Rare-earth garnets with the general formula RE_3Al_5O_12 (RE = La–Lu, Y) are investigated in the MO LCAO approximation. The phonon spectrum of Y_3Al_5O_12 at the Γ point is calculated. Fundamental vibrations in the structure of Y_3Al_5O_12 are assigned based on an analysis of ab initio calculated displacement vectors. The elastic constants of the RE_3Al_5O_12 crystals are determined. Calculations are performed in terms of the density functional theory. The need to use hybrid functionals, which take into account the contribution of the nonlocal exchange into the Hartree–Fock formalism, is shown. The description of inner shells of the rare-earth ion down to 4 f inclusive by a pseudopotential (4 f -in-core) is shown to ensure significant reducing the computer cost, with the description accuracy of the structure and the lattice dynamics being preserved. The CRYSTAL program, intended for ab initio calculations of periodic structures, is used.