Vector Model for Indirect Proton Hyperfine Interactions in π-Electron Radicals

Significant information on unpaired electron distributions in molecules, liquids, and solids can sometimes be obtained from magnetic resonance studies of electron-nucleus hyperfine interactions. A particularly important problem in this field involves the proton hyperfine interactions that have been observed in the electron magnetic resonance spectra of a large number of aromatic radicals in solution [1-3]. The problem concerns the theoretical basis for a proposal that the observed hyperfine splitting aN due to aromatic proton N in an aromatic radical can be used to estimate the unpaired spin density ρN at carbon atom N to which proton N is attached, according to the equation [2,4-6] aN = QρN. (1) A semiempirical value of Q derived from observed splittings aN and theoretically calculated spin densities ρN is -30 ± 5 gauss, or -85 ± 15 Mc [2]. Equation (1) has been derived recently using molecular orbital theory with arbitrary π-π configuration interaction [6]. We show here how the simple proportionality in equation (1) may be derived using the Dirac vector model