Encoding the Core Electrons with Graph Concepts

Abstract

The core electron problem of atoms in chemical graph studies has always been considered as a minor problem. Usually, chemical graphs had to encode just a small set of second row atoms, i.e., C, N, O, and F, thus, graph and, in some cases, pseudograph concepts were enough to "graph" encode the molecules at hand. Molecular connectivity theory, together with its side-branch the electrotopological state, introduced two "ad hoc" algorithms for the core electrons of higher-row atoms based, mainly, on quantum concepts alike. Recently, complete graphs, and, especially, odd complete graphs have been introduced to encode the core electrons of higher-row atoms. By the aid of these types of graphs a double-valued algorithm has been proposed for the valence delta, deltav, of any type of atoms of the periodic table with a principal quantum number n > or =2. The new algorithm is centered on an invariant suggested by the hand-shaking theorem, and the values it gives rise to parallel in some way the values derived by the aid of the two old "quantum" algorithms. A thorough comparative analysis of the newly proposed algorithms has been undertaken for atoms of the group 1A-7A of the periodic table. This comparative study includes the electronegativity, the size of the atoms, the first ionization energy, and the electron affinity. The given algorithm has also been tested with sequential complete graphs, while the even complete graphs give rise to conceptual difficulties. QSAR/QSPR studies do not show a clear-cut preference for any of the two values the algorithm gives rise to, even if recent results seem to prefer one of the two values.

Model of the physical properties of halides with complete graph‐based indices

“…The choice of a K 7 for iodine is based on the fact that, to date, only odd complete graphs, K p with p = 1, 3, 5, 7, 9 have been successful in QSPR studies. The K p representation for the core electrons allows us to define the following general algorithm for δ v 1–6: Parameter p · r equals the sum of all vertex degrees in a complete graph. The handshaking theorem states that this sum for every type of graph, and pseudo‐graphs, complete graphs inclusive, equals twice the number of connections 21, 22.…”

“…This choice would introduce K 2 vertices for n = 2 atoms, thus redefining all graphs made up of second‐row atoms. Previous work 1–6 has shown that in Eq. (1), q = 1, and p , even if other values cannot be excluded a priori.…”

“…The complete K p graph representation for the core electrons of any atom with principal quantum number n ≥ 2 has recently been used to model many properties of different classes of compounds, 1–6. To date, the preferred “graph” representation for the core electrons was the odd complete graph representation, which gave rise to a two‐valued algorithm for the valence delta number, δ v .…”

“…The valence delta number, δ v , is used to define the valence basis indices of the molecular connectivity (MC) theory 7–10, i.e., the valence molecular connectivity indices, and the I ‐ and E ‐state pseudoconnectivity indices 11–13. The old “quantum” algorithm for δ v was two‐valued, the new “pure” graph algorithm can, instead, be four‐valued 6, 9, 10, and this improves its flexibility, which allows us to derive a more tunable set of values for the molecular connectivity and pseudo‐connectivity indices, among which to choose the best ones for model purposes, in a way comparable to the variable indices 14, but at a different level. The fact that, to date, only odd complete graphs were successful in deriving optimal quantitative structure–property or –activity relationships (QSPR/QSAR) was rather intriguing.…”

“…The fact that, to date, only odd complete graphs were successful in deriving optimal quantitative structure–property or –activity relationships (QSPR/QSAR) was rather intriguing. In fact, recent findings underline the good theoretical quality of sequential complete graphs 6. As there is no mathematical proof of the superiority of odd complete over sequential complete graphs for encoding the core electrons, the only explanation for this preference could be that up to now no specific property for sequential complete graph‐based MC indices has been tested.…”