Introduction

Despite the robustness and wide acceptance of Kohn–Sham density functional theory (DFT) in predicting the properties of zero- to three-dimensional systems, almost three decades ago it was demonstrated that all common functionals, including hybrids, are not able to properly describe long-range interactions, such as van der Waals (vdW) forces1. These dispersive interactions due to pure electronic correlation effects (London dispersion energy) at large interatomic distances are fundamental for several physical and chemical properties of matter and contribute to the description of, to cite some examples, the structures of both organic (e.g., DNA and proteins) and inorganic (e.g., crystals) substances, and the adsorption of molecules onto surfaces. Interestingly, it was shown that standard DFT approaches “under-correlate” even at short distances, i.e., some energy contribution to the total energy is missing also at distances typical of chemical bonds (covalent, ionic)2,3,4,5,6. Thus, the proper treatment of correlation effects via ad hoc corrections to the local DFT functionals is non-trivial even for solid phases where dispersive interactions do not play a leading role.

Recently, we showed that this consideration is true for the low-pressure polymorph of CaCO3, i.e. calcite (trigonal lattice, space group \(R\overline{3}c\), no. 167)7,8,9, and here we extend this approach to aragonite (orthorhombic, s.g. Pmcn, no. 62, Fig. 1). According to a recent experimental and theoretical thermodynamics study10, calcite is the stable phase at room conditions (1 atm, 300 K), whereas aragonite is a high-pressure phase (calcite-aragonite phase transition at 300 K of about 0.3 GPa). From the geological perspective, carbonates are considered possible carbon carriers for the creation of deep carbon reservoirs in the interior of Earth11,12. While calcite is the widest calcium carbonate, formed in sedimentary, igneous and metamorphic rocks, aragonite is mostly the product of biomineralization processes13. For example, the polyps of corals grow and organize aragonite crystals in marine (liquid) environments, forming intricate skeletal structures14, a biologically-driven process that is aided by organic molecules known as templates15. In general, aragonite is by far the most common phase occurring as a biomineral in many invertebrate skeletons and sediments derived from them. Furthermore, aragonite is found in the deposition of hot springs, and in stalactite/stalagmite cave formations13. At the industrial level, calcium carbonates are also important additives used in medicines, building materials, polymers, and many other applications16.

Figure 1
figure 1

Aragonite unit cell, according to the Pmcn (on the left, the one employed in the present work) and to the Pnma (on the right) crystallographic settings. The reference frame was oriented so that the x, y and z Cartesian axes are parallel to the a, b and c lattice vectors, respectively. Calcium, carbon and oxygen atoms in the unit cell are shown as blue, ochre and red spheres, respectively.

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For all these reasons, aragonite and the other CaCO3 phases were thoroughly investigated with different techniques, such as X-ray diffraction (XRD)17,18,19, scanning electron microscopy20,21, cathodoluminescence22,23,24, infrared (IR) and Raman spectroscopies25,26 and also ab initio methods, including Density Functional Theory25,26,27. However, most of the cited theoretical works were conducted with local functionals that do not consider the effects of long-range interactions. An interesting result of previous DFT simulations on aragonite is that the unit cell volume was overestimated with respect to the X-ray diffraction refinements reported in the literature19. The same overestimation was observed by both generalized-gradient approximation functionals and hybrid ones for calcite7. This is a non-expected behaviour because static DFT simulations are carried out at absolute zero Kelvin, namely no thermal effects are included in the computations, hence the theoretical unit cell volume should be smaller than the experimental ones. This is also reflected in the computed vibrational frequencies, most of them being underestimated, i.e., red-shifted in comparison with the experimental IR and Raman data, for both low-pressure and high-pressure calcium carbonate polymorphs.

As previously introduced, we aim at demonstrating that long-range interactions are important for aragonite, considering a combined theoretical and experimental analysis of the infrared and Raman spectra of this high-pressure phase. We adopted DFT for the computational part, using the well-known B3LYP hybrid functional and the two DFT-Dn corrections (n = 2, 3) to add the contributions of van der Waals interactions. Since the dispersion parameters of the B3LYP-D2 approach were recently optimized in the so-called B3LYP-D* method, this latter notation will be employed to refer to the specific DFT-D2 correction. The two methods are formally similar, namely they add an a posteriori contribution (EvdW) to the total energy according to the formula Etot = EDFT + EvdW. The main difference between the two methods resides in the atomic parameters used in the formulations, which are fixed in DFT-D2 and dependent on the geometry of the system in DFT-D3. This means that B3LYP-D3 is slightly less empirical than the B3LYP-D*. These two approaches were employed in a previous work of us28, where we obtained a better description of the unit cell of aragonite by including these missing interactions in the physical treatment via the a posteriori DFT-Dn corrections, with the B3LYP-D3 approach resulting in a cell volume smaller than that resulting from XRD refinements. This is due to a significant shortening of the Ca–O distances and, to lesser extent, of the C–O bond lengths, meaning that the interaction between Ca2+ and CO32− ions increased. Some brief structural results are reported in Table S1 in the Supplementary Materials for the sake of completeness. Since the total energy of the system changes due to the inclusion of the London interactions, also the vibrational frequencies are expected to show variations because they are calculated from the Hessian matrix, i.e., the second derivatives of the energy with respect to the atomic displacements. The IR/Raman results obtained in the present work were assessed using experimental data reported in the literature, and also polycrystalline Raman spectra of powdered aragonite acquired in the context of this study. Also, we investigated the evolution of the vibrational (optical) modes of the mineral with pressure between − 3 GPa and 25 GPa. Up to now, there is only a single theoretical work on this last topic, which reported a striking variation of two modes with the cell volume, suggesting a possible phonon instability at certain PT conditions27. As it will be shown below, we checked if this behaviour is due to the absence of proper treatment of long-range interactions, or if it is subtly related to specific computational parameters of the simulations.

Results and discussion

Infrared and Raman spectra at 0 GPa

In this section, a comparison between the infrared and Raman vibrational modes and spectra of aragonite simulated in this work at equilibrium and the data reported in theoretical/experimental literature is performed to assess the quality of the DFT approach adopted in the present work. It is worth mentioning that the vibrational frequencies were obtained within the harmonic approximation, whereas experimental ones are intrinsically anharmonic. Hence, all DFT results will be systematically blue-shifted, i.e., they will show higher frequency wavenumbers than experimental data.

The aragonite crystallographic cell contains four unit formula (Z = 4) of CaCO3, for a total of N = 20 atoms and 3N = 60 degrees of freedom. Since the mineral belongs to the space group Pmcn (point group D2h, crystal class mmm), three atoms (Ca, C, O1) occupy a 4c site and the O2 atom resides in the Wyckoff position 8d, the total irreducible representation (IRREP) of the degrees of freedom (Γtot) is given by:

$$\Gamma_{{{\text{tot}}}} \, = \,\Gamma_{{\text{a}}} \, + \,\Gamma_{{\text{o}}} \, = \,{9}A_{{\text{g}}} \, + \,{6}A_{{\text{u}}} \, + \,{6}B_{{{\text{1g}}}} \, + \,{9}B_{{{\text{1u}}}} \, + \,{9}B_{{{\text{2g}}}} \, + \,{6}B_{{{\text{2u}}}} \, + \,{6}B_{{{\text{3g}}}} \, + \,{9}B_{{{\text{3u}}}} ,$$

that can be subdivided in acoustic (Γa = B1u + B2u + B3u), infrared-active (8B1u + 5B2u + 8B3u), Raman-active (9Ag + 6B1g + 9B2g + 6B3g) and silent (6Au). The subscripts g and u indicate gerade and ungerade modes that are, respectively, symmetric or antisymmetric with respect to the inversion centre. In addition, the presence of this symmetry element activates the so-called mutual exclusion rule, hence vibrational modes are active only in infrared (IRREPs B1u, B2u and B3u) or in Raman (Ag, B1g, B2g and B3g), but not in both.

The optical modes are further distinguished in 33 “external”, meaning rotation and translation of the carbonate group and calcium ion (labelled as lattice modes, “L”), and 24 “internal”, namely the vibrational modes of the CO32− ions. According to the assignment proposed by White29, and from a graphical inspection of the internal optical modes, the carbonate group vibrations are subdivided into in-plane (labelled as ν4, between 700 cm−1 and 720 cm−1) and out-of-plane (ν2, 850–920 cm−1) O–C–O bending modes, and symmetric (ν1, between 1080 cm−1 and 1090 cm−1) and asymmetric (ν3, about 1440–1600 cm−1) C–O stretching motions.

The B3LYP-D* and B3LYP-D3 results of the infrared vibrational frequencies are reported in Table 1, alongside previous simulations and experimental measurements, whereas Fig. 2a shows the calculated IR spectrum of aragonite. As expected, all the modes are affected by the DFT-Dn corrections, because the total energy calculated for each atomic displacement include EvdW, hence they contribute to the construction of the Hessian matrix. The extent of their contribution is different according to which atoms are involved in the specific vibrational mode, with those at low wavenumbers (involving mostly Ca2+ ions and CO32– rotations) being the more affected (up to about 50 cm−1). Only slight variations were observed for carbonate ions bending and symmetric stretching modes (less than about 10 cm−1 for B3LYP-D3), but the higher variations were noted for the asymmetric stretching modes. Furthermore, considering the B3LYP data as the reference25,26, the B3LYP-D3 frequencies are more blue-shifted than the B3LYP-D* ones. The vibrational results follow what was observed on the equilibrium geometry of aragonite in our previous work28 and, thus, should be more in line with the physics of the system at absolute zero. The optical frequencies are in general good agreement with those of the detailed experimental/theoretical work of Carteret and collaborators25, showing relatively low absolute variations. More into details, the mean absolute deviations (MADs) of the modes with IRREP B1u, B2u and B3u simulated with the B3LYP-D* (B3LYP-D3) approach are 5.8 cm−1 (16.3 cm−1), 20.0 cm−1 (11.5 cm−1) and 7.8 cm−1 (16.0 cm−1), respectively, in line with the previous B3LYP simulations performed without the inclusion of dispersive interactions25. It is interesting noting that the DFT-D2 scheme provided a mix of blue- and red-shifted IR peaks, i.e., the calculated modes were at higher and lower frequency values, respectively, than the experimental ones, whereas the B3LYP-D3 data are systematically blue-shifted but for one mode (vide infra). A satisfactory agreement is also found when considering the previous FTIR results of Weir and Lippincott30 and the very recent ones of Bishop and collaborators31. The present simulations are also in good agreement with those of Ungureanu and co-workers27, who employed the same DFT functional (B3LYP) and basis sets but without any correction for long-range interactions. This could be the source of the observed differences, as in the case of the simulations performed by Carteret and collaborators25.

Table 1 Infrared-active vibrational frequencies (ν, in cm−1) of aragonite, subdivided into irreducible representations (IRREP).
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Figure 2
figure 2

(a) Infrared and (b) Raman spectra of aragonite, as simulated at the B3LYP-D* (filled blue line) and B3LYP-D3 (filled red line) levels of theory. Experimental IR25 and Raman (present work) spectra are reported as black lines for a direct comparison. The small blue and red vertical lines mark the simulated vibrational frequencies with the B3LYP-D* and B3LYP-D3 approaches, respectively, whereas the grey dashed ones highlight the experimental ones. Intensities were normalized to the most intense peak in the spectra.

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However, the #1 and #3 modes of the B2u IRREP calculated with the B3LYP-D* approach deviate from the experimental values25 by about − 78 cm−1 and − 18 cm−1 (red shifts), respectively. The same behaviour was noted for the same modes at the B3LYP level by both Ungureanu et al.27 and Carteret and co-workers25, who obtained red shifts of − 30 cm−1/− 23 cm−1 and − 40 cm−1/− 22 cm−1, respectively. It was suggested that these modes are very sensitive to the unit cell volume variations25, and the DFT-D3 results, for which the unit cell volume is smaller than both the B3LYP(-D*) and the experimental ones, seem to confirm this hypothesis because of the better agreement of the B2u modes #1 and #3 with the polarized IR spectroscopy data (differences of − 15.4 cm−1 and 9.7 cm−1, respectively). This is an important issue, because these lattice modes, especially B2u #1, could lead to phonon (dynamic) instability, as also previously explained by Ungureanu and collaborators27. However, the source of this discrepancy is probably elsewhere, as it will be discussed in the following.

In fact, linked to the IR vibrational data, the static dielectric tensor ϵ0 and its components (see Table 2) provided more insights into this matter. In general, the terms of the high frequency (electronic) dielectric tensor ϵ∞ are underestimated by about 3% (B3LYP-D3) and 5% (B3LYP-D*) with respect to the experimental values25, a common figure when employing hybrid functionals7. The static dielectric tensor is well described by the B3LYP-D3 approach, but strong variations were observed in the xx component of both ϵ0 and its ionic contribution F when using the B3LYP-D* method. This is due to the calculated oscillator strength of the #1 B2u mode, which is very high compared to the experimentally determined value (see Table S2 in the Supplementary Materials). This overestimation, albeit of one order of magnitude lower than the present one at B3LYP-D*, was also obtained in the DFT/B3LYP simulations of Carteret and co-workers25. Hence, the trend in the DFT simulations, conducted with the same functional and localized basis sets, related to the oscillator strength of the #1 B2u mode, was B3LYP-D3 (4.5254) < B3LYP (6.5549) < B3LYP-D* (54.048), with an experimental value of 2.2283.

Table 2 Static dielectric tensor ϵ0, as obtained from the sum of the high-frequency dielectric tensor ϵ∞ and the ionic contribution F (sum of the oscillator strengths fn). Δ% is the percentage difference between the B3LYP-D3 results and the experimental data.
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To investigate these behaviours, it was performed a scan of the potential energy well on the cited B2u optical mode, using both B3LYP-D3 and B3LYP-D* approaches (see Fig. 3). With this method, with X0 vector describing the equilibrium configuration of the system (Cartesian equilibrium coordinates of the atoms), we explored the configurations given by \({\mathbf{X}}_{n} = {\mathbf{X}}_{0} + n\Delta \times {\mathbf{L}}_{1}\), where L1 is the Cartesian eigenvector of the mode B2u mode Q1, n is a positive or negative integer and Δ is the step along Q1. The maximum displacement corresponding to the classical turning points of the fundamental vibrational state is expressed by the equation:

$$V = \frac{1}{2}\omega_{1}^{2} Q_{1}^{2} = \frac{1}{2}k_{1} x_{1}^{2} = \frac{1}{2}\hbar \omega_{1} = E_{0}^{vib,1} ,$$

where Q1 is the displacement along the mass weighted B2u normal mode coordinate #1, ħ is the reduced Planck’s constant, and \(\nu_{1} = {{\omega_{1} } \mathord{\left/ {\vphantom {{\omega_{1} } {2\pi }}} \right. \kern-0pt} {2\pi }}\) is the vibrational frequency. In the above equation, \(k_{1} = \mu_{1} \omega_{1}^{2}\) is the generalized force constant associated to the collective displacement coordinate \(x_{1}\) (length units), \(x_{1} = {{Q_{1} } \mathord{\left/ {\vphantom {{Q_{1} } {\sqrt {\mu_{1} } }}} \right. \kern-0pt} {\sqrt {\mu_{1} } }}\) with \(\mu_{1}\) the reduced mass of the Q1 oscillator. Then, the maximum classical displacement in the fundamental level is given by the formula:

$$\left| {x_{1} } \right| = \sqrt {\frac{\hbar }{{\mu_{1} \omega_{1} }}} .$$
Figure 3
figure 3

Potential energy well of the #1 B2u optical mode (active in infrared) as obtained from (a) the B3LYP-D3 and (b) the B3LYP-D* approaches. The blue line corresponds to the harmonic approximation (see text for details), whereas the red line is the result of the scan performed along the displacement coordinate of the normal mode. The black thick horizontal line marks the fundamental vibrational energy level, calculated from the harmonic approximation (blue line). (c) Graphical representation of the eigenvectors (yellow arrows) of the #1 B2u optical mode, with calcium, carbon and oxygen atoms in the unit cell coloured in blue, ochre and red, respectively.

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In Fig. 3, the displacement is reported as dimensionless normal coordinate \(t_{1} = \left( {{{\omega_{1} } \mathord{\left/ {\vphantom {{\omega_{1} } \hbar }} \right. \kern-0pt} \hbar }} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} Q_{1}\), as provided by the CRYSTAL code, so at ± 1 the potential energy is equal to the energy of the fundamental vibrational state. From the graphs, it can be noted that in both B3LYP-Dn simulations the shape of the potential energy curve calculated by scanning the normal mode coordinate (blue line) is a perfect parabola. In the B3LYP-D3 scan (red line in Fig. 3a), the potential well between − 1 and + 1 almost matches that of the harmonic potential, with a slight deviation at the extremes of this range, hence the computational parameters provide an adequate description of the B2u optical mode. Conversely, the potential curve obtained from the B3LYP-D* simulation (Fig. 3b) deviates from the harmonic behaviour even below the classical displacement ± 1. These observed deviations could be due to some errors in the calculation of the Hessian matrix (second derivatives of the energy with respect to the atomic displacements) with the B3LYP-D* method. A possible source of errors could be the aragonite structure not being sufficiently close to the true minimum of the potential energy, but this hypothesis can be discarded because we adopted very strict convergence criteria during geometry optimization of the mineral in our previous work28. A second possibility could be the presence of large numerical errors during the computation of the Hessian. We are currently investigating this possibility, which is however beyond the scope of the present work. As an approximate solution, from the analysis of the potential energy surface, the force constant k of the harmonic oscillator was recalculated using the new data (red curve), finding a value that is higher than the previous one by an order of magnitude. This led to a vibrational frequency for this the #1 B2u mode of 103.6 cm−1, which is in line with the experimental data (105.4 cm−1). Following the same procedure with the B3LYP-D3 approach, we obtained a frequency of 102.6 cm−1, which is close to the theoretical value of 90.2 cm−1 reported in Table 1.

Regarding the Raman frequencies (see Table 3), as observed for the IR data, the simulations at the DFT/B3LYP-D3 and B3LYP-D* levels were in good agreement with the confocal Raman spectroscopy analysis on powdered aragonite performed in the present work (see Fig. 2b). The agreement can also be extended to previous experimental10,26,34 and theoretical studies26,27.

Table 3 Raman-active vibrational frequencies (ν, in cm−1) of aragonite, subdivided in irreducible representations (IRREP).
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The effect of the long-range corrections on the lattice and carbonate ion modes is similar to that discussed for the infrared signals. The mean deviation (MD) and mean absolute deviation (MAD) of the B3LYP-D* frequencies with respect to the Raman measurements at 80 K of De La Pierre and co-workers26 are low (MD = − 1.1 and MAD = 6.9), and even lower than those resulting from the ab initio simulations carried out by the same authors (MD = − 4.6 and MAD = 7.7). In general, all the lattice and carbonate optical vibrations are within ± 10 cm−1 from the corresponding experimental values but for four low-frequency lattice modes (#4 Ag, #1 and #2 B1g, #1 B3g) and two CO32 vibrations (#9 Ag and #9 B2g) that showed an absolute deviation greater than 10 cm−1. These figures increase when employing the B3LYP-D3 approach, with mean (absolute) deviation of 10.6 (10.7) cm−1 with respect to the experimental data26, and a maximum absolute difference of 32.1 cm−1. Furthermore, as observed in the calculated infrared spectrum, the optical frequencies are systematically blue-shifted because of (i) the harmonic description of the modes and (ii) the smaller unit cell volume of aragonite, hence the increased interactions between the atoms in the structure. Indeed, previous X-ray diffraction determinations showed the aragonite cell volume being about 227 Å3 in room conditions (298 K and 1 atm)19,35, while DFT simulations without any thermal effect resulted in equilibrium geometries with cell volumes of ca. 236 Å3 (B3LYP)25, 230 Å3 (B3LYP-D*)28, and 223 Å3 (B3LYP-D3)28. As also explained in our previous works7,28, the inclusion of the dispersive forces significantly ameliorates the description of the geometry of both calcite and aragonite, resulting in unit cells that are slightly smaller than the experimental data.

Regarding the calculated intensities, it is possible to observe that the infrared signals are generally more intense on the ν3 carbonate ion modes (asymmetric stretching) whereas they are less strong in the spectral range 600–1100 cm−1. In the Raman spectra, the B3LYP-Dn approach tends to overestimate the peak intensity of the ν3 in-plane O–C–O bending modes, and to underestimate the lattice modes (ν < 300 cm−1). In this latter case, on the experimental side, the spectrum could be affected by uncertainties associated to the measurements, whereas the different approximations in the simulations (basis sets, Hamiltonian, DFT-Dn corrections and computational parameters) could be the source of the small differences. In this sense, we agree with the considerations of De La Pierre and collaborators26, who pointed out the basis set is the most probable cause of this discrepancy. In fact, albeit the quality of the chosen basis sets is indeed high for solid state calculations, Raman intensities may require even richer basis sets. However, while in molecular simulations there are very rich basis sets, such as aug-cc-pVTZ, Gaussian-type orbitals with this kind of accuracy have not been developed yet for periodic structures.

As a final note on the vibrational properties of aragonite at equilibrium conditions, the calculated frequencies of the silent Au modes (four lattice and two carbonate ion vibrations, see Table S3 in the Supplementary Materials) are in line with those previously obtained by other authors25,27, with the DFT-Dn corrections providing blue-shifted modes. As observed for the #1 B2u mode active in infrared, the #1 Au lattice mode strongly varies between the different simulation approaches, with the same frequency trend B3LYP-D3 > B3LYP > B3LYP-D*.

Effect of pressure on the infrared and Raman properties of aragonite

The variation of the peaks in the vibrational spectra of aragonite as a function of (hydrostatic) pressure was investigated by calculating the optical frequencies and their IR/Raman intensities in unit cells of the mineral with different volumes. Each unit cell volume corresponds to a pressure value that was determined in a previous study by means of equation of state fitting28. The interested reader may find more information on this topic and detailed structural data of aragonite simulated with both B3LYP-D3 and B3LYP-D* approaches at various compressed/expanded states in the cited work.

The optical frequencies and their related signal (IR/Raman) intensities were calculated for 13 unit cell volumes, three in expansion and ten in compression, between − 3 GPa and 25 GPa. For the sake of brevity, the complete results of this procedure carried out with the B3LYP-D3 and B3LYP-D* approaches are reported in Table S4 and Table S5, respectively, in the Supplementary Materials. However, due to the observed anomalous behaviour of some calculated modes (vide supra), the following discussion will mainly consider the B3LYP-D3 results, and some comments on the B3LYP-D* data will be presented when applicable.

A graphical representation of the evolution of the infrared and Raman spectra as a function of pressure is reported in Fig. 4 and Fig. 5, respectively, as obtained from the B3LYP functional corrected with the DFT-D3 scheme.

Figure 4
figure 4

Evolution of the infrared spectrum of aragonite at different pressures, calculated at DFT/B3LYP-D3 level of theory. The spectra were subdivided in four regions, thus four panels, to avoid graphical cluttering and to better highlight the frequency shift of different IR bands. The vertical grey lines mark the position of the relevant peaks at 0 GPa. Infrared intensities are expressed in arbitrary units.

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Figure 5
figure 5

Evolution of the Raman spectrum of aragonite at different pressures, calculated at DFT/B3LYP-D3 level of theory. The spectra were subdivided in four regions, thus four panels, to avoid graphical cluttering and to better highlight the frequency shift of different IR bands. The vertical grey lines mark the position of the relevant peaks at 0 GPa. Raman intensities are expressed in arbitrary units.

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Qualitatively, it can be noted that the IR spectrum of aragonite, which is dominated by the CO32– modes in the range 2000–400 cm−1, is highly affected by pressure, with all but one optical mode (ν2 out-of-plane O–C–O bending mode with IRREP B1u) that are blue shifted by compression. The intensity of some signals is also enhanced due to (i) increased interatomic interactions, and hence dipole moment variation, (ii) overlapping of the peaks, or both. Similar considerations can be extended to the calculated Raman spectra where the symmetric stretching mode (ν1) of the CO32− ions and lattice modes with ν < 400 cm−1 are the most intense signals. The former peak shows a remarkable blue shift for P > 15 GPa, but no significant change in its intensity was observed. The single signal of the in-plane O–C–O bending mode (ν4) that falls at about 710 cm−1 is composed by two strong (Ag + B1g) and two very weak (B2g + B3g) peaks that overlap. By increasing pressure, the two intense modes are blue shifted but with different ratio (Ag > B1g), resulting in the deconvolution of the single peak in two signals. Interestingly, the #6 Ag optical mode frequency increases up to about 21 GPa, then it decreases by further compression of the unit cell. This mode is related to a ν4 CO32− band, i.e., to O–C–O bending mode, which is graphically reported in Fig.S2 (Supplementary Materials). It is known that aragonite undergoes a phase transition to the post-aragonite structure (space group Pmmn) between 30 and 40 GPa36,37, and probably the change in the trend of this vibrational mode with pressure (at about 25 GPa) is a first sign of structural variations that will occur by further compressing the aragonite unit cell. Indeed, as shown in Fig.S2 (Supplementary Materials), by increasing pressure from 0 GPa up to 25 GPa, the carbonate ions move along the [001] direction. As recently observed by DFT simulations38, this is one of the possible change of the internal geometry of aragonite that leads to the post-aragonite structure. The frequency lowering of the cited vibrational mode could be due to the increased interaction between Ca2+ and CO32− ions that are brought closer due to pressure, which decreased the force constant of the O–C–O angle. Similarly to the ν4 carbonate mode, the lattice vibrations show high variations in the peak positions, intensity and overlapping of the signals.

A fruitful discussion can be also made by quantifying the blue and red shifts of the optical modes as a function of compression/expansion. There are several ways to describe the behaviour of phonon modes with pressure P (or volume Ω), which is of utmost importance when dealing with thermodynamics and thermoelasticity of crystalline phases6,39,40,41,42,43. For example, a simple polynomial function, as proposed by Erba and co-workers44 and recently adopted for several phases43,45,46, is typically sufficient to model the νp(Ω) trend. Here, this functional form was described in terms of the so-called mode-Grüneisen (or mode-gamma) parameters, γp, which are calculated for each pth mode as47:

$$\gamma_{p} = - \frac{{\partial \ln \nu_{p} }}{\partial \ln \Omega } = - \frac{\Omega }{{\nu_{p} }}\frac{{\partial \nu_{p} }}{\partial \Omega }.$$

For each normal mode, the γp parameter was calculated through the analytic first derivative at each volume (pressure) of the second order polynomial resulting from the fitting of the numerical νp(Ω) curves. The resulting mode-Grüneisen parameters γp are reported in Table 4, and graphically shown in Fig. 6. It can be noted that most of the γp values are positive, meaning that the vibrational frequencies increase by reducing the unit cell volume, thus they increase with pressure. There are a few exceptions, i.e., two ν2 CO32– optical modes (B1u + Ag) that showed the opposite behaviour. Lattice modes (L) have large mode-gamma parameters, γp > 1, whereas the carbonate ones have values close to zero (0–0.3), with a mean Grüneisen’s parameter \(\overline{\gamma }\) equal to 1.281. These results are consistent with the crystal structure of aragonite, because the Ca–O interactions of the CaO9 polyhedrons showed a larger variation than the strong C–O bonds of the carbonate group with pressure36,48,49.

Table 4 Mode-Grüneisen parameters γi calculated at the equilibrium geometry for the infrared, Raman and silent optical frequencies of aragonite.
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Figure 6
figure 6

Mode-gamma Grüneisen’s parameters γp calculated at equilibrium volume at the B3LYP-D3 level, as a function of the vibrational frequency νp.

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All these observations are in general good agreement with the previous theoretical analysis performed at the B3LYP level without the inclusion of van der Waals interactions27. In the present work, as expected, the use of a correction for the long-range interactions affected the trends of the variation of the optical mode frequencies with pressure, in particular for the lattice modes, with a mean absolute deviation from the values calculated by Ungureanu et al.27 of about 0.94. Conversely, the effect of the DFT-D3 scheme on the γp of the carbonate ion modes is much lower (\(\left| {\overline{\Delta \gamma }_{p} } \right|\)= 0.04). High differences in the mode-Grüneisen parameters were calculated for four lattice modes, marked with an asterisk in Table 4, which are associated to rotations of the carbonate ions around an axis parallel to the c lattice vector and centred on either the carbon or one of the oxygen atoms of the CO32− group. The highest deviations were observed for the #1 B2u, the one previously discussed, and the #1 Au, whose ∆γp values were − 4.42 and − 8.96, respectively. A graphical representation of the eigenvectors of the three vibrational modes not previously shown is reported in Fig.S1 in the Supplementary Materials.

The very large mode-gamma parameters previously calculated27 showed the occurrence of zone-central (Γ-point) soft phonons with negative frequency values, i.e., the lattice became dynamically instable, when the aragonite unit cell is expanded, for example, by increasing temperature. According to their results, Ungureanu and collaborators27 suggested that there could be a phase transition at P = 0 GPa and T > 600 K. In the present study, the use of B3LYP-D3 approach did not provide any soft mode from –3 GPa to 25 GPa (see Table S4 in the Supplementary Materials), as the associated mode-Grüneisen parameters are small. Instead, our B3LYP-D* simulations reported in the present work provided γp for the #1 B2u and the #1 Au modes that are even larger than those obtained by Ungureanu and co-workers27, namely 28.38 (+ 18.52) and 22.28 (+ 7.32), respectively. This suggests that the aragonite-to-calcite phase transition would occur at lower temperatures than that previously hypothesized27. However, the considerations on any phase transition made upon the B3LYP-D* and B3LYP results have to be carefully addressed in future works.

Furthermore, the present mode-Grüneisen parameters at the B3LYP-D3 level well compare with the experimental ones of Gillet and collaborators50, obtained from Raman spectra collected at different pressure conditions (see Table 4). The same applies to the more recent results of Wang and co-workers51, who analysed by both FTIR and Raman spectroscopies the variation of the optical mode frequencies with pressure and temperature. The authors found that the mode-Grüneisen’s parameters are within 0.9–3.5 for lattice modes (ν < 350 cm−1) and below 0.4 for the carbonate ion vibrations (650–1600 cm−1), which are consistent with the present DFT simulations conducted with the inclusion of long-range interactions.

Aragonite (CaCO3, space group Pmcn) is one of the main polymorphs of calcium carbonate, with important applications in the fields of geology, mineralogy/petrology, biology and materials science. We can summarize the main findings of the present combined ab initio and experimental study on the vibrational (IR/Raman) properties of this mineral as follows:

  • The inclusion of the long-range interactions via the DFT-Dn schemes is relevant to the calculation of the vibrational properties of the mineral. The combination of the hybrid B3LYP functional and the DFT-D3 approach seems providing the best description of the physics of the system, providing accurate crystal-chemical, elastic, and vibrational spectroscopy data. The systematic blue shift, i.e., increased frequency value, of the optical modes is a consequence of both the smaller unit cell volume calculated in static conditions (0 K without any thermal contribution) and the other necessary approximations in the simulations.

  • The very high variability of the #1 B2u (IR) and #1 Au (silent) modes between B3LYP, B3LYP-D* and B3LYP-D3 calculations performed in the present work and in previous studies was analysed in detail. This variability is probably due to the computational parameters and methods employed for the calculation of the potential energy surface of the modes. In this context, it was found in the adopted approaches the trend B3LYP-D* < B3LYP < B3LYP-D3, with the latter providing better results in comparison with the experimental data, as also assessed by the analysis of the harmonic potential curve.

  • The above consideration can be extended to other properties that depend on the lattice dynamics, such as the static dielectric tensor. In fact, the poorly defined oscillator strength of the B2u mode resulted in a dramatic deviation of the xx component of the ionic and static dielectric tensors.

  • The better vibrational results of the B3LYP-D3 approach ameliorated the description of the mode-Grüneisen’s parameters, in particular those of the cited B2u and Au modes, in better agreement with the few available experimental data.

Methods

Computational parameters

The ab initio simulations were carried out with the CRYSTAL17 software32, and within the Density Functional Theory framework. The Becke 3 parameters exchange functional52 and the correlation functional of Lee, Yang and Parr53 were adopted in the hybrid B3LYP formulation, which includes 20% of the exact Hartree–Fock (HF) exchange. In addition, B3LYP adds some non-local contribution to the exchange–correlation terms32. This hybrid functional is a common choice because of its high accuracy in determining the vibrational properties of both molecular and solid systems33,54. To properly consider the effects of long-range interactions, the semiempirical DFT-D2 scheme55, reparametrized according to the B3LYP-D* approach56, and the DFT-D357,58 were employed throughout the work.

CRYSTAL17 employs a linear combination of atomic orbitals (LCAO) to construct the wave function of the system under analysis, and all-electron Gaussian-type orbitals (GTO) were selected for each atom in the aragonite structure among those optimized by Valenzano and co-workers59, named BSD in the cited work, which were adopted in the simulations of both aragonite25,27 and calcite7,8,9. More into details, 8–6511 (21), 6–311 (11) and 8–411 (11) basis sets were used for the description of the atomic orbitals of Ca, C and O, respectively.

The accuracy of Coulomb and exchange series are controlled by five thresholds, which were set to 10–8 (ITOL1 to ITOL4) and 10−16 (ITOL5), meaning that when the overlap between two atomic orbitals is lower than 10−ITOL, the corresponding integral is either discarded or treated with less precision. The first Brillouin zone (i.e., the reciprocal space) is sampled by means of a 6 × 6 × 6 Monkhorst–Pack mesh, corresponding to 64 independent k points60. The CRYSTAL default pruned grid (75 radial, 974 angular points, XLGRID) was used to calculate the total energy of the system, via numerical integration of the electron density over the unit cell volume32. Strict criteria on the self-consistent-field (SCF) convergence on energy was considered for the calculation of the vibrational frequencies, setting the threshold to 10−10 Hartree.

Aragonite structural model

The aragonite model, simulated with the two DFT-Dn schemes and under the effect of hydrostatic compression/expansion has been recently investigated by the authors of the present work in a recent theoretical study, using the same computational framework here adopted28. For the sake of completeness, the simulated unit cell parameters in equilibrium (zero pressure, absolute zero temperature) were a = 4.9544 Å, b = 7.8836 Å and c = 5.7091 Å (unit cell volume Ω = 222.987 Å3) at B3LYP-D3 level, and a = 5.0177 Å, b = 7.9005 Å and c = 5.7901 Å (Ω = 229.535 Å3) at B3LYP-D*.

Calculation of the vibrational spectra

Vibrational frequencies of aragonite were obtained at Γ point at different pressures by diagonalizing the mass-weighted Hessian matrix W (dynamical matrix), whose elements are the second derivatives of the lattice potential with respect to mass-weighted atomic displacements33:

$$W_{\alpha i,\beta j} \left( \Gamma \right) = \frac{{H_{\alpha i,\beta j} }}{{\sqrt {M_{\alpha } M_{\beta } } }},$$

with Hαi,βj the energy second derivative, Mα and Mβ the atomic masses and the subscripts in latin (i, j) and in Greek letters (α, β) the atomic coordinates and the atoms, respectively. The strength of the harmonic oscillators, fp, were calculated for each mode using the mass-weighted effective Born charge vectors \(\vec{Z}_{n}\)25,61:

$$f_{p,ij} = \frac{1}{{4\pi \varepsilon_{0} }}\frac{4\pi }{\Omega }\frac{{\vec{Z}_{n,i} \vec{Z}_{n,j} }}{{\nu_{p}^{2} }},$$
$$\vec{Z}_{p,i} = \sum\limits_{\alpha ,j} {{\mathbf{t}}_{p,\alpha j} Z_{\alpha ,ij}^{*} \frac{1}{{\sqrt {M_{\alpha } } }}} .$$

In the above formulas, ϵ0 is the dielectric permittivity in vacuum, p refers to the pth normal mode, i and j indicate the Cartesian components, and \(Z_{\alpha }^{*}\) is the Born effective charge tensor of atom α, calculated using the Berry phase approach62,63. The term tn,αj represent an element of the eigenvector matrix T that transforms the directions from Cartesian to normal coordinate.

Comparisons with previous experimental data was quantified according to the statistical indices previously adopted for calcite7:

$$\left| \Delta \right|_{\max } = \max \left( {\left| {\nu_{p} - \nu_{p}^{\exp } } \right|} \right)$$
$$MD = \frac{1}{M}\sum\limits_{p = 1}^{M} {\nu_{p} - \nu_{p}^{\exp } }$$
$$MAD = \frac{1}{M}\sum\limits_{p = 1}^{M} {\left| {\nu_{p} - \nu_{p}^{\exp } } \right|}$$

The infrared intensities were calculated analytically in terms of absorbance, using the classical absorption formula as explained by Maschio and co-workers64:

$$A\left( \nu \right) = \frac{1}{3}\sum\limits_{ii = 1}^{3} {\frac{4\pi }{{\lambda \rho }}{\text{Im}} \left[ {n_{ii} \left( \nu \right)} \right]} ,$$

where the terms A(ν), λ, ρ, and n are the infrared absorption, the wavelength of the incident light, the mineral density and the complex refractive index, respectively. The polarization direction is expressed by the subscripts ii. The real and imaginary parts of the refractive index nii were obtained from the following relations:

$$\left\{ {{\text{Re}} \left[ {n_{ii} \left( \nu \right)} \right]} \right\}^{2} - \left\{ {{\text{Im}} \left[ {n_{ii} \left( \nu \right)} \right]} \right\}^{2} = {\text{Re}} \left[ {\varepsilon_{ii} \left( \nu \right)} \right]$$
$$2{\text{Re}} \left[ {n_{ii} \left( \nu \right)} \right] \cdot {\text{Im}} \left[ {n_{ii} \left( \nu \right)} \right] = {\text{Im}} \left[ {\varepsilon_{ii} \left( \nu \right)} \right]$$

The term ϵii(ν) is the complex dielectric tensor, which was computed for each inequivalent polarization direction according to a classical Drude-Lorentz model:

$$\epsilon_{ii} \,\left( \nu \right)\, = \,\epsilon_{\infty ,ii} \, + \,\sum\limits_{p} {\frac{{f_{p,ii} \nu_{p}^{2} }}{{\nu_{p}^{2} - \nu^{2} - i\nu d_{p} }}} .$$

with ϵ∞ the optical (high frequency) dielectric tensor, νp the transverse optical frequency, fp the oscillator strength and dp the damping factor of the pth vibrational mode. As suggested by Maschio and co-workers64, to obtain a band broadening of the peaks similar to those typically found in experimental samples, the damping factor (i.e., the full width at half maximum of each vibrational mode) was set to 8.

The Raman intensities were instead calculated within the Placzek approximation65 assuming the mineral as a polycrystalline powder, by using the optical vibrational modes by means of a pseudo-Voigt functional form66,67:

$$A\left( \nu \right) = \eta L\left( \nu \right) + \left( {1 - \eta } \right)G\left( \nu \right),$$

where, in this case, A(ν) represents the Raman intensity and L(ν) and G(ν) are given by:

$$L\left( \nu \right) = \sum\limits_{p} {\frac{{I_{p} }}{\pi }\frac{{{{\varphi_{p} } \mathord{\left/ {\vphantom {{\varphi_{p} } 2}} \right. \kern-0pt} 2}}}{{\left( {\nu - \nu_{p} } \right)^{2} + \left( {{{\varphi_{p} } \mathord{\left/ {\vphantom {{\varphi_{p} } 2}} \right. \kern-0pt} 2}} \right)^{2} }}}$$
$$G\left( \nu \right) = \sum\limits_{p} {2\sqrt {\frac{\ln 2}{\pi }} \frac{{I_{p} }}{{\varphi_{p} }}\exp \left[ { - \frac{{4\ln 2\left( {\nu - \nu_{p} } \right)^{2} }}{{\varphi_{p}^{2} }}} \right]}$$

with Ip the computed Raman intensity and φp the full width at half maximum for the pth vibrational mode, and η the Lorentz factor. A pure Lorentzian form, corresponding to η = 1, was employed to obtain the typical sharp peaks of Raman spectra32.

All the tensorial properties related to the intensity of the peaks in the infrared and Raman spectra, i.e., the dielectric tensor, and the polarizability, were calculated analytically through a Couple Perturbed Kohn–Sham (CPKS) approach68,69. Finally, the static dielectric tensor was calculated as the sum of the high-frequency dielectric tensor ϵ∞ and the vibrational contribution, the latter expressed as the sum of all the oscillator strengths, \(F_{{ij}} = \sum {_{n} f_{{n,ij}} }\)25.

Aragonite sample and Raman spectroscopy analyses

The aragonite CaCO3 sample used for the present study was a single-crystal specimen provided by the Mineralogical Museum “Bombicci” of the University of Bologna, Italy. The sample was grounded to a fine powder, with crystallite sizes below 1 μm, for the characterization of the vibrational properties via Raman spectroscopy. The Raman spectra were collected at room temperature (~ 20 °C) using a WITec alpha300R confocal Raman microscopy system, made of an optical microscope and an ultra-high throughput UHTS 300 VIS spectrometer with CCD camera and gratings of 600 g/mm. A green laser beam (532 nm) was used as excitation light, setting a power of 30 mW to prevent heating the sample and possible alterations on the aragonite phase (e.g., inducing phase transitions and decarbonisation of the sample). The laser beam was focused on the sample with a 20× Zeiss microscope objective with a low numerical aperture objective (NA = 0.40) to avoid optical artefacts. The backscattered Raman spectra were collected in confocal mode between 100 and 1700 cm−1, with a resolution of about 2.7 cm−1 and an acquisition time of 10 min. The Rayleigh scattering line was removed by an edge filter.