In the full-electron and pseudopotential basis of localized atomic orbitals with the exchange-correlation functional PBE and the dispersion correction D3 of the CRYSTAL code, the crystal structure, dielectric constant tensor, LO-TO mode splitting, infrared reflection spectra and Raman scattering of the mineral belousovite and its synthetic analogues with formula AZnSO₄X (A= K, Rb, Cs, Tl; X= Cl, Br, I). The presence of SO₄, ZnO₃X tetrahedra and AO_nX_m polyhedra in the structure is shown, which appear in different regions of the vibrational spectra. In polarized reflection spectra, SO4 modes have high-intensity bands ν₁ and ν₃ from 900 to 1160 cm^-1, formed by vibrations of nonequivalent oxygen atoms. In the lattice region below 150 cm^-1, vibrations involving cation atoms are distinguished. Individual modes with a high level of LO-TO splitting have been identified. In the Raman spectrum, the most intense modes will also be ν₁,ν₃. The first group consists of four modes, where the most intense one at 900-920 cm^-1 is formed largely by vibrations of individual nonequivalent oxygen atoms, and the other three by sulfur atoms and other atoms. On the contrary, modes with wavenumbers above 1100 cm^-1 have an overwhelming contribution from O4 oxygen atoms with the shortest S-O and A-O distances. For these and other modes, linear dependences of wave numbers on interatomic distances, average atomic masses and radii of cations A and anions X have been established. Keywords: belousovite, synthetic crystals, ab initio, dielectric constant, infrared reflectance spectra, Raman spectra.

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