA (graphite), C₆-DIA (diamond), and C₆-LON (lonsdaleite), boron nitrides B₃N₃-GRB, B₃N₃-DIA, and B₃N₃-LON, Ni₃As₃-NIC, B₆(OH)₁₆, zeolite K₃(Al₂Si₄O₁₂(OH)-LIT, Na₂ZrSi₂O₇-PKL (parakeldyshite), La₃Ga₅GeO₁₄ (LGG), and La₃GaGe₅O₁₆ (LAN). It is for the first time that structural invariants are recognized in topologically different crystal structures of chemical systems. Bifurcations in the evolution pathways of precursor clusters (structural branching points) have been determined for the formation of three-dimensional periodic structures. Frequency analysis carried out for the topological and symmetry pathways in the formation and evolution of clusters (primary chain S ₃ ¹ -microlayer S ₃ ² -microframework S ₃ ³ ) elucidated new crystal-formation trends in diverse chemical systems at the microscopic level.     

Generalized topological spaces in evolutionary theory and combinatorial chemistry

The search spaces in combinatorial chemistry as well as the sequence spaces underlying (molecular) evolution are conventionally thought of as graphs. Recombination, however, implies a nongraphical structure of the combinatorial search spaces. These structures, and their implications for search process itself, are heretofore not well understood in general. In this contribution we review a very general formalism from point set topology and discuss its application to combinatorial search spaces, fitness landscapes, evolutionary trajectories, and artificial chemistries.     

Analytic geometrical derivatives of second-order molecular properties from perturbation theory

Assuming the Born-Oppenheimer approximation for molecular wavefunctions satisfies the Hellmann-Feynman theorem, Rayleigh-Schrödinger perturbation theory is employed to develop an analytic formula for derivatives of expectation values and second-order properties with respect to nuclear coordinates.     

Density functional theory study of geometrical structures and electronic properties of silica nanowires

Silica nanowires are expected to possess structural diversity like bulk silica. We modeled three silica nanowires based on the side-shared two-membered rings, spiro-united two-membered rings, and three-membered rings, respectively. By performing density functional theory calculations, we studied their geometrical structures and electronic properties with and without the presence of external electric field. It is found that the stability of silica nanowires increases with length and diameter. As indicated by calculated large HOMO-LUMO gaps, silica nanowires are expected to be good insulating materials. The energy gaps, however, gradually decrease with applied electronic field and finally close, resulting in the breakdown of the insulating nanowires. Moreover, it is shown that the breakdown threshold remarkably increases with the nanowire diameter. These significant findings from the present calculations for the simplest silica nanowires will provide relevant insight into the structures and properties of much more complicated real silica nanowires.     

Conformational aspects of glutathione tripeptide: electron density topological & natural bond orbital analyses

Glutathione tripeptide (γ-glutamyl-cysteinyl-glycine) is a flexible molecule and its conformational energy landscape is strongly influenced by forming intramolecular hydrogen bond, its charge and the environment. This study employs DFT-B3LYP method with the 6-31+G (d,p) basis set to carry out conformational analysis of neutral, zwitterionic, cationic, and anionic forms of glutathione. In analyzing the structural characteristics of these structures, intramolecular hydrogen bonds were identified and characterized in details by topological parameters such as electron density ρ(r) and Laplacian of electron density $ \nabla^{2} $ ρ(r) from Bader’s atom in molecules theory. Charge transfer energies based on natural bond orbital analysis are also considered to interpret these intramolecular hydrogen bonds. Our results show that these hydrogen bonds are partially electrostatic and partially covalent in nature, in which the covalent contribution increases as the stabilization energy of hydrogen bond increases. Furthermore, the back bone and side chain (Ramachandran map) orientations of various ionic forms of glutathione have been studied and conformation of each constitution of glutathione tripeptide (i.e., Glu, Cys, and Gly moieties) was determined. In most species side chain conformation were found to be hindered gauche–gauche orientation by intramolecular hydrogen bonds.     

Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs

Fibonacci numbers are studied with respect to the topological theory of benzenoid hydrocarbons. These numbers are identified as the number of Kekulé structures of nonbranched all-benzenoid hydrocarbons, the number of matchings of paths, the number of independent sets of vertices of paths, the number of nonattacking rooks of certain rook boards, as well as the number of Clar structures of certain benzenoid hydrocarbons. Fibonacci numbers were also identified as the number of conjugated circuits of certain benzenoid hydrocarbons and thus they were also related to the structure-resonance model. Maximal independent sets of caterpillar trees are also shown to be Fibonacci numbers.     

Physical aspects of charge transfer theory

The possible shapes of the ground (U_I) and first excited (U_II) adiabatic potentials are discussed in connection with the electron transfer problem. If U_I has two minima the electron transfer process is adiabatic, regardless of the value of the gap 2Γ between U_I and U_II; the transfer rate W has the form W = B exp(?E_A/T), the pre-exponential factor B can be influenced by the size of the gap, especially when Γ is small. For the fluctuation model of the medium in the case of purely adiabatic transfer (large Γ) the value of W corresponds to diffusional penetration through the potential barrier. A barrier of arbitrary shape is considered and an expression for W obtained, which matches with the corresponding expression for the “hindered adiabatic” case of small Γ.     

Diatomic interaction energies in the topological theory of atoms in molecules

Summary A partitioning of theab initio total energy into one-center and two-center terms is proposed. The partitioning scheme is developed using the auxiliary function (2, 1; 1, 2) = γ(2, 1)γ(1, 2) and the topological theory of atoms in molecules. It is shown that this scheme can be used at theoretical levels beyond Hartree-Fock. The numerical results indicate that the two-center terms follow the experimental trend of the dissociation energies for a series of related compounds.     

Analysis of the delocalization in the topological theory of chemical bond

The topological analysis of the gradient field of the electron localization function provides a convenient theoretical framework for the partition of the molecular space into basins of attractors having a clear chemical meaning. The basin populations are evaluated by integrating the one-electron density over the basins. The variance of the basin population provides a measure of the delocalization. The behavior of the core C(X) and protonated valence basins V(X, H) populations were investigated. The analysis of the population variance in terms of cross-contributions is presented for aromatic and antiaromatic systems, hypervalent molecules and hydrogen-bonded complexes. For hypervalent molecules this analysis emphasizes the importance of the ionic resonance structures.     

Diatomic interaction energies in the topological theory of atoms in molecules

Summary.  A partitioning of the ab initio total energy into one-center and two-center terms is proposed. The partitioning scheme is developed using the auxiliary function L˜(2, 1; 1, 2)=γ(2, 1)γ(1, 2) and the topological theory of atoms in molecules. It is shown that this scheme can be used at theoretical levels beyond Hartree–Fock. The numerical results indicate that the two-center terms follow the experimental trend of the dissociation energies for a series of related compounds. Received March 5, 1996/Final revision received August 19, 1996/Accepted August 29, 1996     

Generation of the eigenvectors of the topological matrix from graph theory

The technique of describing the characteristic polynomial of a graph is here extended to construction of the eigenvectors. Recurrence relations and path tracing are combined to generate eigenvector coefficients as polynomial functions of the eigenvalues. The polynomials are expressed as linear functions of Chebyshev polynomials in order to simplify the computational effort. Particular applications to the Hückel MO theory, including heteroatom effects, are shown.     

Some new aspects of π-electron theory

An approximate method is proposed for treating the non-orthogonality of atomic functions and correlation of -electrons by means of perturbation theory and Pariser-Parr-Pople (PPP) method as a zero approximation. The formula derived for the ground state energy of -systems is suitable to describe a number of effects, which the PPP theory, in principle, cannot account for. In particular, it is shown how such intermolecular phenomena like exchange repulsion and dispersion interaction can be accounted for in MO method.The ground state energies of -electron systems are calculated with and without an account of -electron correlation. If for the atomic functions non-orthogonality is taken into consideration the experimental values of these energies can be described using a spectroscopic set of standard calculation parameters. Probably, within the framework of this method, it is possible to describe simultaneously and consistently the spectroscopic, thermochemical and kinetic data.

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Zusammenfassung Für die Behandlung der Nicht-Orthogonalität von Atomfunktionen und die Korrelation der -Elektronen wird eine Näherungsmethode vorgeschlagen, bei der die Pariser-Parr-Pople-(PPP)-Methode als nullte Näherung in einer Störungsrechnung verwendet wird. Die Formel für den Grundzustand von -Systemen ist für die Beschreibung einiger Effekte geeignet, die in der PPP-Beschreibung nicht berücksichtigt werden. Insbesondere wird gezeigt, wie solche intermolekularen Phänomene wie Abstoßung infolge Austausch und Dispersions-Wechselwirkung bei der MO-Methode berücksichtigt werden können. Die Energie des Grundzustandes von -Elektronensystemen wird mit und ohne Berücksichtigung der -Elektronenkorrelation berechnet. Wenn für die Atomfunktionen die Nicht-Orthogonalität berücksichtigt wird, so können die experimentellen Werte dieser Energie mit Hilfe eines spektroskopischen Satzes von Standardparametern erhalten werden. Vermutlich ist es möglich, im Rahmen dieser Methode, die spektroskopischen, thermochemischen und kinetischen Daten einheitlich und konsistent zu beschreiben.

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Résumé On propose une méthode approchée pour traiter la non orthogonalité des fonctions atomiques et la corrélation des électrons au moyen de la théorie des perturbations utilisant l'approximation de la méthode de Pariser-Parr-Pople comme ordre zéro. La formule obtenue pour l'énergie du fondamental permet de décrire un certain nombre d'effets dont la théorie PPP ne peut rendre compte. En particulier on montre comment certains phénomènes intermoléculaires comme la répulsion d'échange et la dispersion peuvent être pris en considération.L'énergie de l'état fondamental des systèmes d'électrons est calculée avec et sans corrélation. Si la non orthogonalité des fonctions d'onde atomiques est incluse, les valeurs expérimentales de ces énergies peuvent être décrites en utilisant un ensemble «spectroscopique» de paramètres standards. Dans le cadre de cette méthode il est probablement possible de décrire simultanément les propriétés spectroscopiques, thermochimiques et cinétiques.

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I thank Dr. I. E. Chlenov for his assistance in the calculations of -energies. I am grateful to Dr. W. H. Eugen Schwarz for useful discussion.     

A mean-field approach of the theory of topological constraints in polymer melts and networks

A new mean-field concept is discussed for the topological constraints in polymer melts and networks. The constraining potential is assumed to be determined by the dependence of the free energy of the surrounding chains on the position of the considered chain. The concept is applied to polydisperse systems, in which a new criterion is derived to describe the onset of the influence of entanglement effects.     

95Mo nuclear magnetic resonance parameters of molybdenum hexacarbonyl from density functional theory: appraisal of computational and geometrical parameters

Solid-state (95)Mo nuclear magnetic resonance (NMR) properties of molybdenum hexacarbonyl have been computed using density functional theory (DFT) based methods. Both quadrupolar coupling and chemical shift parameters were evaluated and compared with parameters of high precision determined using single-crystal (95)Mo NMR experiments. Within a molecular approach, the effects of major computational parameters, i.e. basis set, exchange-correlation functional, treatment of relativity, have been evaluated. Except for the isotropic parameter of both chemical shift and chemical shielding, computed NMR parameters are more sensitive to geometrical variations than computational details. Relativistic effects do not play a crucial part in the calculations of such parameters for the 4d transition metal, in particular isotropic chemical shift. Periodic DFT calculations were tackled to measure the influence of neighbouring molecules on the crystal structure. These effects have to be taken into account to compute accurate solid-state (95)Mo NMR parameters even for such an inorganic molecular compound.     

Statistical theory for hydrogen bonding fluid system of A_aD_d type (I): The geometrical phase transition

The influence of hydrogen bonds on the physical and chemical properties of hydrogen bonding fluid system of AaDd type is investigated from two viewpoints by the principle of statistical mechanics. In detail, we proposed two new ways that can be used to obtain the equilibrium size distribution of the hydrogen bonding clusters, and derived the analytical expression of a relationship between the hydrogen bonding free energy and hydrogen bonding degree. For the nonlinear hydrogen bonding systems, it is shown that the sol-gel phase transition can take place under proper conditions, which is further proven to be a kind of geometrical phase transition rather than a thermodynamic one. Moreover, several problems associated with the geometrical phase transition and liquid-solid phase transition in nonlinear hydrogen bonding systems are discussed.     

Iron-dextran complex: geometrical structure and magneto-optical features

Molecular mass of the iron-dextran complex (M(w)=1133 kDa), diameter of its particles (～8.3 nm) and the content of iron ions in the complex core (N(Fe)=6360) were determined by static light scattering, measurements of refractive index increment and the Cotton-Mouton effect in solution. The known number of iron ions permitted the calculation of the permanent magnetic dipole moment value to be μ(Fe)=3.17×10(-18) erg Oe(-1) and the determination of anisotropy of linear magneto-optical polarizabilities components as Δχ=9.2×10(-21) cm(3). Knowing both values and the value of the mean linear optical polarizability α=7.3×10(-20) cm(3), it was possible to show that the total measured CM effect was due to the reorientation of the permanent and the induced magnetic dipole moments of the complex. Analysis of the measured magneto-optical birefringence indicated very small optical anisotropy of linear optical polarizability components, κ(α), which suggested a homogeneous structure of particles of spherical symmetry.     

Some aspects of the DNA hypochromic effect theory

The expressions for the polymer absorption band hypochromism and oscillator strength are studied in frames of the first-order perturbation theory. The physically justified approximations for a polymer hypochromism calculation are indicated. The uniform interpretation of the effect origin is given independently on the approximation used. In frames of the perturbation theory the DNA hypochromism formula is obtained, from which the known experimental dependences of hypochromism on chain length and polymer helicity degree follows directly. The analytic expression of the DNA hypochromism dependence on AT pairs content is obtained. It is established that in most cases for natural DNA the nucleotide sequence does not influence in practice the value of the hypochromic effect.