First-Principles Study of Structural, Elastic, and Optical Properties of Trigonal CaCO₃ Under Pressure

Abstract

Calcium carbonate (CaCO₃) has attracted considerable attention owing to its structural versatility and broad applications in materials science and geochemistry. In this study, we employed Density Functional Theory (DFT) simulations to systematically investigate the structural, elastic, and dynamic properties of trigonal CaCO₃ under hydrostatic pressures ranging from 0 to 1.2 GPa. The optimized lattice constants closely align with previous theoretical and experimental values, thereby confirming the reliability of the computational approach. Mechanical stability was validated across the entire pressure range, with elastic constants and moduli demonstrating gradual increases under compressive strain. Elastic anisotropy was rigorously quantified using universal anisotropy indices, three-dimensional surface visualizations, and directional projections of elastic moduli. These analyses revealed pronounced pressure-dependent anisotropy. Furthermore, optical properties, including refractive indices and dielectric functions, were analyzed to clarify pressure-induced variations in electromagnetic interactions. These findings offer valuable insights into the pressure behavior of CaCO₃, advancing its potential applications in advanced functional materials and geophysical research.

1. Introduction

Calcium carbonate (CaCO₃), a ubiquitous natural mineral, plays a pivotal role in the CaO-CO₂ system [1]. Crystallographic analysis reveals its polymorphism, with three anhydrous forms—calcite (trigonal), aragonite (orthorhombic), and vaterite (hexagonal)—and three hydrated forms: ikaite (CaCO₃·6H₂O), monohydrocalcite (CaCO₃·1H₂O, MHC), and calcium carbonate hemihydrate (CaCO₃·½H₂O, CCHH) [2]. Widely extracted from limestone or synthesized via carbonation, CaCO₃ is extensively used as a functional filler in the plastics and rubber industries, where it enhances mechanical stiffness, reduces production costs, and improves the thermal stability of composite materials [3,4]. In construction, CaCO₃ facilitates cement carbonation hardening and lime mortar production [5,6]. Furthermore, CO₂-sequestered CaCO₃ suspensions reduce emissions while enhancing the performance of cement-based materials [7]. Recent biomedical advances leverage its biocompatibility for pH-responsive drug delivery systems and bone graft composites [8,9]. Additionally, CaCO₃ synthesized in situ within polyacrylamide hydrogel films exhibits high transparency, excellent toughness, and tunable optical anisotropy, endowing the material with unique optical properties suitable for applications such as information encryption [10].

The physical properties of calcium carbonate (CaCO₃) have garnered significant attention from theorists due to its wide range of applications in materials science and geochemistry. The crystallographic parameters of the R-3c phase of the CaCO₃ compound were reported as a = b = 4.970 Å, c = 16.845 Å, and the formation enthalpy is ΔH_f = −2.53 eV/atom in the open quantum materials database record [11]. Furthermore, the lattice parameters and enthalpies of generation of the same phases were recorded as a = b = 5.01 Å, c = 16.99 Å, ΔH_f = −2.688 eV/atom in the material project [12]. Notably, A. K. SINGH et al. [13] observed that calcite-I transforms into calcite-II at a pressure of 1.45 GPa and a temperature of 25 °C. Similarly, D. Vo Thanh and A. Lacam [14] reported this transformation at 1.46 GPa. Furthermore, Merrill et al. [15] found the same phase transition in calcite through X-ray diffraction experiments at 1.5 GPa and 20 °C. M.G. Brik [16] employed first-principles calculations to investigate the structural, electronic, optical, and elastic properties of calcite (CaCO₃) and further analyzed the effects of hydrostatic pressure on these properties. Prencipe et al. [17] successfully simulated the vibrational spectrum of calcite using ab initio quantum mechanical calculations and compared it with experimental data, demonstrating the reliability of the computational approach. Using X-ray diffraction and thermal expansion measurements, Chessin et al. [18] determined the oxygen atom positions, thermal parameters, and lattice constants in calcite, analyzing their temperature dependence to gain insights into the carbonate ion’s structure and properties. Pilati et al. [19] demonstrated the successful application of lattice dynamics to calcite, revealing insights into its vibrational and thermodynamic properties through a combination of crystallographic data and empirical potentials. Maslen et al. [20] utilized synchrotron radiation X-ray diffraction to reveal that the electron density and optical anisotropy of natural rhombohedral CaCO₃ are closely related to its crystal structure and electron density distribution.

While the behavior of trigonal CaCO₃ (calcite) under pressure has been studied, existing research often lacks the comprehensive scope required to fully unravel its complex response. To avoid phase transitions in calcite, this study focuses on the 0–1.2 GPa pressure range and distinguishes itself by conducting a systematic, multi-faceted investigation, explicitly incorporating detailed analysis of its elastic anisotropy and pressure-induced optical properties. This holistic approach is designed to uncover novel insights beyond those offered by previous, more limited examinations. Understanding these interlinked structural, anisotropic elastic, and optical changes is crucial for accurately modeling deep Earth processes and for exploiting pressure as a tool to discover unique functional behaviors relevant to advanced materials design. Our findings are anticipated to provide distinct perspectives and deeper understanding, thereby informing innovative applications in geophysics and materials science.

2. Method

For this study, Density Functional Theory (DFT), as implemented in the CASTEP code [21,22], was employed to perform calculations on CaCO₃. The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) [23] and norm-conserving pseudopotentials [24] were used to describe electronic exchange–correlation interactions and ion–electron interactions, respectively. A plane-wave energy cutoff of 630 eV was applied throughout all calculations. Sampling of the irreducible Brillouin zone (BZ) was conducted using a Monkhorst–Pack k-point grid of 8 × 8 × 2. During geometry optimization with CASTEP, the lattice parameters were constrained to maintain trigonal symmetry (a = b, α = β = 90°, γ = 120°). The key parameters employed include an electronic energy convergence tolerance of 5.0 × 10⁻⁶ eV/atom, alongside ionic convergence criteria of a maximum force of 0.01 eV/Å, maximum stress of 0.02 GPa, and maximum displacement of 5.0 × 10⁻⁴ Å. These parameters ensured excellent convergence accuracy.

3. Results and Discussion

3.1. Structural Properties

A representation of the trigonal crystal system CaCO₃ with the space group R-3c is presented in Figure 1. The CaCO₃ crystal comprises three distinct atomic sites, namely C at 6a, Ca at 6b, and O at 18e. The initial step involved optimizing the crystal structure. The optimized equilibrium lattice constants (a, c), density (ρ), and volume (V) of CaCO₃ are presented in

Table 1. Lattice parameters, density, and volume of CaCO₃ crystal structures at different pressures.

Pressure (GPa)		a (Å)	c (Å)	ρ (g/cm3)	V (Å3)
0	Present	5.0473	17.2499	2.62032	380.564
	Ref. [16]	5.0533	17.3260	-	383.2
	Ref. [17]	5.0492	17.3430	-	382.9
	Ref. [25]	5.045	17.504	-	385.9
	Ref. [18]	4.993	16.9169	-	365.24
	Ref. [19]	4.991	17.062	-	368.07
	Ref. [20]	4.988	17.068	-	367.76
	Ref. [26]	4.984	17.121	-	368.31
0.4	Present	5.0392	17.2127	2.63441	378.530
0.8	Present	5.0361	17.1373	2.64919	376.418
1.2	Present	5.0302	17.0881	2.66313	374.447

, alongside corresponding experimental values and theoretical simulations for comprehensive comparison and analysis.

Table 1 delineates the structural parameters, volume (V), and density (ρ) of CaCO₃ across various pressure conditions. It is evident from the table that the density of the aforementioned material undergoes a substantial increase as the pressure rises, whereas the volume experiences a gradual decline. Through geometrical optimization at 0 GPa, the lattice constants of CaCO₃ were ascertained to be a = b = 5.0473 Å and c = 17.2499 Å. Under high-pressure conditions, a notable decrease in the lattice constant is observed. The computational findings presented in Table 1 are predominantly in harmony with those documented in prior theoretical studies [16,17,25] and experimental investigations [18,19,20,26]. Some discrepancies in the calculated results can be attributed to the selection of different parameters. Consequently, it can be inferred that the theoretical data presented in this study is feasible, and the parameters chosen are appropriate, thereby providing a suitable basis for further data analysis.

The pressure dependence of the lattice parameters is shown in Figure 2a. The lattice compression of CaCO₃ exhibits remarkable anisotropy, with linear compressibility coefficients of $K_a$ = 2.695 TPa⁻¹ for the a-axis and $K_c$ = 8.127 TPa⁻¹ for the c-axis. The compressibility of $K_c$ is significantly higher than that of $K_a$, indicating that the crystal is more easily compressed along the c-axis. These coefficients are calculated using the formula [27]:

$$K_i = -{\displaystyle \frac{1}{a_{i0}}}{({\displaystyle \frac{d{a}_i}{\mathit{dP}}})}_T$$

(1)

The pressure-dependent evolution of unit cell volume is presented in Figure 2b, where open circles represent interpolated values derived from Birch–Murnaghan equation of state fitting. The equation is expressed as [28]:

$$P = {\displaystyle \frac{3}{2}}{B}_0(x^{- {\displaystyle \frac{7}{3}}}-x^{- {\displaystyle \frac{5}{3}}}\left)\right[1+(B^{’}-4\left)\right(x^{- {\displaystyle \frac{2}{3}}}-1\left)\right]$$

(2)

where x = V/V₀ denotes relative volume change, $B_0$ represents the bulk modulus ($B_0 = -V{(\mathit{dP}/\mathit{dV})}_T$), and $B^{’}$ corresponds to its pressure derivative ($B^{’} = -V{(d{B}_0/\mathit{dP})}_T$). The calculated results are $B_0$ = 72.3(5) GPa and $B^{’}$ = 2.89(2); this is in good agreement with the values obtained by M.G. Brik ($B_0$ = 66.917 GPa, $B^{’}$ = 2.698) [16].

3.2. Elastic Properties

A comprehensive investigation was conducted to evaluate the structural stability of CaCO₃ and to gain further insight into its anisotropic behavior under hydrostatic pressure. The trigonal crystal has six independent elastic components, namely $C_{11}$, $C_{12}$, $C_{13}$, $C_{14}$, $C_{33}$, and $C_{44}$. Furthermore, the computed elastic constants must conform to the stipulated mechanical stability conditions [29].

$$C_{44}-P > 0,{ C}_{11}-P-\left|C_{12}+P\right| > 0$$

(3)

$$(C_{33}-P)\left(C_{11}+C_{12}\right)-2{(C_{13}+P)}^2 > 0$$

(4)

$${(C}_{44}-P)\left(C_{11 }-C_{12}-2P\right)-2C_{14}^2 > 0$$

(5)

As demonstrated by the data in

Table 2. The elastic constants $C_{\mathit{ij}}$ (GPa) of CaCO₃ were calculated at respective pressures of 0, 0.4, 0.8, and 1.2 GPa.

Pressure (GPa)		${\mathit{C}}_{11}$ (GPa)	${\mathit{C}}_{12}$ (GPa)	${\mathit{C}}_{13}$ (GPa)	${\mathit{C}}_{14}$ (GPa)	${\mathit{C}}_{33}$ (GPa)	${\mathit{C}}_{44}$ (GPa)
0	Present	140.38	51.76	49.12	16.17	79.89	32.41
	Ref. [30]	153	55	50	13	82	37
	Ref. [31]	146.26	59.70	50.76	−20.76	85.31	34.05
0.4	Present	141.01	52.80	50.52	16.66	80.36	32.93
0.8	Present	145.09	54.77	51.86	17.63	82.01	33.31
1.2	Present	148.29	58.48	55.59	17.66	83.45	34.64

and Figure 3, CaCO₃ satisfies the mechanical stability criteria across the 0–1.2 GPa pressure range, thereby indicating that CaCO₃ is mechanically stable within the aforementioned pressure range.

The mechanical attributes of materials are often characterized by the bulk modulus ($B$), shear modulus ($G$), and Young’s modulus ($E$). Subsequently, the Reuss–Voight–Hill approximation is employed to determine the bulk modulus ($B$), shear modulus ($G$), and Young’s modulus ($E$) of CaCO₃ [32,33,34]. The values of $B$ and $G$ are determined through the application of the following expressions [35]:

$$B = {\displaystyle \frac{B_V + B_R}{2}}, G = {\displaystyle \frac{G_{V }+ G_R}{2}}$$

(6)

where

$$B_V={\displaystyle \frac{2C_{11}+C_{33}+2C_{12}+4C_{13}}{9}}, B_R={\displaystyle \frac{1}{(2S_{11}+S_{33})+2{(S}_{12}+2S_{13})}}$$

(7)

$$G_V={\displaystyle \frac{\left(2C_{11}+C_{33}\right)-(C_{12}+2C_{13})+3(2C_{44}+\frac{C_{11}-C_{12}}{2})}{5}}$$

(8)

$$G_R={\displaystyle \frac{15}{4\left(2S_{11}+S_{33}\right)-4(S_{12}+2S_{13})+6(S_{44}+S_{11}-S_{12})}}$$

(9)

The Young’s modulus ($E$) can be defined in terms of the bulk modulus ($B$) and the shear modulus ($G$) as follows [35]:

$$E = {\displaystyle \frac{9\mathit{BG}}{3B+G}}$$

(10)

The hardness ($H_V$) of orthorhombic CaCO₃ is calculated by [36]; the units of G, B, and H_V are all GPa:

$$H_V = 0.92{({\displaystyle \frac{G}{B}})}^{1.137}{G}^{0.708}$$

(11)

As illustrated in

Table 3. Calculated bulk modulus $B$ (GPa), shear modulus $G$ (GPa), Young’s modulus $E$(GPa), Poisson’s ratio $v$, $\mathrm{and} B/G$ of CaCO₃ under pressures varying between 0 and 1.2 GPa.

Pressure	${\mathit{B}}_{\mathit{V}}$ (GPa)	${\mathit{B}}_{\mathit{R}}$ (GPa)	$\mathit{B}$ (GPa)	${\mathit{G}}_{\mathit{V}}$ (GPa)	${\mathit{G}}_{\mathit{R}}$ (GPa)	$\mathit{G}$ (GPa)	$\mathit{E}$ (GPa)	$\mathit{\nu }$	$\mathit{B}/\mathit{G}$	${\mathit{H}}_{\mathit{V}}$
0	73.404	67.705	70.555	35.869	28.789	32.329	84.361	0.3007	2.1757	4.4382
0.4	74.450	68.679	71.564	35.896	28.766	32.331	84.298	0.3037	2.2135	4.3676
0.8	76.573	70.386	73.479	36.605	28.866	32.735	85.508	0.3061	2.2447	4.3367
1.2	79.931	73.195	76.563	36.860	29.201	33.031	86.634	0.3114	2.3179	4.2079

, the values of $B$, $G,$ and $E$ for CaCO₃ at 0–1.2 GPa are presented. It can be observed that the modulus of elasticity ($B$, $G$, $E$) of CaCO₃ exhibits a gradual and uninterrupted increase with rising pressure within the specified pressure range.

The Poisson’s ratio ($v$) and the $B/G$ ratio can serve as reliable indicators of the brittle or ductile nature of a material [37]. A Poisson’s ratio of less than 0.26 and a $B/G$ ratio of less than 1.75 indicate brittleness. As evidenced in Table 3, the ductility of CaCO₃ increases gradually with increasing pressure. Vickers hardness ($H_V$) is a useful criterion for distinguishing between soft and hard materials [36]. A material is considered to exhibit soft characteristics if its Vickers hardness is less than 10 GPa, while a material with a hardness above this threshold is classified as hard [37]. Table 3 illustrates that CaCO₃ exhibits softening behavior as the pressure increases.

The longitudinal wave velocity V_l, transverse wave velocity V_t, and mean wave velocity V_m of CaCO₃ at pressures of 0–1.2 GPa have been calculated according to Navier’s equations [38,39] and displayed in

Table 4. Calculated wave velocities $V_l$, $V_T$, and $V_m$ ($\mathrm{km}/\mathrm{s}$) of CaCO₃ under pressures of 0, 0.4, 0.8, and 1.2 GPa.

Pressure	${\mathit{V}}_{\mathit{m}}$$(\mathbf{km}/\mathbf{s}$)	${\mathit{V}}_{\mathit{l}}$$(\mathbf{km}/\mathbf{s}$)	${\mathit{V}}_{\mathit{t}}$$(\mathbf{km}/\mathbf{s}$)
0	3.9241	6.5861	3.5125
0.4	3.9149	6.5976	3.5032
0.8	3.9294	6.6492	3.5152
1.2	3.9395	6.7295	3.5218

. As shown in Table 4, the V_l, V_t, and V_m of CaCO₃ are calculated to be 6.5861 km/s, 3.5125 km/s, and 3.9241 km/s at 0 GPa pressure, respectively.

$$V_l=\sqrt{(B+{\displaystyle \frac{4}{3}}G){\displaystyle \frac{1}{\mathsf{\rho }}}}, V_T=\sqrt{{\displaystyle \frac{G}{\mathsf{\rho }}}}$$

(12)

$$V_m={[{\displaystyle \frac{1}{3}}({\displaystyle \frac{2}{V_T^3}}+{\displaystyle \frac{1}{V_l^3}})]}^{-{\displaystyle \frac{1}{3}}}$$

(13)

Studies of mechanical properties in materials typically consider the influence of microfractures and lattice deformations on their behavior, with elastic anisotropy recognized as a critical factor influencing both phenomena. Consequently, analyzing elastic anisotropy is essential for improving the structural resilience of engineered materials. The description of elastic anisotropy can be achieved through the utilization of the anisotropy index ($A^U$) and anisotropy percentage ($A_B$, $A_G$). The aforementioned parameters can be derived from the following equation [40,41]:

$$A_B = {\displaystyle \frac{ B_{V }- B_R}{B_{V }+ B_R}}, A_G = {\displaystyle \frac{G_V - G_R}{G_{V }+{ G}_R}}, A_E={\displaystyle \frac{E_V - E_R}{E_V +{ E}_R}}$$

(14)

$$A^U=5{\displaystyle \frac{G_V}{G_R}}+{\displaystyle \frac{B_V}{B_R}}-6 \ge 0$$

(15)

In this context, $B_V$ and $B_R$ represent the bulk modulus of the Voigt and Reuss models, respectively, while $E_V$ and $E_R$ denote the Young’s modulus of the aforementioned models. Similarly, $G_V$ and $G_R$ signify the shear modulus of the Voigt and Reuss models. The values of $A_B$, $A_E$, and $A_G$ are employed to ascertain the extent of anisotropy in the bulk, Young’s, and shear moduli of the materials, respectively. The calculated values of $A_B$, $A_E$, and $A_G$ for different pressures are presented in

Table 5. Calculated elastic anisotropic indexes $A_B$, $A_G$, $A_E$, and $A^U$ under different pressures.

Pressure	${\mathit{A}}_{\mathit{B}}$	${\mathit{A}}_{\mathit{G}}$	${\mathit{A}}_{\mathit{E}}$	${\mathit{A}}^{\mathit{U}}$
0	0.0403	0.1095	0.0974	1.3138
0.4	0.0404	0.1103	0.1012	1.3233
0.8	0.0421	0.1159	0.1069	1.4035
1.2	0.0439	0.1182	0.1085	1.4284

. In the case of elastically isotropic materials, the values of $A_B$, $A_E$, and $A_G$ are all equal to zero. Conversely, a value of one for $A_B$, $A_E$, and $A_G$ indicates the maximum possible elastic anisotropy. From the table, it can be observed that the bulk modulus, Young’s modulus, and shear modulus of CaCO₃ exhibit varying degrees of anisotropy. $A^U$ represents the universal elastic anisotropy index for all crystalline phases, with $A^U$ = 0 indicating isotropic behavior in single crystals. The degree of anisotropy in the material is quantified by the magnitude of the deviation of $A^U$ from 0. It is apparent that the anisotropy of CaCO₃ grows progressively with rising pressure.

The linear compressibility, Poisson’s ratio, shear modulus, and Young’s modulus of CaCO₃ at pressures of 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa were visualized in three-dimensional (3D) and two-dimensional (2D) formats using the ELATE program [42], enabling a comprehensive analysis of elastic anisotropy. For isotropic crystals, 3D representations adopt a perfectly spherical geometry, whereas anisotropic crystals display deviations from spherical symmetry that correlate directly with the degree of elastic anisotropy. Figure 4, Figure 5, Figure 6 and Figure 7 illustrate the 3D plots of these properties (linear compressibility, Poisson’s ratio, shear modulus, and Young’s modulus) at the specified pressures. All four properties exhibit marked deviations from spherical symmetry, confirming pronounced anisotropic behavior in CaCO₃ under pressure. In contrast, isotropic materials would retain circular/spherical geometries; thus, observed deviations underscore the material’s directional dependence.

However, the anisotropy observed at varying pressures cannot be fully characterized solely through three-dimensional surface structures. To systematically quantify the elastic anisotropy of CaCO₃, two-dimensional projections of these 3D representations were analyzed. Figure 8, Figure 9, Figure 10 and Figure 11 present the projections of linear compressibility, Poisson’s ratio, shear modulus, and Young’s modulus on the xy, xz, and yz planes across the studied pressure range. These projections enable a clearer visualization of directional elastic behavior. Notably, the xy-plane projections of linear compressibility and Young’s modulus exhibit circular symmetry (indicative of isotropy), while pronounced irregularities on the xz and yz planes confirm anisotropy in these orientations. The remaining elastic moduli (shear modulus and Poisson’s ratio) display marked anisotropy across all planes (xy, xz, and yz) under pressure.

While Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 visualize the anisotropy of CaCO₃, the pressure-induced changes are too subtle to discern clearly in these plots. To deepen our understanding of the elastic anisotropy in CaCO₃ at pressures of 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa, we performed calculations of the directional elastic modulus along the three main axes. The computed data is tabulated in

Table 6. The ${\beta }_{\mathit{min}}$ (TPa⁻¹), ${\beta }_{\mathit{max}}$ (TPa⁻¹), ${\upsilon }_{\mathit{min}}$, ${\upsilon }_{\mathit{max}}$, $G_{\mathit{min}}$ (GPa), $G_{\mathit{max}}$ (GPa), $E_{\mathit{min}}$ (GPa), and $E_{\mathit{max}}$(GPa) of 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa.

Pressure (GPa)	${\mathit{\beta }}_{\mathit{min}}$ (TPa−1)	${\mathit{\beta }}_{\mathit{max}}$ (TPa−1)	${\mathit{v}}_{\mathit{min}}$	${\mathit{v}}_{\mathit{max}}$	${\mathit{G}}_{\mathit{min}}$ (GPa)	${\mathit{G}}_{\mathit{max}}$ (GPa)	${\mathit{E}}_{\mathit{min}}$ (GPa)	${\mathit{E}}_{\mathit{max}}$ (GPa)
0	2.9235	8.9229	0.0369	0.6934	21.129	55.591	54.773	131.99
0.4	2.8502	8.8602	0.0378	0.7044	20.943	56.093	54.023	133.27
0.8	2.7378	8.7317	0.0232	0.7238	20.638	57.839	55.090	137.37
1.2	2.5155	8.6311	0.0413	0.7243	21.385	58.160	53.551	139.26

. The deviations of the linear compressibility $\beta$, Poisson’s ratio $v$, shear modulus $G,$ and Young’s modulus $E$ of CaCO₃ at 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa were compared using the ratios ${\beta }_{\mathit{max}}$/${\beta }_{\mathit{min}}$, ${\upsilon }_{\mathit{max}}$/${\upsilon }_{\mathit{min}}$, $G_{\mathit{max}}$/$G_{\mathit{min},}$ and $E_{\mathit{max}}$/$E_{\mathit{min}}$. An increase in the value of ${\beta }_{\mathit{max}}$/${\beta }_{\mathit{min}}$ results in a corresponding increase in linear compressibility anisotropy, as do the ratios of the other principal components: ${\upsilon }_{\mathit{max}}$/${\upsilon }_{\mathit{min}}$, $G_{\mathit{max}}$/$G_{\mathit{min}}$ and $E_{\mathit{max}}$/$E_{\mathit{min}}$. At 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa, the ${\beta }_{\mathit{max}}$/${\beta }_{\mathit{min}}$ is 3.05, 3.11, 3.19, and 3.43, respectively. Similarly, the values of $G_{\mathit{max}}$/$G_{\mathit{min}}$ are 2.63, 2.68, 2.72, and 2.80, while the values of $E_{\mathit{max}}$/$E_{\mathit{min}}$ are 2.41, 2.47, 2.49, and 2.60. It is apparent that the values of β, $G,$ and $E$ tend to increase overall when the pressure is between 0 and 1.2 GPa. The analysis of the above data indicates that the degree of anisotropy of $\beta$, $G,$ and $E$ increases when the pressure varies from 0 to 1.2 GPa.

3.3. Electronic Properties

A thorough discussion of the electronic structure is essential for understanding the relationship between the crystal structure and optical properties of CaCO₃. Figure 12 presents the pressure-dependent band structures of CaCO₃, where the horizontal red line indicates the Fermi level within the Brillouin zone. As shown in Figure 12, the principal energy dispersion range of the band structure expands significantly with increasing pressure. The calculated band gap at ambient pressure is 6.312 eV, which aligns well with the experimental result of 6 eV [43].

To further analyze the electronic characteristics, we performed density of states (DOS) calculations, as illustrated in Figure 13. The DOS profiles reveal the orbital-specific electronic distributions and, when combined with partial density of states (PDOS) and band structure analyses, enable identification of atomic contributions to the conduction and valence bands. Figure 13 demonstrates that the lower valence band primarily consists of O-2p orbitals with minor contributions from C-2p and O-2s states, while the upper valence band is dominated by O-2p states. The conduction band within the 5–10 eV energy range is predominantly composed of Ca-3d states, while the 10–15 eV and 15–20 eV intervals exhibit primary contributions from Ca-3p and C-2p orbitals, respectively. Notably, while pressure variations do not alter the fundamental orbital composition of the valence and conduction bands, they induce energy range modifications; however, under the four applied pressures, the changes are very small and visually indistinguishable, so only Figure 13 is shown.

3.4. Optical Properties

The interaction between a material’s structure and electromagnetic radiation is directly related to its optical properties, which are crucial in determining its photophysical behavior and impact on absorption, transmission, and reflection spectra. From an optoelectronic perspective, the interaction with visible light is particularly significant. The response to incident radiation is wholly contingent upon a multitude of energy-dependent (frequency) optical parameters, including the real and imaginary components of the dielectric constant ${\mathsf{\epsilon }}_1$($\mathsf{\omega }$) and ${\mathsf{\epsilon }}_2$($\mathsf{\omega }$), the real component of the refractive index $n$($\mathsf{\omega }$), and so forth. Furthermore, the imaginary part of the optical index $k$($\mathsf{\omega }$); the loss function $L$($\mathsf{\omega }$); the real and imaginary parts of the photoconductivity, respectively; the reflectance $R$($\mathsf{\omega }$); and the absorption coefficient $\mathsf{\alpha }$($\mathsf{\omega }$) are also of significance. The theoretical underpinnings and computational methodologies enable deriving these critical optical constants from first principles [44]. In this investigation, we have focused on the optical parameters when the electric field polarization vector is aligned with the [100] direction and the light propagates along the [001] direction. The results obtained for the calculations of CaCO₃ at 0 GPa, 0.4 GPa, 0.8 GPa, and 1.2 GPa are presented in Figure 14 and Figure 15.

The figures demonstrate how the real and imaginary parts of the dielectric function, the refractive index, and the extinction coefficient of CaCO₃ vary with pressure. These variations are illustrated in Figure 14. The phase velocity of the electromagnetic wave within the sample is dictated by the real component of the refractive index, while the measured attenuation is given by the imaginary component. As illustrated, the refractive index $n$($\mathsf{\omega }$) exhibits a similar trajectory to that of ${\mathsf{\epsilon }}_1$($\mathsf{\omega }$), while the imaginary part of the optical index $k$($\mathsf{\omega }$) displays a comparable trend to that of ${\mathsf{\epsilon }}_2$($\mathsf{\omega }$). In accordance with the principles of optics, the real and imaginary parts of the refractive index are inextricably linked to the dielectric constant, as demonstrated by the following equation: ${\mathsf{\epsilon }}_1\left(\mathsf{\omega }\right) ={ n}^2\left(\mathsf{\omega }\right)-k^2\left(\mathsf{\omega }\right)$ and ${\mathsf{\epsilon }}_2\left(\mathsf{\omega }\right) = 2n\left(\mathsf{\omega }\right)k\left(\mathsf{\omega }\right)$ [45].

The rate of decrease in the real part ${\mathsf{\epsilon }}_1\left(\mathsf{\omega }\right)$ of the dielectric function governs the magnitude of the imaginary part ${\mathsf{\epsilon }}_2\left(\mathsf{\omega }\right)$ peak, which directly correlates with optical absorption and quantifies the extent of electron excitation transitions. The peak position corresponds to the photon energy, with the variation as shown in Figure 14d. Comparative analysis reveals a pronounced increase in photon energy with elevated pressure. By integrating band structure (Figure 12) and density of states (Figure 13) analyses, the dielectric peaks in CaCO₃ are linked to electronic transitions between valence and conduction bands. Figure 14d reveals three distinct absorption peaks, with the feature centered at ~7 eV demonstrating markedly enhanced intensity relative to the other two. This spectral characteristic is predominantly attributed to electronic transitions from O-2p to Ca-3d orbitals. The observation aligns consistently with orbital-resolved DOS profiles, where O-2p states dominate the valence band edge, while Ca-3d orbitals contribute significantly to the conduction band at higher energy levels.

The reflectivity is defined as the ratio of the energy of the reflected wave to the energy of the incident wave. The reflectivity spectrum is illustrated in Figure 15a. The reflectivity exhibits peaks at 5 eV, 20 eV, and 30 eV, with the highest at 20 eV under 1.2 GPa and at 30 eV under 0 GPa. The peak at 6 eV is likely due to interband transitions or excitonic effects consistent with the intrinsic bandgap of calcite. The peaks at 20 eV and 30 eV may arise from collective electron oscillations (plasmons) or many-body excited states. Further theoretical calculations and high-precision experiments are required to validate this phenomenon.

In the analysis of a material’s optical characteristics, the energy-dependent loss function is a crucial parameter, as depicted in Figure 15c. The energy loss function, $L(\mathsf{\omega }) =-\mathrm{Im}({\displaystyle \frac{1}{{\mathsf{\epsilon }}_2(\mathsf{\omega })}})$, quantifies the energy dissipated by a material per unit of frequency. It facilitates comprehension of the shielded excitation spectrum, particularly the collective excitation generated by fast charges traversing a solid. Energy loss reaches a maximum at a particular incident light frequency (energy), which is designated as the bulk screened plasma frequency. As illustrated in Figure 15c, a pronounced electromagnetic loss peak is evident. This suggests that the fundamental excitation spectrum is relatively monochromatic.

Figure 15b illustrates the absorption spectrum. The absorption coefficient indicates the level of penetration that light at a particular wavelength can undergo within the material before it is absorbed, and it also offers insights into the potential for optimal solar energy conversion efficiency. A comparison of Figure 15b,d reveals a striking similarity between the absorption coefficient curves and the real parts of the photoconductivity curves, particularly in terms of their overall trend. The curves of $\mathsf{\alpha }$($\mathsf{\omega }$) in the 0–40 eV range exhibit no discernible broad peaks, indicating that the coupling between the electronic leaps in the material and the lattice vibrations or other electrons is markedly weak and contributes to the propagation of the photons in CaCO₃.

4. Conclusions

This study employs first-principles calculations to systematically investigate the structural, elastic, dynamic, and optical properties of trigonal calcium carbonate (CaCO₃) under hydrostatic pressures of 0–1.2 GPa. Optimized lattice parameters align closely with established theoretical and experimental values, confirming the reliability of the computational approach. Calculated elastic constants and moduli show steady increases with pressure while adhering to the Born–Huang criteria for mechanical stability, verifying CaCO₃′s robust mechanical integrity under pressure conditions. By quantifying elastic anisotropy through the universal anisotropy index, three-dimensional elastic modulus surfaces, and two-dimensional projections, we demonstrate pronounced pressure-dependent anisotropy, which intensifies progressively with compression. Additionally, pressure-induced variations in optical properties clarify the modulation mechanisms of electromagnetic interactions. These findings establish theoretical foundations for understanding CaCO₃’s structural responses, mechanical behavior, and optical functionalization potential in pressure environments. The computational insights presented here may guide future experimental studies and inform strategies for optimizing this material’s performance.