Theory of the Temperature Dependences of the Hyperfine Interactions in the Methyl Radical

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Abstract
The temperature dependences of the hyperfine interactions in the ESR spectrum of the methyl radical are examined theoretically using a semiempirical nonionic valence-bond theory. K-shell polarization and incomplete following of the molecular vibrations by the carbon sigma orbitals are accounted for. The interactions are calculated as functions of the nuclear positions in the out-of-plane bending mode (A2″), and are averaged over the ground vibrational state. By a modest adjustment of semiempirical integral parameters and the degree of the sigma orbital following, four calculated hyperfine coupling constants (H in CH3, D in CD3, 13C in 13CH3 and in 13CD3) are fitted to the values observed at 97°K. The vibrational averaging is then extended to several of the lower excited states. By assuming a Boltzmann population of these states, the interactions are then calculated as functions of temperature from 0°K to 500° for CH3 and from 0° to 700° for CD3 without further parameter adjustment. Thus, by fitting four coupling constants at a single temperature and assuming a definite quantum-mechanical model, we predict the coupling constants over a wide temperature range. The relationship between the temperature dependences of the interactions and the isotope effects is discussed. The contribution of each orbital in the molecule to each hyperfine interaction is calculated, and it is found that the unpaired pi electron plays an insignificant role in the temperature dependence of the proton interaction, but that it dominates the carbon−13 temperature dependence. The calculations indicate that at room temperature the absolute value of the proton interaction in 12CH3 should decrease as the temperature increases by the amount (1.35±0.03)×10−3 G/deg, and the carbon−13 interaction in 13CH3 should increase by (1.29±0.08)×10−2 G/deg. Unfortunately, the only observation of a temperature coefficient is not directly comparable to the theoretical results reported here, since in that measurement water was the solvent. The result was (2.1±0.2)×10−3 G/deg for the proton in 12CH3. In addition to the solvent effect, several other possible sources for the discrepancy are discussed, including vibrations other than A2″, and our use of an approximate normal coordinate for the A2″ mode. Use of a value for the force constant recently suggested by Lester, Andrews, and Pimentel is found to markedly decrease the calculated temperature coefficients.

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