We report an ab initio calculation of the potential surface, quantum structures, and zero-point energies of CH₅⁺ and CH₂D₃⁺ in full dimensionality. This potential energy surface is a very precise fit to 20 633 ab initio energies and an even larger data set of potential gradients, obtained at the MP2/cc-pVTZ level of theory/basis. The potential, which exactly obeys the permutational symmetry of the five hydrogen atoms, is used in diffusion Monte Carlo (DMC) calculations of the fully anharmonic zero-point energies and ground-state wave functions of CH₅⁺ and CH₂D₃⁺. Bond length distributions are obtained from the DMC ground state and are compared to those resulting from classical molecular dynamics simulations, which are performed at the quantum zero-point energy for roughly 300 picoseconds.