Properties of the Latent Roots of a Matrix: The Estimation of <i>π</i>-Electron Energies

Abstract

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.