Topological Atomic Charges, Valencies, and Bond Orders

Abstract

Theorems and corollaries are proved and relationships are derived for the atomic and bond moments of energy in molecules and solids or, in terms of graph theory, for the self-returning walks SRWs that begin in graph vertexes and edges, respectively. Topological atomic charges, valencies, and bond orders are introduced by means of the limit distributions of the SRWs. A hierarchical system of topological rules on atomic and bond reactivity is presented proceeding from the alternating positive and negative contributions to the atomic and bond moments. Some light is thus shed on the topological control of chemical reactivity.

From Infection Clusters to Metal Clusters: Significance of the Lowest Occupied Molecular Orbital (LOMO)

“…Since ∑ i SRW i k corresponds to the total number of SRWs of length k on the molecular graph, the right-hand side of eq can be viewed as the limit for the relative number of SRWs. Bonchev and co-workers proposed that SRW i k in the molecular graph can be interpreted as electron motion near atom i . , Thus, LOMO coefficients are expected to have something to do with the relative frequencies of such electron motion. This further can be correlated with Feynman’s idea of accounting for all possible electron paths .…”

“…Bonchev and co-workers proposed that SRW i k in the molecular graph can be interpreted as electron motion near atom i . 18 , 21 Thus, LOMO coefficients are expected to have something to do with the relative frequencies of such electron motion. This further can be correlated with Feynman’s idea of accounting for all possible electron paths.…”

“…Nagao, Nishikawa, and Aono demonstrated that the transition amplitude in the Feynman path integral method can be obtained by summing contributions from all independent trajectories . Each SRW in a molecular graph would be interpreted as an independent Feynman’s electron trajectory …”

“…They further developed their method based on the SRW, proposing topological atomic charges, valencies, and bond orders. 21 Estrada and Knight suggested in their seminal book of network theory 22 that the LOMO distribution has something to do with centrality. Recently, Redzepovic and co-workers have applied spectral graph theory to benzenoid hydrocarbons, finding that the π-electron energies of these molecules can be related with the energies of the HOMO and LOMO.…”

“…For example, Bonchev and co-workers pointed out that the expansion coefficients of the LOMO of a molecule are related with the number of self-returning walks (SRWs) on its molecular graph. − Since SRWs play an important role in the characterization of the center of a graph, the LOMO coefficients may be of importance for determining atomic environments in molecules. They further developed their method based on the SRW, proposing topological atomic charges, valencies, and bond orders . Estrada and Knight suggested in their seminal book of network theory that the LOMO distribution has something to do with centrality.…”