The calculated crystal structures of BaTiO₃ in the cubic (Pm3m), orthorhombic (Amm2, C2v), and tetragonal (P4mm, C4v) phases are illustrated in Fig. 1. The Wyckoff positions of Ba, Ti, and O atoms for the cubic, tetragonal, and orthorhombic phases of BaTiO₃ are summarized in Table 1. In the cubic Pm3m phase, all atoms occupy high-symmetry Wyckoff sites and the Ti atom remains centered within the oxygen octahedron, resulting in a nonpolar paraelectric structure. In contrast, symmetry lowering in the tetragonal (P4mm) and orthorhombic (Amm2) phases displaces the Ti atom from the octahedral center along the polar direction, giving rise to spontaneous ferroelectric polarization. These phase-dependent changes in Wyckoff positions and atomic coordinates reflect the microscopic origin of ferroelectricity in BaTiO₃.

The structures were optimized using the GGA-PBE method to determine the ground-state structural parameters. Optimization curves for all three phases are presented in Fig. 2. These curves summarize the computed lattice parameters for the material in its cubic, orthorhombic, and tetragonal phases, along with a comparison with existing experimental and theoretical data presented in Table 2.

The results in Table 2 show a close agreement between the present results and the corresponding theoretical and experimental results. In addition, Table 2 indicates that the ground state energy (E_o) of the material for all three phases are negative and nearly equal to each other, indicating that there is a slight variation in the stability of the material between the three phases. The resistance to uniform compression is measured by the bulk modulus (B), which is the ratio of the change in pressure to the change in volume. Although there is a slight variation in the bulk modulus across the three phases, the larger the bulk modulus, the greater the incompressibility/greater the force required to compress the material; therefore, the results presented in Table 2 indicate that the orthorhombic phase exhibits the lowest bulk modulus, indicating that it is the most compressible phase, while the cubic phase shows the highest resistance to uniform compression. The way that the bulk modulus of BaTiO₃ changes with respect to applied pressure can be seen by taking the first derivative of the bulk modulus as a function of pressure. It shows how much a material can be compressed under different levels of pressure and ultimately helps determine at what pressure it is most resistant to compaction. As shown in Table 2, the first derivative of BaTiO₃’s bulk modulus (B’) has a positive value, meaning that B rises with increasing pressure and becomes less compressible at higher pressures. We also determined the formation energy (E_for) for all three phases of BaTiO₃ in eV/Atom, all of which have negative values, indicating that all three phases are thermodynamically stable and can therefore be produced experimentally. The orthorhombic phase has the most negative value of E_for, which means it is the most thermodynamically stable phase of the three and thus is most likely to be produced. Although the orthorhombic phase is the most thermodynamically stable based on formation energy, it exhibits a comparatively lower bulk modulus, reflecting enhanced lattice flexibility associated with ferroelectric distortion.

To evaluate the broad applications of the investigated material, it is critical to determine its precise band gap. To accomplish this task, a first-principles method is used to obtain accurate values of band gap and other electronic parameters. The electronic band structures and total density of states (TDOS) of BaTiO₃ in its cubic, tetragonal, and orthorhombic phases were calculated using the GGA-PBE and HSE06 exchange-correlation functionals (Fig. 3). In all phases, BaTiO₃ exhibits a semiconducting character with an indirect band gap, where the valence band maximum (VBM) and conduction band minimum (CBM) occur at different high-symmetry k-points. The GGA-PBE functional underestimates band gaps, while the HSE06 hybrid functional provides significantly improved agreement with experimentally reported optical band gaps (3.3 eV) [50]. A systematic increase in the band gap is observed with decreasing structural symmetry (cubic < tetragonal < orthorhombic), which can be attributed to enhanced lattice distortion and Ti off-centering in the lower-symmetry ferroelectric phases [2]. The bandgap values of the considered compound in all three phases, computed in the present study and in previous theoretical and experimental studies, are presented in Table 3.

The total and projected density of states (TDOS/PDOS) of BaTiO₃ in its cubic, orthorhombic, and tetragonal phases were calculated using the HSE06 hybrid functional and are shown in Fig. 4a–c. In all cases, the Fermi level (E_F) is set to zero energy. A clear band gap separating the valence and conduction regions is observed for all phases, confirming the semiconducting nature of BaTiO₃ and the improved gap description provided by the hybrid functional. For the cubic phase, the valence band region extending from approximately −6 eV up to the valence band maximum is predominantly composed of O-2p states, with a minor contribution from Ti-3d states near the upper valence edge. The conduction band onset is mainly governed by Ti-3d states, while Ba-derived states contribute primarily at much higher energies. This orbital distribution reflects the characteristic Ti–O covalent hybridization in perovskite oxides and is consistent with the paraelectric nature of the high-symmetry cubic structure. The absence of significant Ti-3d weight at the Fermi level further confirms the insulating character of the cubic phase. In the orthorhombic phase, a noticeable redistribution of electronic states near both the valence and conduction band edges is observed. Compared to the cubic phase, the PDOS shows enhanced splitting of the Ti-3d states in the conduction band, attributed to symmetry lowering and increased lattice distortion. The O-2p states remain dominant in the valence band; however, their bandwidth and peak structure indicate stronger anisotropy and modified Ti–O bonding. This enhanced orbital splitting contributes to the larger band gap observed in the orthorhombic phase and reflects stronger electronic localization associated with the ferroelectric distortion. For the tetragonal phase, the PDOS characteristics lie between those of the cubic and orthorhombic phases. The tetragonal distortion lifts the degeneracy of the Ti-3d orbitals, leading to moderate splitting at the conduction band edge. The O-2p dominated valence band exhibits a slightly narrower bandwidth than in the cubic phase, indicating reduced orbital overlap along the polar axis. The hybridization between Ti-3d and O-2p states near the band edges remains pronounced, underscoring the role of Ti off-centering in stabilizing the ferroelectric phase. A comparative analysis across the three phases demonstrates that the degree of Ti-3d orbital splitting and O-2p bandwidth modulation increases with decreasing crystal symmetry (cubic → tetragonal → orthorhombic). This trend directly correlates with the phase-dependent lattice distortions and Ti atom displacement from the center of the oxygen octahedron. Importantly, the HSE06 functional not only corrects the band gap underestimation inherent to GGA-based approaches but also preserves the physically meaningful orbital character and hybridization trends across different phases.

A good understanding of the optical properties of materials is essential to determine how well optoelectronic devices perform. The transverse dielectric function describes how materials respond to linearly varying light fields and how well they store charge under an electric field. It is essential because it describes how the material responds to photon-electron interactions through momentum and energy transfer. Momentum transfer can be thought of as the movement of the interacting particles, while the energy transfer refers to a change in their corresponding energies. With respect to solar radiation and such lower energies, momentum transfer is typically very low (≈0) so that the electric dipole approximation can be used. Under this assumption, the frequency-dependent dielectric function can be formally expressed as [51]: (1)ε(ω)=ε1(ω)+iε2(ω)

Eq. (1) denotes the real (R) and imaginary (Im) parts of the dielectric constants as ε₁(ω) and ε₂(ω), respectively. The components of the dielectric constants can be computed by examining the direct transitions between conduction band states and valence band states. The dielectric function ε(ω) indicates how polarized a material becomes in an electric field and is influenced by the density of states (DOS), the material’s electric polarizability, and the presence of free charges or ions.

As illustrated in Fig. 5a,b, the real component ε₁(ω) and imaginary components ε₂(ω) of the dielectric function were estimated using the HSE06 functional in the energy range of 0–15 eV. This investigation was conducted to explore the optical response of the cubic, orthorhombic, and tetragonal phases of the BaTiO₃ compound. The static real part ε₁ (0), i.e., the dielectric constant for all three phases of the BaTiO₃ compound, is found to be 4.7, 4.4, and 4.5, respectively. The real part of the dielectric function (ε₁(ω)) is responsible for the description of the dispersive behaviour and the response to polarization of the material under consideration. Within the low-energy region (IR to visible range, below ~3.1 eV), ε₁(ω) manifests comparatively elevated positive values for all phases, thereby signifying robust dielectric polarization and normal dispersion. The static dielectric constant (ε₁(0)) is observed to be highest for the orthorhombic phase, followed by the tetragonal and cubic phases. It indicates increased lattice distortion and enhanced polarizability in the lower-symmetry ferroelectric phases. As the photon energy increases towards the UV region, the parameter ε₁(ω) gradually decreases and exhibits pronounced oscillations associated with interband electronic transitions. The crossing of ε₁(ω) through zero at higher energies suggests the onset of plasma-like behaviour and reduced reflectivity in this spectral range. The imaginary part of the dielectric function, ε₂(ω), directly represents optical absorption arising from interband electronic transitions. In Fig. 5b, the prominent peaks for ε₂(ω) are noticed at 5.3 eV for all phases. The peaks in ε₂(ω) exhibit inconsistent features in the visible and UV regions. The affirmative values of ε₂(ω) signify zones of optical absorption. For all three phases, the parameter ε₂(ω) remains almost negligible in the IR and across most of the visible region, thereby confirming the optical transparency of the material below the fundamental band gap. A sharp rise in ε₂(ω) is observed at energies close to the optical absorption edge (~3–4 eV), corresponding to transitions from O-2p dominated valence bands to Ti-3d dominated conduction bands. The absorption onset occurs at the boundary between the visible and UV regions, consistent with BaTiO₃’s wide-band-gap semiconductor nature. A comparative analysis reveals that a decrease in crystal symmetry (cubic → tetragonal → orthorhombic) results in an overall enhancement of both ε₁(ω) in the low-energy region and ε₂(ω) peak intensities in the UV range. This phenomenon underscores the profound interconnection between the crystal structure, ferroelectric distortion, and optical response in BaTiO₃.

The expression for the absorption coefficient α(ω) is as follows [52]: (2)α(ω)=2ω[ε12(ω)+ε22(ω)−ε1(ω)]12

As shown in Fig. 6a, α(ω) represents the fraction of light intensity attenuated per unit distance by a material. The absorption spectra reveal that all three phases have minimal absorption in the IR and most of the visible spectrum, confirming that BaTiO₃ is optically transparent at low photon energies. A sharp rise in the absorption coefficient occurs near the visible–UV boundary (around 3–4 eV), which aligns with the fundamental optical band edge. This onset of absorption is linked to interband electronic transitions from O-2p valence bands to Ti-3d conduction bands, consistent with electronic structure and dielectric analyses. In the UV range, the absorption coefficient quickly climbs to 10⁵–10⁶ cm⁻¹, indicating strong light–matter interaction and efficient photon absorption at higher energies. Notably, UV absorption varies with phase: the orthorhombic phase exhibits slightly higher absorption and more distinct peak structures than the cubic phase, while the tetragonal phase shows intermediate behavior. It suggests increased lattice distortion and stronger Ti–O orbital hybridisation in the lower-symmetry ferroelectric phases, enhancing electronic transitions and broadening absorption features.

A compound’s reflectance and reflectivity spectra indicate whether or not it is an excellent candidate to be an ideal absorber. The reflectivity ‘R(ω)’ and refractivity spectra ‘n(ω)’, which are shown in Fig. 6b,c for the materials under study, can be analyzed to define the optical behavior of a material. These can be expressed mathematically in the following way: (3)n(ω)=(12)[ε12(ω)+ε22(ω)+ε1(ω)]12 (4)R(ω)=[n(ω)−1]2+k2(ω)[n(ω)+1]2+k2(ω)where, (5)k(ω)=|ε(ω)|−ε1(ω)2

R(ω), the reflection coefficient, measures the quantity of electromagnetic (EM) radiation reflected from the approaching surface. Fig. 6b shows how R(ω) varies with energy. The reflectivity spectra further support these observations. In the IR and visible regions, BaTiO₃ exhibits relatively low, slowly varying reflectivity, consistent with its transparency and weak absorption in these energy ranges. As the photon energy increases toward the UV region, the reflectivity rises markedly, with several peaks corresponding to strong interband transitions. Among the three phases, the orthorhombic phase exhibits the highest reflectivity peaks in the UV range, followed by the tetragonal and cubic phases, indicating stronger optical response and greater electronic polarization in the distorted ferroelectric phases. The graph shows that each phase reflects energy between 3 and 15 eV. Nevertheless, the energy reflectance steadily decreases to negligible values around 18 eV, so that the higher the energy of the electromagnetic radiation, the less efficient the material is at reflection.

Refractivity ‘n(ω)’ is another significant optical property that is described by the measurement of the refractive index, and is used to explain how long it takes the light waves to get through a material. The n(0) value of BaTiO₃ for the cubic, orthorhombic, and tetragonal phases is 2.18, 2.09, and 2.12, respectively. In the IR and visible regions (below ~3.1 eV), the refractive index exhibits relatively high, slowly varying values across all phases, indicating strong polarization and low optical losses. Among the three phases, the orthorhombic phase shows the highest refractive index, followed by the tetragonal and cubic phases. This trend reflects the enhanced polarizability and stronger lattice distortion in the lower-symmetry ferroelectric phases. The elevated n(ω) values in this energy range are consistent with the large real part of the dielectric function (ε₁(ω)) previously discussed.

The extinction coefficient k(ω) describes the absorbance of energy and is plotted with incident energy for the three different phases of the material BaTiO₃, as indicated in Fig. 6d. The peak of k(ω) is at 4.85 eV and stronger in the visible region, indicating that the greatest amount of sunlight will be absorbed in the materials of the visible region. The extinction coefficient (k(ω)) remains nearly zero in the IR and most of the visible region, confirming the optical transparency of BaTiO₃ at low energies. A sharp increase in k(ω) is observed near the absorption edge (~3–4 eV), in agreement with the rapid rise in the absorption coefficient and the imaginary part of the dielectric function (ε₂(ω)). In the UV region, k(ω) exhibits several prominent peaks associated with intense interband transitions from O-2p valence states to Ti-3d conduction states. Phase-dependent differences are again evident in the UV range. The orthorhombic phase displays the largest k(ω) values and the most pronounced spectral features, indicating stronger absorption and higher optical losses at high photon energies. The tetragonal phase shows intermediate behavior, while the cubic phase exhibits comparatively weaker absorption features. These trends correlate directly with the degree of lattice distortion and Ti off-centering, which enhance orbital splitting and electronic localization in the lower-symmetry phases. Also, the correlation between k(ω) and the energy is identical to the correlation of ε₂(ω) and the energy, since k(ω) is going to be used to determine how fast the material will absorb the light.

The quasi-harmonic Debye model was employed in the Gibbs2 code [49] to calculate the thermal parameters. The total energies of different primitive cells were compared to determine their macroscopic thermal behavior at different temperatures and pressures. These total energy quanta played a vital role in the computation of the thermal properties [53]. BaTiO₃’s thermodynamic parameters are examined over 0–1000 K in temperature and 0–30 GPa in pressure. Figs. 7–16 illustrate the impacts of temperature and pressure on the following quantities: volume (‘V’), bulk modulus (‘B’), Debye temperature (‘θ_D’), Grüneisen parameter (‘γ’), thermal expansion (‘α’), and heat capacity (‘C_v’). The deviations of V/V₀ with respect to pressure at distinct temperatures for the cubic, orthorhombic, and tetragonal phases of the BaTiO₃ compound are shown in Fig. 7. The V/V₀ ratio decreases with increasing pressure and temperature for all configurations, confirming the reliability of our computed results.

The bulk modulus is another parameter that quantifies the ability of a substance to resist a uniform pressure on its volume. The obtained findings can be attributed to the following: as the temperature of a substance increases, its atoms or molecules have more kinetic energy and become less tightly packed, resulting in crystal expansion. As a result, the material becomes less rigid or more compressible, i.e., its bulk modulus decreases with increasing temperature. Conversely, the particles of a material are brought closer together when higher pressure is applied, because intermolecular forces increase resistance to changes in material volume, leading to higher rigidity and less compressibility; i.e., an increase in applied pressure increases the bulk modulus. Fig. 8 shows the variation in the bulk modulus of the three phases of BaTiO₃ with temperature at the applied pressure, where the bulk modulus increases with increasing pressure and decreases with increasing temperature. The three phases of BaTiO₃ have bulk moduli of 158.84, 160.23, and 151.99 GPa at 0 K and zero pressure; these values also agree with those obtained from structural optimizations (Table 2). Fig. 9 illustrates the correlation between the bulk modulus and temperature at different pressures for all three phases of BaTiO₃. The graph shows that the Bulk modulus, B, which measures resistance to homogeneous compression, decreases with increasing temperature. The trend is observed in all the pressure conditions analyzed in this study. It is also noted that increasing pressure increases the Bulk modulus at a given temperature. As indicated by the calculated bulk modulus values, the orthorhombic phase is comparatively more compressible than the cubic and tetragonal phases, reflecting enhanced lattice flexibility associated with ferroelectric distortion. The way in which a material’s dimensions vary in response to temperature deviations is determined by its thermal expansion coefficient. It measures how much a material’s size changes fractionally for every unit change in temperature.

The computation of thermal expansion (×10⁵ K⁻¹) with pressure at different temperatures for the cubic, orthorhombic, and tetragonal phases of the BaTiO₃ compound is shown in Fig. 10. At 0 K, the thermal expansion coefficient remains constant for all three phases, regardless of applied pressure, indicating that it is unaffected by pressure at absolute zero. However, as the temperature rises, the ‘α’also increases, which means that the material expands more with rising temperature. Conversely, as pressure increases, ‘α’ decreases, indicating that higher pressure reduces the rate of expansion with temperature. The deviations of thermal expansion ‘α’ (×10⁵ K⁻¹) concerning temperature at different pressures for cubic, orthorhombic, and tetragonal phases of the BaTiO₃ compound have been shown in Fig. 11. The calculated thermal expansion values are 3.389 × 10⁻⁵ K⁻¹, 3.803 × 10⁻⁵ K⁻¹, and 3.372 × 10⁻⁵ K⁻¹ for the three phases of BaTiO₃ at 300 K and zero pressure. Between 0 and 200 K, the graph ascends dramatically. At 300 K, it steadily rises with a slight slope, especially at high pressures.

A noteworthy component that can be articulated at the macrocosmic and microcosmic levels is the Grüneisen parameter (γ). It is related to many thermal parameters in the macroscopic framework, such as heat capacity, isothermal bulk modulus, and thermal expansion coefficient (α). Still, it is also related to realizing the atom’s vibrational frequency in the microscopic framework [54]. It is a measure of how the vibrational frequencies of phonons change with the volume of the crystal. Thermal expansion and specific heat are linked through γ. As shown in Fig. 12, for the cubic, orthorhombic, and tetragonal phases of BaTiO₃, a decrease is clearly observed as pressure increases in all three phases, suggesting that higher pressures reduce the sensitivity of vibrational modes to volume changes.

On the other hand, as temperature increases, γ increases, suggesting that vibrational modes of materials are more responsive to changes in temperature. The increase of both parameters is consistent across all three materials and supports the correlation with pressure and temperature. In addition, the relationship between pressure and temperature will be examined. Fig. 13 provides evidence for the relationship between the Grüneisen parameters and temperature-pressure for all three phases, with an increasing trend of Grüneisen parameter for both when temperature increases and pressure decreases. The values and comparison of the Grüneisen parameter at 0 GPa of BaTiO₃ are provided in Table 4.

The acoustic Debye temperature (‘Θ_D’) significantly influences the quasi-harmonic Debye model, which describes how the behaviour of materials depends on temperature, particularly in the context of lattice vibrations and thermal properties. The temperature-dependent Θ_D value is needed to correctly model a material’s heat capacity, thermal expansion, and other thermodynamic properties over a wide range of temperatures. The variation of Θ_D for temperature and pressure is shown in Fig. 14. It shows that, as pressure increases, Θ_D increases, indicating that the vibrational frequencies of the lattice increase with additional pressure. In addition, Θ_D decreases as the temperature increases, which shows that the vibrational frequencies of the lattice of a material decrease with increasing temperature. Thus, there is an inverse relationship between Θ_D and temperature and a direct relationship between Θ_D and pressure. The variation of Θ_D with temperature at different applied pressures is shown in Fig. 15 and illustrated as increasing with increasing pressure and decreasing with increasing temperature, which supports the conclusion that the particles’ vibrational frequency is affected by both temperature and pressure, but in opposite manner. The Θ_D for BaTiO₃ at 0 GPa is shown in Table 5.

When seeking to characterize the vibrational properties of a given material for use in various contexts, it is also essential to account for the heat capacity associated with both constant pressure (C_p) and constant volume (C_v) for that particular substance. Fig. 16 illustrates the relationship between the C_p and C_v for the considered compound as a function of pressure and temperature. As shown in Fig. 16, C_v increases rapidly with temperature, below 400 K, and is proportional to T³. The increase starts to taper off at temperatures >400 K. At >400 K, C_v stabilizes and reaches the maximum limit for all phases, known as the ‘Dulong-Petit limit’. The calculated heat capacity at 300 K and 0 GPa for the three different phase states of BaTiO₃ were 90.58 J/mol·K for the cubic phase, 101.97 J/mol·K for the orthorhombic phase, and 88.80 J/mol·K for the tetragonal phase. These values provide essential information on the thermal properties of a material and its thermodynamic behavior.

The thermodynamic trends obtained from the quasi-harmonic Debye model are fully consistent with the experimentally established phase stability sequence of BaTiO₃. Experimentally, BaTiO₃ undergoes well-known temperature-driven phase transitions from the rhombohedral/orthorhombic phase at low temperatures to the tetragonal phase at intermediate temperatures and finally to the cubic paraelectric phase at high temperatures. Our calculations reveal that the orthorhombic phase exhibits the highest bulk modulus and relatively lower thermal expansion at low temperatures, indicating enhanced lattice rigidity and thermodynamic stability in this regime. As temperature increases, the bulk modulus decreases while thermal expansion, Grüneisen parameter, and heat capacity increase, reflecting progressive lattice softening and enhanced anharmonic phonon activity. These effects are most pronounced for the tetragonal and cubic phases, consistent with their experimentally observed stabilization at elevated temperatures. Moreover, the higher Debye temperature of the cubic and tetragonal phases under pressure indicates stiffened phonon modes, which agrees with experimental observations of pressure-induced stabilization of higher-symmetry phases in BaTiO₃. Although the quasi-harmonic Debye approach does not explicitly capture soft-mode ferroelectric instabilities responsible for first-order phase transitions, the smooth evolution of bulk modulus, thermal expansion, Debye temperature, and Grüneisen parameter with temperature and pressure provides a thermodynamically coherent framework that qualitatively supports the experimentally observed phase stability hierarchy. Thus, the present thermodynamic results are in good agreement with experimental phase transition trends and validate the reliability of the employed methodology within its known theoretical limits.

Anharmonic effects and phonon free energies are indeed important for a precise description of BaTiO₃ phase stability, particularly close to transition temperatures. While their inclusion may shift transition points and adjust relative free energies, the overall stability trends obtained in the present quasi-harmonic Debye analysis remain physically meaningful and consistent with experimental observations.

BaTiO₃ is a prototypical ferroelectric material in which soft optical phonon modes and strong anharmonic lattice dynamics drive phase transitions. Consequently, strictly harmonic or quasi-harmonic descriptions are known to be insufficient for capturing the microscopic origin of these transitions, particularly in the immediate vicinity of the orthorhombic–tetragonal and tetragonal–cubic phase boundaries. In this sense, the quasi-harmonic Debye approximation does not explicitly account for soft-mode condensation, higher-order phonon–phonon interactions, or dynamical disorder effects that dominate near critical temperatures.

Nevertheless, the quasi-harmonic Debye model remains a valid and widely used approximation for describing the global thermodynamic response of BaTiO₃ over broad temperature and pressure ranges, provided that the results are interpreted away from the immediate transition region. The method incorporates volume-dependent vibrational contributions to the free energy through the Debye temperature and Grüneisen parameter, enabling a physically consistent description of macroscopic quantities such as the bulk modulus, thermal expansion, heat capacity, and Debye temperature. These quantities evolve smoothly with temperature and pressure and are not solely governed by soft optical modes.

Importantly, numerous first-principles studies have demonstrated that while explicit anharmonic phonon treatments are required to reproduce ferroelectric transition temperatures quantitatively, quasi-harmonic approaches reliably capture relative phase stiffness, compressibility trends, and pressure-induced stabilization of higher-symmetry phases in BaTiO₃. In the present work, the agreement between quasi-harmonic bulk moduli and those obtained from static structural optimizations further supports the approach’s internal consistency.