Theory of the Proton Hyperfine Splittings of Pi-Electron Free Radicals. I. The CH Fragment

Abstract

Using first‐order perturbation theory, a general equation is derived for the Fermi contact hyperfine splittings of any free radical. This equation is essentially the same as that obtained previously by other workers, but restrictions concerning (1) planarity of the radical, (2) sigma–pi separability, (3) choice of spin eigenfunctions, and (4) choice of orbitals used to form excited configurations have been removed. The results of calculations of the proton hyperfine splitting ${\mathrm{(a}}^H)$ of the prototype pi‐electron hydrocarbon radical, the ·CH fragment, are presented. A seven‐electron wavefunction with a minimum basis set of Slater atomic orbitals is employed. The CH bonding orbital incorporates ${\mathrm{sp}}^2$ hybridization and an electronegativity parameter which is optimized by variational minimization of the ground configuration energy. Admixture of three configurations is considered; two represent the single excitation from CH bonding to antibonding orbital and the third the corresponding double excitation. The splitting is computed using first‐ and second‐order perturbation theories, invariant‐perturbation theory, and the variation method. Sensitivity of $a^H$ to variations in (1) values of the orbital exponents (2) choice of spin eigenfunctions, (3) value of the electronegativity parameters, and (4) CH bond angle is investigated. The calculated splitting varies from about −25 to −50 G and depends most critically on the hydrogen $\text{1s}$ orbital exponent. A 10% maximum variation in $a^H$ is found in first order depending on the choice of spin eigenfunctions. Admixture of the doubly excited configuration increases ${\mathrm{|\hspace{0.17em}a}}^H\mathrm{\hspace{0.17em}|}$ by ≈ 10%. The optimum value of the CH bond‐electronegativity parameter is quite close to that previously chosen on intuitive grounds; $a^H$ is not very sensitive to this parameter. The above calculations constitute an improvement in rigor over previous work on the CH fragment in that atomic orbital overlap is not neglected, the hydrogen $\text{1s}$ and carbon $\text{2s}$ orbitals are orthogonalized to the carbon $\text{1s}$ orbital, and arbitrary assumptions regarding CH bond polarity have been eliminated.