Proton Magnetic Resonance of the CH3 Group. IV. Calculation of the Tunneling Frequency and of <i>T</i>1 in Solids

The thirty lowest eigenvalues of the Mathieu equation have been calculated numerically to nine places for a hindered rotor with threefold symmetry attached to a rigid framework. The results are tabulated for eight values of IV0/2ℏ2 ranging from 56.25 to 115.3125. The splittings of the torsional states of the CH3 group have been used to calculate the average tunneling frequency, νt, as a function of temperature for barrier heights of 825 cm—1 to 2770 cm—1. Some useful relations are developed between the eigenvalues and torsional splittings for the 2- and 3-nodal potential functions. The BPP theory is used to calculate the proton spin-lattice relaxation time, T1, for CH3 groups undergoing hindered reorientations in solids and also over-all molecular tumblings, assuming Debye spectral density functions for both motions. The manner in which the tunneling affects T1 is considered at some length and two ways of introducing it are described. The coupling among the three protons within a given CH3 group is discussed briefly in relation to T1.