Self-consistent field theory of the electron spin distribution in π-electron radicals

In a radical the single determinant wave function which gives the lowest electronic energy is one where electrons of opposite spins occupy different sets of molecular orbitals and thereby lower the 'exchange' part of the energy. The unpaired spin density may be negative in a wave function of this type and is easily found by perturbation theory if certain exchange integrals are small. We show that the spin density is then almost the same as one finds by combining the usual single determinant function (with one unpaired electron, and 2n paired ones in n orbitals) with its singly excited doublet configurations. In alternant hydrocarbons the Pariser and Parr theory leads to a simple formula for the spin density: ρr = cro 2 - λ Σ s πrscso 2, where cro is the Hückel coefficient of the odd orbital on atom r, πrs is the mutual polarizability of atoms r and s, and λ is a constant derived from the theory. The observed hyperfine structure in the electron resonance spectra of naphthalene, anthracene, perylene, diphenyl, phenanthrene, and pyrene negative ions agrees well with this formula, provided the constant Q defined by McConnell is about -24·2 gauss. The non-alternant negative ions of acenaphthylene and fluoranthene also agree, but not acepleiadylene. The theory predicts identical spin densities in corresponding positive and negative ions. In neutral alternant radicals we find negative spin densities on all the unstarred atoms of triphenylmethyl, perinaphthenyl, allyl, and benzyl. The calculated negative densities are a little too small to fit the spectra of the first two radicals, but fit just as well as valence bond ones.