Mobius Molecules and Graphs

Mobius molecules and graphs are discussed. A brief discussion of the generalized graphs is also given. The extended Sachs formula for Mobius graphs (and generalized graphs) is .reported and some examples discussed. The parity of the algebraic structure of Mobius molecules are defiined

An Approximate Spectral Density for the Estimation of so me Topological Indices of Alternant Systems

A symmetric two-delta-function model spectral density is used to estimate several topological indices of alternant hydrocarbons, namely: the total n-electron energy (E.), the modified topological index (Z), the HOlVIO-LUMO separation (XHL) and the spectral radius of adjacency matrix (R). It is found, that the invariants defined by integration (like E. and Z) are reproduced much better than the invariants defined as the Iimiting values of the spectral distribution (like XHL and R). The reason for the well known linear dependence between Er. and lnZ, is discussed

Graph-theoretical Search for Benzenoid Polymers with Zero Energy Gap

Recently structural features which characterize the energy gap in polymeric conjugated hydrocarbons within the framework of the simple Hiickel MO theory have been specified using graphtheoretical methods. It has been shown that molecules of interest as potential units in polymers with zero energy gap have to satisfy certain structural conditions which relate the number of selected »excited« valence structures to that of the Kekule structures for the system. As in the previous work we consider units separated by essentially single bonds. Here we elaborate on the search for benzenoid systems which can satisfy the requirement. A necessary condition that a system may have zero energy gap is that the repeating benzenoid unit has a non-prime number (K) of Kekule valence structures. Then K could be factored: K1 • K2 ... Km. (Generally, there are more factorizations of K and our procedure needs to be carried out over all of them). Fragments Fi that have the number of Kekule structures given by K; are recognized, and we try to superimpose all of them over the skeleton of monomer unit. (Our procedure needs to be carried out for all possible fragmentations corresponding to a given factorization) . If it is possible to cover the monomer unit with all the fragments leaving at least two positions available for linking the units in polymer form, then the energy gap of such a polymer is zero. In a number of examples it is illustrated how the actual search is performed. A list of benzenoid systems of interest as units in a polymer with zero energy gap is given. The search is quite efficient and speedy

Technical and logical methods for improving the process of urban planning in Serbia

The subject of the paper is an analysis of the methodology for developing urban plans, considered in a normative, organizational and interest context. Based on current legislation defining the content and procedure for adopting a plan, and the institutional framework that defines the participants in the planning process, a basic methodological model for a planning solution was formed, which was then improved in the context of the collaborative planning paradigm. Starting from the assumption that harmonizing the different interests represents the "grey zone" of planning in Serbia, the paper explores various methodological steps that would give a space for better cooperation between all stakeholders, and therefore contribute to the improvement of procedures for developing plans and the quality of the planning solutions themselves. On the basis of this research, a methodology for urban planning is defined as a logical and technical method of successively configuring a planning solution in a normative, organizational and interest context. Through analysis of the application of the methodological model in practice and a case study, it was confirmed that the method of producing a plan that includes timely and meaningful cooperation can reconcile the interests of the different stakeholders in planning

Some Integrals for Molecular Properties and Relativistic Effects over Hermite-Gaussian Functions

Formulas for some integrals over Hermite-Gaussian functions occurring in the calculations of the molecular first and second order properties as well as relativistic corrections arising in the Breit hamiltonian are discussed. It is shown that all these molecular properties integrals can be reduced to the integrals already encountered in the minimum energy calculations. More specifically, the one-electron and two-electron integrals involving (l/r1 j)" operator, where j denotes either the coordinates of a nucleus or the coordinates of the electron 2 and n is an integer, are expressed in terms of nuclear attraction and Coulomb repulsion integrals, respectively. Therefore the electric and magnetic properties of molecules can be computed with little additional effort if the Hermite-Gaussian basis set is employed. The same conclusion holds for the matrix elements arising in the pseudo-potential calculations involving the Bonifacic- Huzinaga model potential which in turn give a fair description of the heavy atoms inner-shell electrons. Since the Hermite- Gaussian functions are particularly advantageous for atomic orbitals with higher angular momentum quantum numbers (f, g, h etc.) their use is expected to be preferable in molecules involving heavy atoms. The relativistic effects are of great importance for the latter and it is gratifying that the corresponding integrals over Hermite-Gaussians can be expressed in a closed form