DFT calculations for CH3+ within a constrained cage of water molecules permit the controlled manipulation of distances rax and req to “axial” and “equatorial” waters. Equatorial CH···O interactions catalyze methyl transfer (MT) between axial waters. Variation in rax has a greater effect on CH bond lengths and stretching force constants in the symmetric SN2-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 rax and req. In particular, translation of CH3+ in the axial direction is always coupled to cage motion. With longer rax, CH3+ translation is coupled with asymmetric CO bond stretching, but with shorter rax, it is also coupled with OP (equivalent to the umbrella mode of trigonal bipyramidal O···CH3+···O); the magnitude of the imaginary MT frequency increases steeply as rax 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.
More detail can easily be written here using Markdown and $\rm \LaTeX$ math code.