The potential surface for the three methyl rotations in the tertiary-butyl cation, (CH3)(3)C+

The intricate symmetry of the potential energy surface for the three methyl rotation dimensions in the tertiary-butyl cation, (CH3)(3)C+, is examined with the presentation of analytic three-dimensional potential functions for the first time, The simplest function is a four-parameter model which is reminiscent of of a form commonly used in the studies of two-rotor systems. This function, arising purely from symmetry and proper Fourier series truncation, dictates the existence of four ''canonical'' stationary-point conformations, whose ab initio energies are sufficient to determine its parameter values. The simplicity of the surface leads to mirror-image relations concerning the normal mode curvatures at the canonical stationary points, and these relations are shown to be approximately valid on 33-dimensional and nonanalytic ab initio surfaces. A 14-parameter function is also developed, and values for Its parameters were determined from fitting to ab initio energies and tested against normal mode frequencies at the canonical locations. Kinetic (inertial) terms are presented as well. To consider the neglected coupling of the pure torsional coordinates with other internal coordinates along the internal rotation pathways, total energy distribution (TED) analyses of ab initio harmonic force fields were performed, and contributions of from 5% to 40% from other internal coordinates were observed in the internal rotation normal modes at various stationary points. This suggests that further improvement of three-dimensional functions beyond those presented here may be of Limited use without incorporation of bending potential terms.