Topological analysis of five-vertex clusters of group IVA elements

SummaryNet sign analyses of eigenvectors and eigenvalues of five-vertex chemical graphs were performed. The five-vertex graphs are denoted asG5m, where the first subscript 5 stands for the number of vertices andm for the number of edges.G54 is the path with 5 vertices, hence 4 edges, isomorphic with the hydrogen-depleted graph ofn-pentane. In most cases, the ordering according to the net sign is found to be similar to the ordering according to the energy eigenvalues. Applications of net sign analysis to the investigation of ground-state geometry of five-vertex clusters of carbon, silicon and germanium are also discussed.

[1]  Krishnan Raghavachari,et al.  Structure, stability, and fragmentation of small carbon clusters , 1987 .

[2]  D. Mcginty.  Molecular dynamics studies of the properties of small clusters of argon atoms , 1973 .

[3]  A. Harget,et al.  Ground states of .sigma.-bonded molecules. X. Extension of the MINDO/2[modified intermediate negelect of differential overlap/2] method to compounds containing nitrogen and/or oxygen , 1970 .

[4]  J. Tomkin,et al.  Carbon, nitrogen, and oxygen abundances in main-sequence stars. II. 20 F and G stars , 1981 .

[5]  E. Kaxiras.  Structural model for a covalently bonded Si45 cluster , 1989 .

[6]  T. P. Martin,et al.  Mass spectra of Si, Ge, and Sn clusters , 1985 .

[7]  L. J. Schaad,et al.  Hueckel molecular orbital .pi.-resonance energies. Heterocycles containing divalent sulfur , 1973 .

[8]  K. Raghavachari.  Theoretical study of small silicon clusters: Equilibrium geometries and electronic structures of Sin (n=2–7,10) , 1986 .

[9]  Tománek,et al.  Calculation of magic numbers and the stability of small Si clusters. , 1986, Physical review letters.

[10]  J. Pople,et al.  Structures of small carbon clusters: Cyclic ground state of C6 , 1986 .

[11]  L. Hall,et al.  Molecular connectivity in chemistry and drug research , 1976 .

[12]  David H. Magers,et al.  Stability and properties of C4 isomers , 1988 .

[13]  B. K. Rao,et al.  Physics and chemistry of small clusters , 1987 .

[14]  Ante Graovac,et al.  Topological Approach to the Chemistry of Conjugated Molecules , 1977 .

[15]  M. R. Zakin,et al.  Chemistry on Molecular Surfaces: Reactions of Gas Phase Clusters , 1988 .

[16]  J. Eyler,et al.  Sequential ion-molecule reactions in acetylene , 1981 .

[17]  Hans Bock,et al.  HMO model and its application , 1976 .

[18]  V. Anicich,et al.  Ion-molecule reactions of hydrocarbon ions in C2H2 and HCN , 1986 .

[19]  Hans‐Herbert Schmidtke,et al.  LCAO Description of Symmetric Molecules by Unified Theory of Finite Graphs , 1966 .

[20]  T. George,et al.  The Hückel model for small metal clusters. I. Geometry, stability, and relationship to graph theory , 1987 .

[21]  Ivan Gutman,et al.  Estimation of the number of benzenoid hydrocarbons , 1988 .

[22]  R. Hoffmann.  An Extended Hückel Theory. I. Hydrocarbons , 1963 .

[23]  G. Pacchioni,et al.  Silicon and germanium clusters. A theoretical study of their electronic structures and properties , 1986 .

[24]  N. Trinajstic,et al.  Graphical description of möbius molecules , 1976 .

[25]  Myung-Hwan Whangbo,et al.  Orbital Interactions in Chemistry , 1985 .

[26]  G. Schmid.  Developments in transition metal cluster chemistry — The way to large clusters , 1985 .

[27]  Feng-Yin Li,et al.  Topological analysis of eigenvalues of particle in one- and two-dimensional simple quantal systems: Net sign approach , 1991 .

[28]  K. Balasubramanian,et al.  The ionization potentials of Agn and Aun and binding energies of Agn, Aun, Agn+ and Aun+ (n = 1–4) , 1989 .

[29]  Klaus Rüdenberg.  A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. II. The Two‐Center Exchange Integrals , 1951 .

[30]  W. J. Stevens,et al.  Polarizabilities of alkali clusters , 1989 .

[31]  P. Mezey.  Cluster topology and bounds for the electronic energy , 1985 .

[32]  Sven J. Cyvin,et al.  The number of Kekulé structures for primitive coronoids (cycloarenes) , 1988 .

[33]  C. C. J. Roothaan,et al.  A Study of Two‐Center Integrals Useful in Calculations on Molecular Structure. I , 1951 .

[34]  G. Pacchioni,et al.  Calculated properties of alkali metal clusters with fivefold symmetry , 1984 .

[35]  R. King.  Graph theory in the study of metal cluster bonding topology: Applications to metal clusters having fused polyhedra , 1986 .

[36]  E. Herbst,et al.  Laboratory measurements of ion-molecule reactions pertaining to interstellar hydrocarbon synthesis , 1983 .

[37]  M. Ross,et al.  Production of large carbon cluster ions by laser vaporization , 1986 .

[38]  A. C. Day,et al.  Comment on a graph-theoretical description of heteroconjugated molecules , 1977 .

[39]  J. C. Phillips.  Chemical bonding, kinetics and the approach to equilibrium structures of simple metallic, molecular, and network microclusters , 1986 .

[40]  J. Koutecký,et al.  Electronic and geometric structure of LI4 and Na4 clusters , 1980 .

[41]  H. Kroto.  Tilden Lecture. Semistable molecules in the laboratory and in space , 1982 .

[42]  Michael Doob,et al.  Spectra of graphs , 1980 .

[43]  R. B. King,et al.  Chemical bonding topology of bare post-transition-metal clusters: analogies between condensed-phase and gas-phase species , 1988 .

[44]  R. King.  Chemical applications of topology and group theory. 23. A comparison of graph‐theoretical and extended Hückel methods for study of bonding in octahedral and icosahedral boranes , 1987 .

[45]  Freeman,et al.  Photofragmentation of Mass-Resolved Si2-12+ clusters. , 1985, Physical review letters.

[46]  Structure and stability of lithium (Li4 and Li6) clusters , 1983 .

[47]  Robert F. Curl,et al.  The formation of long carbon chain molecules during laser vaporization of graphite , 1987 .

[48]  N. L. Sharma.  Electronic States and Geometrical Structures of Hubbard Clusters with an Impurity , 1987 .

[49]  W. C. Herndon.  Resonance theory. VI. Bond orders , 1974 .

[50]  P. Botschwina,et al.  A theoretical investigation of C5 , 1989 .

[51]  N. Trinajstic.  Hückel Theory and Topology , 1977 .

[52]  S. Unger.  Molecular Connectivity in Structure–activity Analysis , 1987 .

[53]  R. Etters,et al.  Phase transitions in small clusters of atoms , 1977 .

[54]  Ivan Gutman.  Topological analysis of Eigenvalues of the adjacency matrices in graph theory: A difficulty with the concept of internal connectivity , 1988 .

[55]  E. Bright Wilson,et al.  Symmetry, nodal surfaces, and energy ordering of molecular orbitals , 1975 .

[56]  J. Koutecký,et al.  Theoretical aspects of metal atom clusters , 1986 .

[57]  H. F. King,et al.  Structures and energies for C4 , 1986 .

[58]  Shyi-Long Lee,et al.  Net sign approach in graph spectral theory , 1990 .

[59]  Robert R. Lucchese,et al.  Topological analysis of eigenvectors of the adjacency matrices in graph theory: The concept of internal connectivity , 1987 .

[60]  H. Günthard,et al.  Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen , 1956 .

[61]  R. Hettich.  Structural investigations of aluminum cluster ions, Aln- (n = 3-50) , 1989 .

[62]  D. Ewing,et al.  Structures of C5 , 1987 .

[63]  T. George,et al.  Large silicon clusters: confirmation of Phillips' conjecture , 1988 .

[64]  Ivan Gutman,et al.  Topological analysis of the eigenvectors of the adjacency matrices in graph theory: Degenerate case , 1989 .

[65]  E. Heilbronner,et al.  Slide rule computation of Hückel molecular orbitals , 1967 .

[66]  D. Ewing,et al.  Structures and properties of linear Cn molecules , 1982 .

[67]  T. P. Martin.  Compound clusters. II. Intermetallics and phosphides , 1985 .

[68]  Pulsed Ge+n microcluster concentration spectra , 1986 .

[69]  H. Zimmerman.  Moebius-Hueckel concept in organic chemistry. Application of organic molecules and reactions , 1971 .

[70]  M. Dewar,et al.  Ground States of σ‐Bonded Molecules. IV. The MINDO Method and Its Application to Hydrocarbons , 1969 .

[71]  Donald M. Cox,et al.  Production and characterization of supersonic carbon cluster beams , 1984 .

[72]  Dragoš Cvetković,et al.  Conjugated molecules having integral graph spectra , 1974 .

[73]  R. Zahradník,et al.  Calculations of absolute values of equilibrium and rate constants. 9. MINDO/2 study of equilibrium carbon vapor , 1977 .

[74]  A. Jansen,et al.  Hartree-fock-slater linear combination of atomic orbital calculations of the valence electron-distribution in neutral and charged IR clusters , 1989 .

[75]  U. Landman,et al.  Electron Localization and Excitation Dynamics in Small Clusters , 1987 .

[76]  Kenneth S. Pitzer,et al.  LARGE MOLECULES IN CARBON VAPOR , 1959 .

[77]  Roald Hoffmann,et al.  Extended hückel theory—v : Cumulenes, polyenes, polyacetylenes and cn , 1966 .

[78]  K. Ruedenberg.  Quantum Mechanics of Mobile Electrons in Conjugated Bond Systems. I. General Analysis in the Tight‐Binding Formulation , 1961 .

[79]  Q. Fan,et al.  Theoretical study of linear Cn (n=6−10) and HCnH (n=2−10) molecules , 1989 .

[80]  R. Smalley,et al.  FT‐ICR probes of silicon cluster chemistry: The special behavior of Si+39 , 1987 .

[81]  R. B. King,et al.  Chemical applications of group theory and topology. 7. A graph-theoretical interpretation of the bonding topology in polyhedral boranes, carboranes, and metal clusters , 1977 .

[82]  William C. Herndon,et al.  Simplified molecular orbitals for organic molecules , 1971 .

[83]  R. B. King,et al.  Chemical applications of topology and group theory , 1985 .