Computational Modeling of a Caged Methyl Cation: Structure, Energetics, and Vibrational Analysis

DFT calculations for CH3+ within a constrained cage of water molecules permit the controlled manipulation of distances re), and r(eq) to "axial" and "equatorial" waters. Equatorial CH center dot center dot center dot O interactions catalyze methyl transfer (MT) between axial waters. Variation in r(ax), has a greater effect on CH bond lengths and stretching force constants in the symmetric S(N)2-like transition structures than variation in req. In-plane bending frequencies are insensitive to these variations in cage dimensions, but axial interactions loosen the out-of-plane bending mode (OP) whereas equatorial interactions stiffen it. Frequencies for rotational and translational motions of CH3+ within the cage are influenced by r(ax) and r(eq). In particular, translation of CH3+ in the axial direction is always coupled to cage motion. With longer r(ax), CH3+ translation is coupled with asymmetric CO bond stretching, but with shorter r(ax) it is also coupled with OP (equivalent to the umbrella mode of trigonal bipyramidal O center dot center dot center dot CH3+center dot center dot center dot O); the magnitude of the imaginary MT frequency increases steeply as r(ax) diminishes. This coupling between CH3+ and its cage is removed by eliminating the rows and columns associated with cage atoms from the full Hessian to obtain a reduced Hessian for CH3+ alone. Within a certain range of cage dimensions, the reduced Hessian yields a real frequency for MT. The importance of using a Hessian large enough to describe the reaction coordinate mode correctly is emphasized for modeling chemical reactions and particularly for kinetic isotope effects in enzymic MT.