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

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, Xi (i=1, 2, 3), the hyperfine constant aC has the form aC=(SC+ ∑ i=13QCXiC)ρπ+ ∑ i=13QXiCCρiπ,where ρπ and ρiπ(i=1,2,3) are the pi-electron spin densities on atoms C and Xi, respectively. The contribution of the 1s electrons is determined by SC and that of the 2s electrons by the Q's, where QBCA 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 SC= —12.7 gauss, QCHC=19.5 gauss, QCC′C=14.4 gauss, and QC′CC= —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 QCHH is somewhat larger in CHC2 than in CH3, which suggests a theoretical justification for some of the variation in the experimental ``QCHH'' 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.