Theoretical Interpretation of Carbon-13 Hyperfine Interactions in Electron Spin Resonance Spectra

Abstract

A quantitative theory of the isotropic electron‐nuclear spin interactions of carbon 13 in pi‐electron radicals is presented and applied to the hyperfine splittings observed in the electron spin resonance spectra of these substances. The splittings arise from sigma‐pi interactions which polarize both the 1s and 2s electrons. The 1s‐orbital spin polarization is shown to contribute a term of negative sign with a magnitude comparable to that from the 2s electrons. For an sp² hybridized carbon atom that is bonded to three atoms, X_i (i=1, 2, 3), the hyperfine constant a^C has the form $$a^{\mathrm{C}}{\mathrm{=(S}}^{\mathrm{C}}+{\displaystyle \sum _{\text{i=1}}^3}{Q}_{{\mathrm{CX}}_i^C}{\mathrm{)\rho }}^{\pi }+{\displaystyle \sum _{\text{i=1}}^3}{Q}_{{\mathrm{X}}_i{\mathrm{C}}^{\mathrm{C}}{\rho }_i^{\pi },}$$ where ρ^π and ${\rho }_i^{\pi }\text{(i=1,2,3)}$ are the pi‐electron spin densities on atoms C and X_i, respectively. The contribution of the 1s electrons is determined by S^C and that of the 2s electrons by the Q's, where Q_BC^A is the sigma‐pi parameter for the nucleus of atom A resulting from the interaction between the bond BC and the pi‐electron spin density on atom B. Calculations for a planar CHC₂ fragment model yield S^C= —12.7 gauss, Q_CH^C=19.5 gauss, Q_CC′^C=14.4 gauss, and Q_C′C^C= —13.9 gauss. The theory predicts both the magnitude and sign of the hyperfine splittings and is readily applied to a variety of compounds. Excellent agreement is obtained with the available experimental data. For the methyl radical, the measured C¹³ splitting is shown to be consistent with a planar model and limits the deviation from planarity to ≲5°. The theory provides a useful criterion for the validity of approximate wave functions that is illustrated by a comparison of various theoretical treatments for the naphthalene negative ion and triphenylmethyl. The sigma‐pi interaction parameters are shown to depend on the bond length, the type of hybridization (including the angles between sigma bonds), and on the nature of the bonding atoms. For pi‐electron systems, the results demonstrate that the magnitude of the sigma‐pi exchange energy is a small fraction of the total energy. It is also noted that the proton parameter Q_CH^H is somewhat larger in CHC₂ than in CH₃, which suggests a theoretical justification for some of the variation in the experimental ``Q_CH^H'' required to fit measured proton splittings. The form of the theory is readily extended to the treatment of hyperfine splittings from nuclei other than C¹³.