The eigenvalues λi of an Hermitian matrix A of order n satisfy the inequality
i = 1n | λi | ≤ n1/2N,
where N is the Frobenius norm of A. Let n and ν be, respectively, the numbers of atoms and bonds in the conjugated system of a hydrocarbon, and let E be the corresponding Hückel matrix. Then the Hückel π‐electron energy is nα + εβ, where [2ν + n(n − 1)| detE |2/n]1/2 ≤ ε ≤ (2nν)1/2. Analogous bounds are obtained for π‐electron energies calculated by Wheland's method. An approximation for the Hückel π‐electron delocalization energy (DE) in the closed‐shell ground state of a hydrocarbon is DE / β = an1/2 − n + r, where a ≈ 1.30, and r = 0 or 1, according as n is even or odd.
R. Daudel, R. Lefebvre, and C. Moser, Quantum Chemistry (Interscience, New York, 1959), p. 178.
A. Streitweiser, Molecular Orbital Theory for Organic Chemists (Wiley, New York, 1961), p. 108.
If the conjugated system has one or more heteroatoms, the last part of (8) becomes
Tr(E) = lht
when αl = α+hlβ is the Coulomb parameter for atom l. Equation (9) is still valid if a sensible choice is made for the heteroatom Coulomb and exchange integrals. Note also that Eq. (9) applies to those open‐shell systems in which εi = 0 for singly occupied orbitals.
More precisely, the mean value of (grgs−g2) is the small term (g2−〈g2〉)/(t−1), where 〈g2 is the mean‐square gr.
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