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Abstract
A new, simple, relation is established between the total π-electron energy and the HOMO-LUMO gap, applicable to benzenoid hydrocarbons.
Highlights
W ITHIN the simple Hückel molecular orbital (HMO) model, there are two energy-based stability criteria – the total π-electron energy and the energy difference between the highest occupied and lowest unoccupied molecular orbital, the so-called HOMO-LUMO gap.[1,2,3] If the eigenvalues of the molecular graph are λ1 ≥ λ2 ≥ ≥ λn, the HMO total π-electron energy2,3 is given by n/2 ∑ Eπ = 2 λi . (1)i =1 whereas the eigenvalues λn/2 and λn/2+1 correspond, respectively, to the energies of HOMO and LUMO
In Eqs. (1) and (2), and later in the text, it is assumed that n, the number of π-electrons is even
The total π-electron energy is related to the thermodynamic stability of the underlying conjugated compounds, for details see.[4,5]
Summary
W ITHIN the simple Hückel molecular orbital (HMO) model, there are two energy-based stability criteria – the total π-electron energy and the energy difference between the highest occupied and lowest unoccupied molecular orbital, the so-called HOMO-LUMO gap.[1,2,3] If the eigenvalues of the molecular graph are λ1 ≥ λ2 ≥ ≥ λn , the HMO total π-electron energy (in β-units) is given by n/2. (1) and (2), and later in the text, it is assumed that n, the number of π-electrons (and the number of carbon atoms in the underlying conjugated molecule, and the number of vertices of the respective molecular graph) is even. The total π-electron energy is related to the thermodynamic stability of the underlying conjugated compounds, for details see.[4,5] The structure-dependency of Eπ is nowadays well understood, see[5,6,7] and the references cited therein. In formulas (3) and (4), and later in this text, n denotes the number of vertices and m the number of edges of the underlying molecular graph (i.e., the number of carbon atoms and carbon-carbon bonds of the respective conjugated molecule). The approximations (3) and (4) are prohibitively complicated The fact that their mathematical forms are completely different casts doubts on their reliability. Which is based on the classical McClelland approximation[23] and a recently established result by Oboudi.[24]
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