Comprehensive Density Functional Theory Studies of Vibrational Spectra of Carbonates

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.


Introduction
Carbonates form an extensive class of chemical compounds containing the carbonate ion CO 2− 3 as the main structural element [1,2]. In nature, carbonates are found in many minerals and play a crucial role in the carbon exchange of our planet [2][3][4][5][6][7][8]. Carbonate compounds are widely used in the construction industry, optics, and nanotechnology [9][10][11][12][13][14][15]. Appearance of the carbonate species due to chemical interaction with the atmosphere agents was detected at the surface of many oxide materials widely used in optical and electronic technologies [16][17][18][19][20]. In recent years, many artificial crystals of complex carbonates that have no analogues in nature were created [1]. Due to the specific features of the crystal structure, such materials, in many cases, are characterized by high birefringence, nonlinear optical properties, and transparency in the ultraviolet spectral range, which makes them perspective materials for use in optical devices in the UV range [21][22][23][24][25][26][27]. In this aspect, it is of particular importance to study the physicochemical properties of carbonates-in particular, the relationship between their structural and spectroscopic characteristics. According to this approach, this work is aimed at a systematic study of the relationship between the crystal structure and vibrational characteristics of a set of crystals of simple and binary anhydrous carbonates known in nature. Such minerals form several crystal-chemical families, which makes it possible to study the effects effects of cation substitution on the wavenumbers of vibrational modes within the framework of a single structural type. In the near future, with the accumulation of experimental data of complex carbonates, this algorithm can be extended to new families of complex artificial crystals.
Double carbonates are also known in the structure of norsethite (BaMg(CO 3 ) 2 ) [44] (Figure 1c). BaMn(CO 3 ) 2 does not exist in nature, but it was synthesized in [45]. The norsethite structure is described in c-space group R-3c symmetry with doubled c-axis, which corresponds to different rotations of carbonate groups [46]. As the temperature rises in BaMg(CO 3 ) 2 , a phase transition is observed from a phase stable under ambient conditions to a high-temperature structure, which is accompanied by a change in the symmetry R-3c → R-3m. In the R-3m symmetry, the unit cell of BaMg(CO 3 ) 2 consists of the [MgO 6 ] octahedron, [BaO 12 ] polyhedron, and anions. Octahedra and polyhedra are in alternating layers, they are located exactly one above the other, parallel to the [001] direction and are separated by triangular groups CO 2− 3 . Natural isostructural orthorhombic carbonates are aragonite (CaCO 3 ) (Figure 1d), strontianite (SrCO 3 ), cerussite (PbCO 3 ), and witherite (BaCO 3 ) [47,48], listed in the order of increasing the size of the cation. In a crystal structure with the space group Pmcn (Z = 4), layers of 9-coordinated cations M 2+ (M: Ca 2+ , Sr 2+ , Pb 2+ , Ba 2+ ) in an approximately hexagonal close packing alternate with layers of planar CO 2− 3 groups arranged perpendicular to the c-axis. Like calcites, aragonites form solid solutions [49]. Studies of isostructural orthorhombic carbonates are important for understanding phase transition sequences under pressure and temperature and, therefore, can provide insight into a carbon behavior in the Earth mantle [50][51][52][53].
The optical properties of natural carbonates were extensively explored since these widespread and cheap natural minerals can be used as raw resources for infrared technology materials [54]. In this regard, their infrared spectra (IRS) [55] and Raman spectra (RS) [56,57] were previously studied experimentally and then theoretically [58,59]. Single crystals of iron-free magnesite were studied by Raman spectroscopy [30]; calcite and dolomite crystals were studied in [60] and all crystal were studied in [61]. In [62], they were evaluated with the methods of laser Raman spectroscopy and density functional theory (DFT) calculations using the plane wave basis (PW) and pseudopotentials of the Troullier-Martins type (TM PP). Otavite vibrational spectra were measured in experiment [63]. Eight natural carbonate minerals with a calcite structure were studied using Raman spectroscopy [64]. It was shown that changes in the wavenumber of phonon modes of the E g (T) symmetry correlate with the distances between the nearest metal and oxygen atoms M-O and the cation ionic radii. Using a graphical approach, the authors developed the spectroscopic Raman model to calculate the ionic radius of a divalent metal cation present in a mineral.
In [65], studies of the infrared spectra of natural iron-free dolomite were carried out. A combined study of infrared absorption and Raman scattering on a natural dolomite sample CaMg 0.98 Fe 0.02 (CO 3 ) 2 was performed [66]. DFT using exchange correlation potentials in the local density approximation (LDA) and generalized gradient approximation (GGA) in the PW basis with TM PP were used here to interpret the obtained results. The Raman and infrared spectra of cerussite were measured and compared with the spectral characteristics of other minerals of the aragonite family [67]. RS and IRS at high temperature in situ were measured for aragonite, strontianite, cerussite, and witherite at atmospheric pressure [68]. Studies for high and medium temperature infrared absorption and Raman spectroscopy on a synthetic strontianite sample led to the construction of a pressure-temperature phase diagram [69]. In addition, here for the first time, the absorption spectra in the far infrared range were measured for the entire family of aragonite-type carbonates.
The works in which experimental and theoretical spectroscopic studies were performed for a group of carbonates with different structures are of particular interest. Thus, the purpose of the study [70] was to establish the influence of the M 2+ cation type on the shift of positions of the absorption bands of Nanomaterials 2020, 10, 2275 4 of 19 various anhydrous carbonate minerals from the calcite and dolomite families. In this contribution, it is shown that the position of the minima of absorption bands is unique for each chemical composition of carbonates and can be a diagnostic indicator in mineralogy. A selection of the frequencies of intramolecular modes for a large number of carbonates is also found in [71]. In [72], infrared spectra in the wavenumber range of 70-650 cm −1 were presented for 18 common and rare minerals which are quite pure in composition and have a known crystal structure. It is shown that the spectra in the far infrared range of different carbonates from the same structural group have a pronounced similarity, and the observed shifts demonstrate the effect of changing the mass of cations. The vibrational modes of natural minerals-aragonite, calcite, dolomite, magnesite, rhodochrosite, and siderite-that are active in the Raman spectrum were observed, and their pressure and temperature inducing frequency shifts were determined [73].
Thus, there are comparatively numerous experimental and individual theoretical studies of the vibrational spectra of carbonates in which the patterns of their changes during the substitution of cations were established for certain types of structures. However, systematic theoretical studies from a unified standpoint, carried out for all types of carbonate structures and having a predictive power in early works, are absent. The aim of this work is to theoretically study the dependences of the characteristics of the infrared and Raman spectra of carbonates crystallizing in the structures of calcite, dolomite, and aragonite on the radii and masses of metal cations. The parameters of vibrational spectra of carbonates were calculated within the framework of a unified approach based on the known experimental data of crystal structure. Furthermore, the results of our calculations were compared with the available experimental and theoretical parameters of the infrared and Raman spectra of carbonates and, on this basis, the general regularities of changes in the vibrational spectra were determined upon substitution of metal cations. Then, the information can be used as an instrument in the analysis of vibrational spectra of new crystalline compounds and solid solutions. The calculated quantitative dependences can also be used in noncontact nondestructive diagnostics of carbonates by spectroscopic techniques.

Calculation Method
The research of the ordinary optical properties dependences of metal carbonates were carried out within ab initio principles using the Hartree-Fock theory (HF) methods and density functional theory, which are well combined in the CRYSTAL17 program code [74,75]. The hybrid functional B3LYP, which includes the 20% HF exchange with the Becke exchange functional [76] and the LYP correlation functional [77], was used. The basic functions were chosen in the form of a linear combination of localized atomic orbitals of the Gaussian type. We used full-electronic basis sets for carbon, oxygen, magnesium, and calcium atoms from [78] and the gaussian basis sets of double-zeta valence with polarization quality basis set for zinc and cadmium atoms [79,80]. We used pseudopotential basis sets from [81] for strontium and barium, those of [82] for manganese, and those of [83] for lead.
The reciprocal space was sampled using a Monkhorst-Pack [84] grid with 216 independent k-points in the irreducible Brillouin zone for trigonal crystals, and 64 points for orthorhombic crystals. The accuracy of the self-matching procedure was no less than 10 −9 a.u. (1 a.u. = 27.21 eV). The vibrational frequencies of the lattice atoms were calculated using the FREQCALC procedure [85,86]. The phonon harmonic frequencies ω p at the point Г (k = 0, the center of the first Brillouin zone) were obtained from the diagonalization of the mass-weighted Hessian matrix of the second derivatives of energy with respect to atomic displacements u [87,88]: where atoms a and b with masses M a and M b are displaced in the unit cell (index 0) from equilibrium positions along the iand j-Cartesian directions, respectively. The first order derivatives were Nanomaterials 2020, 10, 2275 5 of 19 calculated analytically, whereas the second order derivatives were obtained numerically. The intensity of IR absorption for the ν-vibration was calculated using the Born effective charge tensor Z*, which characterizes the change in dynamics and the electronic configuration of atom displacement. The relative intensities of the Raman peaks were calculated analytically using the extension scheme of the analytical calculation of IR intensity [89]. The proposed technique was previously used to study the ordinary properties of sulfates [90].

Crystal Structure
The crystal structure parameters of calcite, dolomite, norsethite, and aragonite types calculated in this work are shown in Table 1. There is a close agreement between the optimized and experimental parameters of the crystal lattice. Thus, the standard deviation ∆ = As the physical quantities to describe the regularities of changes in the vibrational properties of carbonates from cationic substitution in the lattice, we used the mass of metal atoms (a.m.u.; Mg-23.985, Ca-39.963, Mn-54.938, Zn-63.929, Sr-87.906, Cd-113.904, Ba-137.905, and Pb-207.977) and the effective Shannon ionic radii [91]. The cationic radii of metals are determined by their electronic structure and depend on the coordination environment. The last filled electron shell of the magnesium ion is 2p 6 , and the radius of Mg 2+ surrounded by six nearest neighbors is 0.72 Å. Similarly, for calcium: 3p 6 , 1.00 Å. In zinc and cadmium, the filled shells are 3d 10 and 4d 10 , and the radii are 0.74 and 0.95 Å, respectively. In aragonite, each Ca 2+ ion is already surrounded by nine oxygen atoms and, therefore, its effective radius is 1.18 Å. In the case of strontium and barium, the radii for the 9-fold environment are 1.31 and 1.47 Å, and for 12-1.44, 1.61 Å. The electronic configuration of lead [Xe] 4f 14 5d 10 6s 2 6p 2 distinguishes it from other elements; therefore, the radius of 9-coordinated Pb 2+ is 1.35 Å, which is larger than that of strontium, but smaller than that of barium. Transition metals have partially filled 3d shells with the number of electrons from 5 to 8, and decreasing radii for Mn 2+ (0.83 Å), Fe 2+ (0.78 Å), and Co 2+ (0.745 Å). Following [33], we write the chemical formula of an arbitrary solid solution as M1 X1 M2 X2 M3 X3 CO 3 , X1 + X2 + X3 = 1. Then, the average radius of the cation is determined as For dolomite, the cation radius is 0.86 Å, and for norsethite-1.165 Å. The average atomic mass of metals is calculated in a similar way.
It is convenient to describe the change in crystal cell parameters or frequencies (intensities) of the vibrational spectra y on the radii of cations or their masses (r) using the linear dependence y(r) = y 0 + y 1 ·r, where y 0 is the value of the function at r = 0, y 1 -derivative of the function y, characterizing the rate of change of the corresponding value. The obtained calculated data y(r i ), I = 1, N are approximated by a linear dependence (fit), and the accuracy of this procedure is controlled by

Vibrational Spectra
The rhombohedral cell of calcite contains ten atoms, and 30 possible vibrational modes can be classified for it, according to irreducible representations of the point group as: Г tot = A 1g (R) + 3A 1u + 3A 2g + 3A 2u (IR) + 4E g (R) + 6E u (IR). A 1g and 4E g modes are active in Raman spectra (R), 3A 2u and 5E u modes are active in infrared (IR), A 1u and 3A 2g modes are spectroscopically inactive, and 1A 2u and 1E u modes are acoustic. Nine translational modes will refer to the symmetry A 2g + A 1u + A 2u + E g + 2E u , six librational modes to A 2g + A 2u + E g + E u , and twelve internal vibrations to A 1g + A 2g + A 1u + A 2u + 2E g + 2E u . Modes of the A 2u symmetry have polarization E||z and modes of E u symmetry have E⊥z polarization. Internal vibrations of E u symmetry are of the ν 4 type (in-plane bending), A 2u modes are of the ν 2 type (out-of plane bending), and E u symmetry are of the ν 3 type of symmetric stretching. In the Raman spectrum, the ν 4 (in-plane asymmetric stretching) mode has E g symmetry, the ν 1 symmetric stretch mode has A 1g symmetry, and a ν 3 asymmetric stretch type has E g symmetry.

Optical Spectra of Crystals with a Calcite Structure
The infrared absorption spectra (IRS) and Raman scattering spectra (RS) of calcite-structured carbonates calculated in this work, obtained by Gaussian broadening of normal long-wavelength vibrations, are shown in Figure 2. The obtained wavenumbers of vibrations active in the IRS of calcites, together with the available experimental data, are given in Table S1; for the vibrations active in RS, they are given in Table S2 in the Supplementary materials. A good agreement was observed between the calculated vibration wavenumbers and the experimentally measured values. Thus, the average root-square deviations for eight IRS-active wavenumbers obtained by the B3LYP method for magnesite and determined experimentally in [92] and [58] do not exceed 4.0% and 3.3%, respectively. There is also a good agreement with the calculated data of the authors of [93] (4.7%) and [94] (3.5%). The root-mean-square deviations for the wavenumbers of five vibrations active in RS, obtained by the B3LYP method in this work, are 1.3% (1.0%) in the experiment in [95] for magnesite (calcite), 1.5% (1.4%) for the experiment in [58], 1.6% (1.1%) for the experiment in [61], and 1.6% (1.7%) for the experiment in [73]. In MgCO3, the most intense mode (5132 km/mol) in IRS corresponds to the internal vibration ν3 of Eu symmetry with the wavenumber of 1424 cm −1 . Taking its intensity as 100%, for the ν2 vibration with a wavenumber of 874 cm −1 , we obtained 4%, and for the ν4 (746 cm −1 ) mode, even less-0.9%. In calcite, the wavenumbers corresponding to the vibrations ν3, ν2, and ν4 are 1400, 875, and 712 cm −1 , and their intensities are 5447 km/mol (100%), 3%, and 0.5%, which practically do not differ from magnesite. For ZnCO3 and CdCO3, the structures of the spectra in the high-frequency region remain similar to magnesite. Thus, for internal modes, there is a linear correlation between the change in the wavenumber and the radius of the cation RM.  In MgCO 3 , the most intense mode (5132 km/mol) in IRS corresponds to the internal vibration ν 3 of E u symmetry with the wavenumber of 1424 cm −1 . Taking its intensity as 100%, for the ν 2 vibration with a wavenumber of 874 cm −1 , we obtained 4%, and for the ν 4 (746 cm −1 ) mode, even less-0.9%. In calcite, the wavenumbers corresponding to the vibrations ν 3 , ν 2 , and ν 4 are 1400, 875, and 712 cm −1 , and their intensities are 5447 km/mol (100%), 3%, and 0.5%, which practically do not differ from magnesite. For ZnCO 3  242 cm −1 (5%). For CaCO 3 , the lattice modes are shifted to the low-wavenumber region, and their intensities decrease. In the region of lattice vibrations of ZnCO 3 , the most intense modes will be E u symmetry with wavenumbers of 287 cm −1 , 212 cm −1 , and only then with A 2u symmetry: 348 cm −1 , 176 cm −1 . In CdCO 3 , this trend continues. Thus, for translation modes of E u symmetry, a dependence on the cation mass is observed: ω(cm −1 ) = 306.0 − 1.37·M (0.915). For the rest of the lattice modes, the best linear dependence was established for the cation radius: for rotational ones: In RS, the intense line (taken as 100%) in Figure 2 is due to vibrations of the ν 1 type, and falls on 1099 cm −1 in MgCO 3 , and 1087 cm −1 in calcite. This mode has a significant polarization dependence [89]: the xx and yy components are ten times larger than the zz component. Internal vibrations of ν 4 type also have a noticeable intensity: for magnesite-with a wavenumber of 737 cm −1 , for calcite-711 cm −1 , and also for ν

Vibrational Spectra of Crystals with a Dolomite Structure
The IRS and RS of carbonates with the dolomite structure are given in Figure 3, and Tables S3 and S4 of the accompanying materials show the wavenumbers of normal long-wave vibrations of crystals with the dolomite and norsethite structures, calculated by the B3LYP method and measured experimentally.
The examination of these tables shows that there is a satisfactory agreement between the wavenumbers of lattice [69] and internal [61] vibrations of natural dolomite calculated and measured in IRS. The root-mean-square deviation is 8.7 and 1.0%, respectively. For the wavenumbers of vibrations active in RS, the root-mean-square deviation of the results of this calculation from the experimental values is 1.4%. In crystals with a dolomite structure, the picture of theoretical spectra does not differ significantly from the calcite spectra. In IRS CaMg(CO 3 ) 2 , the most intense mode (5318 km/mol, 100%) is the ν 3 mode at 1416 cm −1 . The internal vibration ν 2 with a wavenumber of 877 cm −1 has an intensity of 3.5%, and for vibration ν 4 at 727 cm −1 , the intensity is close to 1%. Unlike calcite, the vibration ν 1 of the A u symmetry is allowed by symmetry; however, its intensity is practically zero. The most intense (19%) in the region of lattice vibrations is the E u symmetry mode with a wavenumber of 337 cm −1 . Modes of the same symmetry, but with a much lower intensity, appear at 257 (2.5%) and 167 cm −1 (4%). Nanomaterials 2020, 10, x FOR PEER REVIEW 9 of 19 binary carbonate CaMn(CO3)2 differs from other crystalline dolomites in the parameters of its vibrational spectra.
In IRS of the BaMg(CO3)2 crystal (Figure 4) in the region of intramolecular vibrations, the most intense (5194 km/mol) vibration will be ν3 with a wavenumber of 1439 cm −1 . Against this background, the ν2 vibration with a wavenumber of 878 cm −1 and the intensity of 3% is almost imperceptible, moreover, the ν4 vibrations (694 cm −1 , 0.5%) and ν1 allowed here (1125 cm −1 , 0.2%) practically do not appear. In the region of lattice vibrations, vibrations of Eu symmetry with wavenumbers of 315, 200, and 106 cm −1 stand out in intensity, while less intense vibrations of A2u symmetry have wavenumbers of 347 and 115 cm −1 . The first of these less intense vibrations corresponds to the displacements of magnesium atoms in antiphase with the anions, and the second corresponds to the displacements of barium atoms. Magnesium atoms are also involved in the formation of this mode, and they shift synchronously with the anion. In RS of BaMg(CO3)2, vibrations of anion atoms will also dominate: ν1 with a wavenumber of 1126 cm −1 (its intensity is taken as 100%), ν2 of the same symmetry and intensity of 3%, as well as doubly degenerated ν4 (697 cm −1 ) and ν3 (1444 cm −1 ) with intensities of 21% and 3%, respectively. For lattice vibrations, the A1g symmetry mode with a wavenumber of 284 cm −1 and Eg symmetry modes with wavenumbers of 108 and 254 cm −1 will be noticeable, of which the first is rotational, and the second is translational vibration.  In CdMg(CO 3 ) 2 , the most intense (5788 km/mol, 100%) vibration will be ν 3 with a wavenumber of 1407 cm −1 , and for lattice vibration with a wavenumber of 338 cm −1 , the intensity is 16%. A similar picture is observed in CaZn(CO 3 ) 2 , where the intensity of the ν 3 vibration is 5928 km/mol (100%), and the intensities of two lattice vibrations with wavenumbers of 290 and 310 cm −1 are 11% and 3.5%, respectively. The situation is different in CaMn(CO 3 ) 2 , where the intensity of the ν 3 mode is much lower-1121 km/mol (100%), and against its background, the relative intensities of other ν 2 and ν 4 vibrations increased to 22 and 6%, respectively.
The RS of dolomite will be dominated by a fully ν 1 symmetric vibration with a wavenumber of 1097 cm −1 . Its full intensity is taken as 100%. Then, the intensities of the ν 4 , ν 2 , and ν 3 modes will be 15%, 0.2% and 9%, respectively. In the region of lattice vibrations, the most intense are the E g symmetry modes with wavenumbers of 296 cm −1 (13%) and 175 cm −1 (3%). In CdMg(CO 3 ) 2 , the ν 1 mode does not change in wavenumber and remains most intense. The wavenumber of the ν 3 mode decreases, but its intensity sharply increases to 49%. In CaMn(CO 3 ) 2 , the intensity of the ν 3 mode becomes maximum (taken as 100%), while for the ν 1 vibration it is only 15%. Thus, as for IRS, the binary carbonate CaMn(CO 3 ) 2 differs from other crystalline dolomites in the parameters of its vibrational spectra.
In IRS of the BaMg(CO 3 ) 2 crystal (Figure 4) in the region of intramolecular vibrations, the most intense (5194 km/mol) vibration will be ν 3 with a wavenumber of 1439 cm −1 . Against this background, the ν 2 vibration with a wavenumber of 878 cm −1 and the intensity of 3% is almost imperceptible, moreover, the ν 4 vibrations (694 cm −1 , 0.5%) and ν 1 allowed here (1125 cm −1 , 0.2%) practically do not appear. In the region of lattice vibrations, vibrations of E u symmetry with wavenumbers of 315, 200, and 106 cm −1 stand out in intensity, while less intense vibrations of A 2u symmetry have wavenumbers of 347 and 115 cm −1 . The first of these less intense vibrations corresponds to the displacements of magnesium atoms in antiphase with the anions, and the second corresponds to the displacements of barium atoms. Magnesium atoms are also involved in the formation of this mode, and they shift synchronously with the anion. In RS of BaMg(CO 3 ) 2 , vibrations of anion atoms will also dominate: ν 1 with a wavenumber of 1126 cm −1 (its intensity is taken as 100%), ν 2 of the same symmetry and intensity of 3%, as well as doubly degenerated ν 4 (697 cm −1 ) and ν 3 (1444 cm −1 ) with intensities of 21% and 3%, respectively. For lattice vibrations, the A 1g symmetry mode with a wavenumber of 284 cm −1 and E g symmetry modes with wavenumbers of 108 and 254 cm −1 will be noticeable, of which the first is rotational, and the second is translational vibration.

Vibrational Spectra of Crystals with Aragonite Structure
The calculated spectra of infrared absorption and Raman light scattering of calcium, strontium, lead, and barium carbonates with aragonite structure are shown in Figure 5. The wavenumbers of normal long-wave vibrations of crystals with the aragonite structure calculated by the B3LYP method, together with the available experimental and theoretical data, are summarized in Tables S5-S8 of the Supplementary Materials. The comparison of wavenumbers calculated by the B3LYP method with the experimental values shows that the root-mean-square deviation for four ν1-ν4 wavenumbers in four carbonates is, according to [68] In the IRS of aragonite, the most intense vibrations are of the ν3 type with the B2u, B3u symmetry, wavenumbers of 1448 and 1480 cm −1 and intensities of 4523 and 4727 km/mol. In Figure 4, they correspond to a broad intense band with the maximum at 1462 cm −1 (1461 cm −1 in [68]). For convenience of comparison, the intensity of vibration of B3u symmetry is taken as 100%. In SrCO3, the maximum intensity of the B3u mode at 4757 km/mol is taken as 100%, in BaCO3, the B2u symmetry modes at 4953 km/mol, and in PbCO3: 6241 km/mol. Thus, the intensity ν3 increases with the atomic mass of the metal, and the position of the maximum in the series changes according to the law: Unlike calcite, in the infrared spectra of aragonite, the vibration of the ν1 type will be active due to the modes of symmetry B3u, B1u. In the spectrum shown in Figure 4, they correspond to a weak (0.2%) band with a maximum at 1090 cm −1 , the wavenumber of which shifts towards lower values

Vibrational Spectra of Crystals with Aragonite Structure
The calculated spectra of infrared absorption and Raman light scattering of calcium, strontium, lead, and barium carbonates with aragonite structure are shown in Figure 5. The wavenumbers of normal long-wave vibrations of crystals with the aragonite structure calculated by the B3LYP method, together with the available experimental and theoretical data, are summarized in Tables S5-S8 of the Supplementary Materials. The comparison of wavenumbers calculated by the B3LYP method with the experimental values shows that the root-mean-square deviation for four ν 1 -ν 4 wavenumbers in four carbonates is, according to [68], 1.5% for IRS; 0.8% for RS; for IRS [55]-2.0%, and for RS [57]-0.9%. to 2 and 3 cm −1 . This behavior of vibrational modes of the ν4 type is consistent with experimental data [68], where it was found that two peaks are observed in aragonite (CaCO3) and strontianite (SrCO3), and only one for cerussite (PbCO3) and witherite (BaCO3). There is a good ordinary dependence of the peak position on the atomic mass of the metal: in the experiment it is ω(cm −1 ) = 707.5 − 0.128·M, and in the calculation: ω(cm −1 ) = 715.4 − 0.119·M, with the correlation coefficients 0.944 and 0.977, respectively. Figure 5. Calculated infrared spectra (IR) (top) and Raman spectra (bottom) of intramolecular ν1, ν2, ν3, ν4, and lattice vibrations of calcium (black, squares), strontium (red, circles), barium (blue, triangles), lead carbonates (lilac, rhombuses) with aragonite structure. For comparison, the experimental spectra of aragonite and cerussite are given [68] (Reproduced with permission from [68]; Copyright Springer Nature, 2020).
In the Raman spectrum of aragonite (CaCO3), ν1 vibration of Ag symmetry with a wavenumber of 1078 cm −1 is the most intense (taken as 100%). Vibrations ν4 of the Ag and B1g symmetries with intensities of 9% each are also noticeable, which form the maximum in the spectrum shown in Figure  5 at 702 cm −1 . In BaCO3, the ν1 vibration maximum shifts to 696 cm −1 , and its intensity increases. In addition, the width of this peak increases. Oscillations of the ν3 type in CaCO3 have B3g (1465 cm −1 ) and B2g (1595 cm −1 ) symmetries, and an intensity of 5%. In SrCO3, these are vibrations at wavenumbers of 1450 and 1565 cm −1 with intensities of 4%. In BaCO3, they shift to lower values of 1429 and 1528 cm −1 , and the intensities increase to 9 and 13%. Oscillations of the B2g symmetry correspond to the displacements of atoms to the C-O bonds along the b axis, whereas B3g-along the a axis, perpendicular to the layers of anions and cations.
The Raman spectrum of PbCO3 differs from the spectra of other carbonates with the aragonite structure. Here, the most intense (taken as 100%) is the vibration of the ν3 type of B2g symmetry with a wavenumber of 1486 cm −1 [68]. Three other vibrations of this type have B3g, Ag, B1g symmetries, similar wavenumbers of 1394, 1383 and 1380 cm −1 and intensities of 50%, 16%, and 25%, respectively. They form a second maximum in RS at 1388 cm −1 . The ν1-type vibration with a wavenumber of 1067 cm −1 has an intensity of 61%. Modes of the same Ag symmetry are also dominant in the formation of the ν2 band, the position of which in aragonites obeys the law: ω(cm −1 ) = 895.6 − 0.117·M with a high correlation coefficient of 0.977.
In the IRS of aragonite, the most intense vibrations are of the ν 3 type with the B 2u , B 3u symmetry, wavenumbers of 1448 and 1480 cm −1 and intensities of 4523 and 4727 km/mol. In Figure 4, they correspond to a broad intense band with the maximum at 1462 cm −1 (1461 cm −1 in [68]). For convenience of comparison, the intensity of vibration of B 3u symmetry is taken as 100%. In SrCO 3 , the maximum intensity of the B 3u mode at 4757 km/mol is taken as 100%, in BaCO 3 , the B 2u symmetry modes at 4953 km/mol, and in PbCO 3 : 6241 km/mol. Thus, the intensity ν 3 increases with the atomic mass of the metal, and the position of the maximum in the series changes according to the law: ω(cm −1 ) = 1480.2 − 0.365·M (cm −1 ) with a correlation coefficient of 0.991.
Unlike calcite, in the infrared spectra of aragonite, the vibration of the ν 1 type will be active due to the modes of symmetry B 3u , B 1u . In the spectrum shown in Figure 4, they correspond to a weak (0.2%) band with a maximum at 1090 cm −1 , the wavenumber of which shifts towards lower values with increasing atomic mass: ω(cm −1 ) = 1090.6 − 0.127·M. The intensity of vibrations of the ν 2 type of B 1u symmetry is much higher than that of ν 1 vibrations, it decreases with an increase in the atomic mass of the metal, and its wavenumber practically does not change: ω(cm −1 ) = 893.3 − 0.065·M. In the experimental spectra, this dependence has the form: ω(cm −1 ) = 866.2 − 0.123·M. Vibrations of the ν 4 type in CaCO 3 and SrCO 3 correspond to the modes of symmetries B 3u , B 2u with distances between wavenumbers of 13 and 6 cm −1 . In BaCO 3 and PbCO 3 , the distances between wavenumbers decrease to 2 and 3 cm −1 . This behavior of vibrational modes of the ν 4 type is consistent with experimental data [68], where it was found that two peaks are observed in aragonite (CaCO 3 ) and strontianite (SrCO 3 ), and only one for cerussite (PbCO 3 ) and witherite (BaCO 3 ). There is a good ordinary dependence of the peak position on the atomic mass of the metal: in the experiment it is ω(cm −1 ) = 707.5 − 0.128·M, and in the calculation: ω(cm −1 ) = 715.4 − 0.119·M, with the correlation coefficients 0.944 and 0.977, respectively.
In the Raman spectrum of aragonite (CaCO 3 ), ν 1 vibration of A g symmetry with a wavenumber of 1078 cm −1 is the most intense (taken as 100%). Vibrations ν 4 of the A g and B 1g symmetries with intensities of 9% each are also noticeable, which form the maximum in the spectrum shown in Figure 5 at 702 cm −1 . In BaCO 3 , the ν 1 vibration maximum shifts to 696 cm −1 , and its intensity increases. In addition, the width of this peak increases. Oscillations of the ν 3 type in CaCO 3 have B 3g (1465 cm −1 ) and B 2g (1595 cm −1 ) symmetries, and an intensity of 5%. In SrCO 3 , these are vibrations at wavenumbers of 1450 and 1565 cm −1 with intensities of 4%. In BaCO 3 , they shift to lower values of 1429 and 1528 cm −1 , and the intensities increase to 9 and 13%. Oscillations of the B 2g symmetry correspond to the displacements of atoms to the C-O bonds along the b axis, whereas B 3g -along the a axis, perpendicular to the layers of anions and cations.
The Raman spectrum of PbCO 3 differs from the spectra of other carbonates with the aragonite structure. Here, the most intense (taken as 100%) is the vibration of the ν 3 type of B 2g symmetry with a wavenumber of 1486 cm −1 [68]. Three other vibrations of this type have B 3g , A g , B 1g symmetries, similar wavenumbers of 1394, 1383 and 1380 cm −1 and intensities of 50%, 16%, and 25%, respectively. They form a second maximum in RS at 1388 cm −1 . The ν 1 -type vibration with a wavenumber of 1067 cm −1 has an intensity of 61%. Modes of the same A g symmetry are also dominant in the formation of the ν 2 band, the position of which in aragonites obeys the law: ω(cm −1 ) = 895.6 − 0.117·M with a high correlation coefficient of 0.977.
In the IRS of aragonite, in the region of lattice vibrations, there is a maximum at 245 cm −1 , formed by the B 3u symmetry mode with an intensity of 13%, and the main maximum with an intensity of 23% at 194 cm −1 (B 2u symmetry). For SrCO 3 , four peaks are observed at 129, 175, 192 and 215 cm −1 with intensities of 3, 18, 15 and 9%, while in BaCO 3 the intense band is at 157 cm −1 (B 1u and B 3u ) with a low-wavenumber band at 146 cm −1 , and high-wavenumber at 171 cm −1 shoulders. In PbCO 3 , the main features of the spectrum are shifted to the low-wavenumber region. There are two main peaks at 82 cm −1 with an intensity of 17%, and a peak at 103 cm −1 and an intensity of 18%. This structure of the IRS in the lattice region is consistent with the results of measurements [72], where it was found that the broad band at 263 cm −1 in the structure of aragonite shifts to longer wavelengths with an increase in the atomic number of the metal: up to 227 cm −1 in strontianite, then up to 205 cm −1 in witherite, and, finally, up to 136 cm −1 in cerussite.
For lattice vibrations of aragonite in RS, there will be three main bands with maxima at 152 cm −1 , 209 cm −1 and 250, 276 cm −1 . There is a good agreement between the calculated and experimental data [68]. In SrCO 3 , there are two intense bands with maxima at 150 and 191 cm −1 , formed by the modes of symmetries B 3g , B 2g , and a weak band at 252 cm −1 . In the BaCO 3 spectrum, the maximum of the first band is at 153 cm −1 , the second at 181 cm −1 with a shoulder at 198 cm −1 , and the third at 237 cm −1 . The second maximum is formed by modes with B 2g symmetry with pronounced yz polarization. The spectrum of PbCO 3 contains a large number of bands of low intensity, not exceeding 6%.
In [68], it was found that RS and IRS confirm the general trend that each of the internal modes is shifted to lower wavenumbers in the following order: aragonite → strontianite → witherite → cerussite. The coefficients of linear interpolation of the wavenumbers of intramolecular (ν 4 , ν 1 , ν 3 ) vibrations active in RS for carbonates with the aragonite structure, obtained from experimental data [68] and theoretical calculations using the B3LYP method are summarized in Table 2. In the calculations, each type of vibration was determined as the average of the individual modes, which are shown in Table S9. For lattice vibrations, similar linear dependences are also obtained, as recorded in Table S10. As a check of the obtained formulas, we will use the available data [49], where the Raman spectra of (CaCO 3 Tables S8 and S9 gives: 1466, 1077, 703, 262, 183, and 158 cm −1 , that is, the root mean square deviation of the calculated and experimentally determined wavenumbers is about 2.7%.

Conclusions
In this work, the Hartree-Fock theory and the electron density functional in the form of a hybrid B3LYP functional in the basis of a linear combination of atomic orbitals by means of the CRYSTAL17 program code are used to calculate the structure and normal long-wavelength vibrations of MgCO 3 , CaCO 3 , ZnCO 3  Thus, ab initio predictions, with a relatively low computational capacity, can reproduce the full vibrational spectra of crystalline compounds of material science interest, and, on the basis of ordinary spectral dependences, predict their features for solid crystalline solutions. The obtained quantitative dependences of the characteristics of vibrational modes can be used in non-contact non-destructive diagnostics of carbonates by optical methods.
Author Contributions: Y.N.Z.: Conceptualization, data curation, paper writing; V.V.A.: Conceptualization, paper writing. Both authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.