Theory of Isotropic Hyperfine Interactions in π-Electron Radicals

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Indirect proton hyperfine interactions in π-electron radicals are first discussed in terms of a hypothetical CH fragment which holds one unpaired π electron and two σ-CH bonding electrons. Molecular orbital theory and valence bond theory yield almost identical results for the unpaired electron density at the proton due to exchange coupling between the π electron and the σ electrons. The unrestricted Hartree-Fock approximation leads to qualitatively similar results. The unpaired electron spin density at the proton tends to be antiparallel to the average spin of the π electron, and this leads to a negative proton hyperfine coupling constant. The theory of indirect proton hyperfine interaction in the CH fragment is generalized to the case of polyatomic π-electron radical systems; e.g., large planar aromatic radicals. In making this generalization there is introduced an unpaired π-electron spin density operator, ρN, where N refers to carbon atom N. Expectation values of the spin density operator ρN are called ``spin densities,'' ρN, and can be positive or negative. In the simple one-electron molecular orbital approximation a π-electron radical always has a positive or zero spin density at carbon atom N, 0≤ρN≤1. In certain π-electron radical systems; e.g., odd-alternate hydrocarbon radicals, the spin densities at certain (unstarred) carbon atoms are negative when the effects of π—π configuration interaction are included in the π-electron wave function. The previously proposed linear relation between the hyperfine splitting due to proton N, aN, and the unpaired spin density on carbon atom N, ρN,aN=QρNis derived under very general conditions. Two basic approximations are necessary in the derivation of this linear relation. First, it is necessary that σ—π exchange interaction can be treated as a first-order perturbation in π-electron systems. Second, it is necessary that the energy of the triplet antibonding state of the C–H bond be much larger than the excitation energies of certain doublet and quartet states of the π electrons. This derivation of the above linear relation makes no restrictive assumptions regarding the degree of π—π or σ—σ configuration interaction. The validity of the above approximations is discussed and illustrated by highly simplified calculations of the proton hyperfine splittings in the allyl radical, assuming the π—π configuration interaction—and hence the negative spin density on the central carbon atom—to be small. Isotropic hyperfine interactions in molecules in liquid solution can also arise from spin-orbital interaction effects, and it is shown that these effects are negligible for proton hyperfine interactions in aromatic radicals.