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  • 1
    ISSN: 1432-2234
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Description / Table of Contents: Zusammenfassung Bei der Aufspaltung der d-ZustÄnde in einem elektrostatischen Feld, welches erstens von einem Cyclopentadienyl-Ring und zweitens von vier an den Ecken eines Quadrates angeordneten Liganden hervorgerufen wird, ist die effektive Symmetrie D 4v — trotz der C s-Symmetrie des Moleküls. Für das d 2-System werden die Linearkombinationen der Slaterdeterminanten und die Elemente der Störmatrix angegeben. Das Ganze lÄ\t sich z. B. auf die Diskussion von Tricarbonyl-π-cyclopentadienylÄthylmolybdÄn anwenden, dessen Struktur der des behandelten Modells sehr Ähnlich ist.
    Abstract: Résumé La décomposition des orbitales d dans le champ d'un groupe cyclopentadiényl et de quatre ligandes arrangés en carré est étudiée. Pour les systèmes d 1 et d 2, la symétrie effective est C 4v , celle de la molécule n'étant que C s. Pour d 2, on donne les combinaisons linéaires adapté à la symétrie C 4v , et la matrice des énergie perturbatrices. Les résultats s'appliquent à la discussion du cyclopentadiényl-éthyl-molybdène tricarbonyle dont la structure ressemble le modèle étudié.
    Notes: Abstract The splitting of d orbitals in an electrostatic field due to a cyclopentadienyl ring and four ligands arranged at the corners of a square is investigated. It is shown that, although the symmetry of the molecule is C s, the effective symmetry for the d 1 and d 2 systems is G 4v. The linear combinations for the d 2 system in C 1v symmetry are given and the matrix elements for the secular determinant are listed. The results may be applied to a discussion of the bonding in tricarbonyl-π-cyclopentadienylethyl-molybdenum, whose structure is related to the idealized model considered.
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  • 2
    ISSN: 1432-2234
    Keywords: Characteristic polynomial-Graph isomorphism ; Isospectral graphs ; Graph recognition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract The characteristic polynomial of a structure (molecule or a graph) is usually expressed as a function ofx. Here we explore an alternative representation of characteristic polynomials expressed in terms ofL n , the characteristic polynomials of linear chains havingn atoms. While the new forms of the characteristic polynomials are mathematically equivalent to the old forms, they appear to reflect selected structural similarities among homologous molecules better. Besides arriving at general expressions for the form of the characteristic polynomials for numerous families of compounds previously unavailable, the approach is of some interest for the old problem of graph isomorphism and graph recognition in cases of structures which can be associated with a homologous series.
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  • 3
    ISSN: 1432-2234
    Keywords: Graph theory ; chemical graphs ; degenerate rearrangements ; isomerization of octahedral XY 6 complexes ; isomerization of tetragonal-pyramidal XY 4 complexes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract Isomerization mode defines the process of interconversion of one isomer into another. Several mechanisms are conceivable for degenerate rearrangements and, in general, lead to a distinctive network of relations between participating isomers. Here we consider selected modes which are complementary in the sense that if mode 1 transforms an isomer A into B, C, D etc., then mode 2 transforms the same isomer A into X, Y, Z, etc., which includes all isomers not comprised by the first mode. Physico-chemical complementarity can be translated into mathematical complementarity of associated chemical graphs. This allows us to use the tool of Graph Theory. One example of graph theoretical use is the theorem that graph G and its complement G have the same automorphism group (i.e., the same symmetry). We have shown that a close examination of a graph and its complement and their components allows us to recognize the automorphism group in some complex cases without resorting to canonical numbering or other involved procedures, and even allows us to determine isomorphism of different processes.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Theoretical chemistry accounts 92 (1995), S. 97-106 
    ISSN: 1432-2234
    Keywords: Graphs ; Restricted random walks
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Summary In the first part of this contribution we outline the construction of a novel matrix associated with a graph, the entries of which give the probability of a random walk over the graphG starting at sitei to reach sitej inD ij steps. HereD ij is the distance between verticesi, j. The derived matrices, to be referred to as restricted random walk matrices and labeled as RRW matrices, are non-symmetric, for trees the entries being of the form 1/p, wherep is an integer equal to 1 or larger. In the second part of the report we consider a few invariants of the RRW matrices. We will illustrate the use of one such invariant in a regression analysis. We consider the variations of the entropies in isometric octanes with skeletal changes. The derived regression, based on a single descriptor, yields the standard error of 1.26 cal K−1 mol−1 that is the smallest yet reported in the literature.
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  • 5
    ISSN: 1432-2234
    Keywords: Characteristic polynomial ; Chebyshev polynomial ; Topological index ; Structure factor ; Graph
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z G ) was discussed.
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  • 6
    ISSN: 1432-2234
    Keywords: Characteristic polynomial ; Chebyshev polynomial ; Topological index ; Structure factor ; Graph
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z G ) was discussed.
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  • 7
    ISSN: 1432-2234
    Keywords: Chemical graph theory ; Composition ; Alkanes ; Molecular chemical shift
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract The concept of composition as the counterpart to partition is introduced and advocated for the discussion of molecular properties. In the partition approach an observable (experimental) quantity is fragmented into contributions which are non-observable but which hopefully maintain constancy for fragments (bonds) in similar environments and thus facilitate comparison of data. With the composition as an approach the role of “fragments” and “whole” are reversed: one starts with a collection of observable fragment properties (e.g., atomic chemical shifts of NMR spectra) and then constructs an abstract non-observable quantity representing the collection of fragments as a “whole”. If a so-derived quantity for different molecules shows some regularity, the initial loss of information in condensation of independent fragment data is compensated by insight into novel structural correlations. The approach is illustrated first by ordering isomers (e.g., nonanes C9H20) with respect to their content of special graph invariants p 2 and p 3 (numbers of paths of length two and length three, respectively) and then showing that the constructed global quantity derived from individual carbon-13 NMR chemical shifts shows a regular variation with p 2 and p 3, very similar to isomeric variations of numerous thermodynamic properties of nonanes. Subsequently it is outlined how the difference (p 2 p 3) leads to a correlation for mean carbon-13 chemical shifts in octanes and nonanes, taken as an illustration for the approach. It is expected that the outlined approach opens new avenues for data reduction and the search for structure-property correlations.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Theoretica chimica acta 92 (1995), S. 97-106 
    ISSN: 0040-5744
    Keywords: Key words: Graphs ; Restricted random walks
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Summary.  In the first part of this contribution we outline the construction of a novel matrix associated with a graph, the entries of which give the probability of a random walk over the graph G starting at site i to reach site j in D ij steps. Here D ij is the distance between vertices i, j. The derived matrices, to be referred to as restricted random walk matrices and labeled as RRW matrices, are non-symmetric, for trees the entries being of the form 1/p, where p is an integer equal to 1 or larger. In the second part of the report we consider a few invariants of the RRW matrices. We will illustrate the use of one such invariant in a regression analysis. We consider the variations of the entropies in isomeric octanes with skeletal changes. The derived regression, based on a single descriptor, yields the standard error of 1.26 cal K-1 mol-1 that is the smallest yet reported in the literature.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    ISSN: 1520-5142
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    s.l. : American Chemical Society
    ISSN: 1520-5142
    Source: ACS Legacy Archives
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
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