Within the framework of the density functional theory (DFT) and the hybrid functional B3LYP by means of the CRYSTAL17 program code, the wavenumbers and intensities of normal oscillations of MgCO3, CaCO3, ZnCO3, CdCO3 in the structure of calcite; CaMg(CO3)2, CdMg(CO3)2, CaMn(CO3)2, CaZn(CO3)2 in the structure of dolomite; BaMg(CO3)2 in the structure of the norsethite type; and CaCO3, SrCO3, BaCO3, and PbCO3 in the structure of aragonite were calculated. Infrared absorption and Raman spectra were compared with the known experimental data of synthetic and natural crystals. For lattice and intramolecular modes, linear dependences on the radius and mass of the metal cation are established. The obtained dependences have predictive power and can be used to study solid carbonate solutions. For trigonal and orthorhombic carbonates, the linear dependence of wavenumbers on the cation radius RM (or M–O distance) is established for the infrared in-plane bending mode: 786.2–65.88·RM and Raman in-plane stretching mode: 768.5–53.24·RM, with a correlation coefficient of 0.87.