Topological Approach to the Chemistry of Conjugated by Ante Graovac

By Ante Graovac

"The moment step is to figure out structure, Le. which atoms are bonded to which and by means of what kinds of bond. the result's ex­ pressed via a planar graph (or the corresponding connectivity mat­ rix) •••• In constitutional formulae, the atoms are represented through letters and the bonds through traces. They describe the topology of the molecule." VLADIMIR PRELOG, Nobel Lecture, December l2;h 1975. within the current notes we describe the topological method of the che­ mistry of conjugated molecules utilizing graph-theoretical strategies. Con­ jugatedstructures will be very easily studied utilizing planar and connec­ ted graphs simply because they replicate within the easy manner the connectivity in their pi-centers. Connectivity is critical topological estate of a molecule which permits a conceptual qualitative realizing, through a non­ numerical research, of many chemical phenomena or a minimum of that a part of phenomenon which relies on topology. this is able to no longer be attainable sole­ ly via numerical (molecular orbital) analysis.

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To basis functions usedJn our example, Is atomic orbitals of H-atoms are represented by the vertices of one particular type (0), and sp3 hybrid orbitals of C-atom by the vertices of another type (I). formed. Then the Hamiltonian matrix of a problem needs to be The various interactions between the orbitals used (usually expressed by means of the molecular integrals) can be described by (the vertices and) the edges of different weights. The corresponding weighted graph which is G3 of Fig. 13. is called the quantum chemical graph.

79b) The symbols in the above fonnula (Sachs theorem) have the following meaning: s denotes a Sachs graph 52 , S. is a set of all Sachs graphs with j vertices of G, c (s) and J r (s) are the total number of canponents and the total number of cycles (rings), respectively, in a Sachs graph s, and a. is, of course, a coefficient of the characterJ istic polynanial we are about to construct. 4. By definition ao=l. For graphs not containing loops a 1=0. When Sj is arpty set (S. = J ~), then a. = O. J Fonnula (2.

X-~). 73) Comparison with eqs. 62) indicates that the coefficients of P(G;x) will also be dependent on the structure (topology) of a graph G. 3. 2. Graph Spectral Properties of Particular Classes of Graphs For certain classes of graphs the graph spectrum (and the eigenvector problem) can be solved in a closed analytical form, most frequently using the symmetry properties or a graph. 74 ) Using these and similar methods the graph spectral-structural analysis was carried out analytically for regular and complete graphs~5a especially for rings~OO for chains}OO stars and other trees~O for radialenes and related molecular graphs;6 as well as for some other classes of graphs~5a The numerical data for trees and other graphs were calculated by Nosal}2la while those for the most important Huckel graphs can be found, for example, in Coulson and Streitwiesser's tables~5 The interval in which the eigenvalues of a graph lie is limited.

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