The vibrational spectrum of the Si-free katoite hydrogarnet (116 atoms in the unit cell) has been calculated at the periodic ab initio quantum mechanical level with the CRYSTAL program, by using a Gaussian type basis set and the hybrid B3LYP Hamiltonian. The harmonic frequencies at the F point have been obtained by diagonalizing the mass-weighted Hessian matrix, that is evaluated by numerical differentiation of the analytical first derivatives of the energy with respect to the atomic Cartesian coordinates. The parameters controlling the numerical differentiation, as well as the numerical integration of the exchange-correlation functional for the self-consistent field (SCF) calculation, are shown to affect the obtained frequencies by less than 3 cm-1. Before diagonalization, the dynamical matrix is transformed to a block diagonal form according to the irreducible representations of the point group, so that the 345 vibrational modes are automatically classified by symmetry. Various tools are adopted (graphical representation, isotopic substitution, "freezing" part of the unit cell) that permit a complete classification of normal modes and, in particular, an analysis of the modes in terms of simple models (octahedra modes, Ca modes, H stretching, bending, rotations). The harmonic OH stretching band (48 modes) is quite narrow (20 cm-1), indicating that the interaction among OH groups is very weak. As the OH stretching modes are known to be totally separable from the other modes and strongly anharmonic, the one-dimensional Schroedinger equation for the anharmonic oscillator is solved numerically for the two extreme situations, corresponding to the vibration of one decoupled OH and of all 48 OH groups moving in phase. The anharmonic frequencies are 3682 and 3673 cm-1, respectively, in good agreement with IR experiments (a single band at 3661 cm-1 with a width at half band height of 33 cm-1) and confirming that the interaction between OH groups is extremely weak.