Structural, Elastic, Electronic and Optical Properties of SrTMO3 (TM = Rh, Zr) Compounds: Insights from FP-LAPW Study

The structural, mechanical, electronic and optical properties of SrTMO3 (TM = Rh, Zr) compounds are investigated by using first principle calculations based on density functional theory (DFT). The exchange-correlation potential was treated with the generalized gradient approximation (GGA) for the structural properties. Moreover, the modified Becke-Johnson (mBJ) approximation was also employed for the electronic properties. The calculated lattice constants are in good agreement with the available experimental and theoretical results. The elastic constants and their derived moduli reveal that SrRhO3 is ductile and SrZrO3 is brittle in nature. The band structure and the density of states calculations with mBJ-GGA predict a metallic nature for SrRhO3 and an insulating behavior for SrZrO3. The optical properties reveal that both SrRhO3 and SrZrO3 are suitable as wave reflectance compounds in the whole spectrum for SrRhO3 and in the far ultraviolet region (FUV) for SrZrO3.


Introduction
Perovskite structure solids are of great interest in materials science due to their simple crystal structure and their different unique properties such as ferromagnetism, ferroelectricity, superconductivity, thermoelectricity and colossal magneto resistance [1]. Recently, many experimental and theoretical works have been devoted to perovskite oxides RE-TM-O 3 (RE represents rare earth and TM represents transition metal elements) [2][3][4].

Computational Details
The computation has been performed by using FP-LAPW method integrated in Wien2k code [11]. In this method, the unit cell is partitioned into two types of regions: (i) atomic-centered muffin-tin (MT) spheres with radius R α ; and (ii) the remaining interstitial region [18]. The expansion of spherical harmonics is defined within a muffin-tin sphere of radius R MT around each nucleus. The Muffin-Tin radius R MT values used are 2.5, 1.95, 1.67 atomic units (a.u) for Sr, Rh and O atoms for the SrRhO 3 compound and 2.5, 1.8 and 1.5 a.u for Sr, Zr and O atoms for the SrZrO 3 compound. The plane wave cut-off parameter R MT × K max is equal to 8. The cut-off energy to separate the core states from valence states is set to be −9 Ry. Inside the sphere, Fourier expanded up to G max = 14 with a cut-off l max = 12 and 35 k-points in the irreducible Brillion zone with a grid of 10 × 10 × 10 Monkhorst-Pack meshes [19] (equivalent to 1000 k points in full Brillion zone) which are used to obtain self-consistency for SrRhO 3 and SrZrO 3 compounds to be better than 0.1 mRy. The Perdew-Burke-Ernzerhof-generalized gradient approximation (PBE-GGA) is used for the exchange correlation potential [20], while the modified Becke-Johnson potential (mBJ-GGA) [21] is also used to improve the energy band gap of the herein studied compounds.

Structural Properties
Strontium Transition metal oxides (SrTMO 3 ) crystallize in the ABO 3 cubic perovskite structure with the space group Pm-3m (221). The crystal structure of SrTMO 3 (TM = Rh, Zr) is shown in Figure 1. We have optimized the lattice parameters (a) of these compounds by minimizing the total energy. The total energy at different unit cell volumes for SrTMO 3 (TM = Rh, Zr) is shown in Figure 2, the volume versus energy is fitted to the Murnaghan equation of state [22] to estimate the ground-state properties of these compounds such as the lattice constant (a), the bulk modulus (B) and the pressure derivatives of the bulk modulus (B'). These obtained values are listed in Table 1 together with the available experimental and theoretical data for comparison. In general, the computed structural parameters are in good agreement with experimental and theoretical data available in the literature. More precisely, the lattice parameters of SrRhO 3 and SrZrO 3 are found to be 3.976 Å and 4.176 Å, respectively. The present calculations show that the lattice parameter of SrRhO 3 (SrZrO 3 ) is 1.43% (1.63%) larger than the experimental value [23,24]. This guarantees the reliability of the present first-principles computations for the lattice parameters of these compounds using the PBE-GGA method. The calculated bulk modulus for SrZrO 3 is 2.11% larger than the experimental value [25].
respectively. The present calculations show that the lattice parameter of SrRhO3 (SrZrO3) is 1.43% (1.63%) larger than the experimental value [23,24]. This guarantees the reliability of the present first-principles computations for the lattice parameters of these compounds using the PBE-GGA method. The calculated bulk modulus for SrZrO3 is 2.11% larger than the experimental value [25].   respectively. The present calculations show that the lattice parameter of SrRhO3 (SrZrO3) is 1.43% (1.63%) larger than the experimental value [23,24]. This guarantees the reliability of the present first-principles computations for the lattice parameters of these compounds using the PBE-GGA method. The calculated bulk modulus for SrZrO3 is 2.11% larger than the experimental value [25].

Electronic Properties
In this section, we study the electronic properties of SrRhO 3 and SrZrO 3 via calculating the energy band structure and density of states. The calculated band structures along the high symmetry lines in Brillion-zone of SrRhO 3 and SrZrO 3 at zero pressure using PBE-GGA approximation [20] are depicted in Figure 3, where the Fermi level is set at zero eV. In SrZrO 3 , the valence band maximum (VBM) occurs along the M-point symmetry line, while the conduction band minimum occurs along the Γ-point symmetry line with energy band gap 3.69 eV, resulting in an indirect energy band gap (M-Γ) semiconductor.

Electronic Properties
In this section, we study the electronic properties of SrRhO3 and SrZrO3 via calculating the energy band structure and density of states. The calculated band structures along the high symmetry lines in Brillion-zone of SrRhO3 and SrZrO3 at zero pressure using PBE-GGA approximation [20] are depicted in Figure 3, where the Fermi level is set at zero eV. In SrZrO3, the valence band maximum (VBM) occurs along the M-point symmetry line, while the conduction band minimum occurs along the Γ-point symmetry line with energy band gap 3.69 eV, resulting in an indirect energy band gap (M-Γ) semiconductor.  The calculated energy band gap is tabulated in Table 2 along with the available previous theoretical value [26]. The calculated energy gap is larger than the theoretical value by 0.32 eV [26]. To the best of our knowledge, there is no experimental value available to compare with. The usual trend of the modified Becke-Johnson potential (mBJ-GGA) method is enlarging the energy band gap values (Eg) of the semiconductor and insulator materials which makes Eg comparable to the experimental results [21]. The band structures of SrRhO3 and SrZrO3 compounds using mBJ-GGA are displayed in Figure 4. For the SrZrO3 compound, the minimum energy gap within mBJ-GGA is still indirect with the same direction as the PBE-GGA approach, but its value increases by about 0.85 eV to become 4.54 eV. SrZrO3 is classified as an insulator within the mBJ-GGA method. SrRhO3 has a metallic nature with no energy gap within the two approaches PBE-GGA and mBJ-GGA.  The calculated energy band gap is tabulated in Table 2 along with the available previous theoretical value [26]. The calculated energy gap is larger than the theoretical value by 0.32 eV [26]. To the best of our knowledge, there is no experimental value available to compare with. The usual trend of the modified Becke-Johnson potential (mBJ-GGA) method is enlarging the energy band gap values (E g ) of the semiconductor and insulator materials which makes E g comparable to the experimental results [21]. The band structures of SrRhO 3 and SrZrO 3 compounds using mBJ-GGA are displayed in Figure 4. For the SrZrO 3 compound, the minimum energy gap within mBJ-GGA is still indirect with the same direction as the PBE-GGA approach, but its value increases by about 0.85 eV to become 4.54 eV. SrZrO 3 is classified as an insulator within the mBJ-GGA method. SrRhO 3 has a metallic nature with no energy gap within the two approaches PBE-GGA and mBJ-GGA.        To make a deeper analysis of the bonding nature of our titled compounds, the wave function obtained from a final Wien2k calculation at the optimal geometry has been analyzed in the critic program (version 1.0). Critic is a full-edged program for the topological analysis of solid-state electron densities This program uses the quantum theory of atom in molecule (QTAIM) to make a topological analysis of the electronic density (ρ) of a crystal. By means of QTAIM, we can automatically locate all critical points (CP's) of the electronic density rising from a nil flux of the electron density gradient To make a deeper analysis of the bonding nature of our titled compounds, the wave function obtained from a final Wien2k calculation at the optimal geometry has been analyzed in the critic program (version 1.0). Critic is a full-edged program for the topological analysis of solid-state electron densities This program uses the quantum theory of atom in molecule (QTAIM) to make a topological analysis of the electronic density (ρ) of a crystal. By means of QTAIM, we can automatically locate all critical points (CP's) of the electronic density rising from a nil flux of the electron density gradient To make a deeper analysis of the bonding nature of our titled compounds, the wave function obtained from a final Wien2k calculation at the optimal geometry has been analyzed in the critic program (version 1.0). Critic is a full-edged program for the topological analysis of solid-state electron densities.
This program uses the quantum theory of atom in molecule (QTAIM) to make a topological analysis of the electronic density ( ) of a crystal. By means of QTAIM, we can automatically locate all critical points (CP's) of the electronic density rising from a nil flux of the electron density gradient condition [27]. The procedure implemented in critic is to divide into disjoint regions Ω (basins) through the real space approaches. Here, we can generate topological schemes in two or tridimensional plots. It is clearly shown from the topological distribution of the charge density in Figure 7A, that the charge distribution in the SrZrO 3 compound is spherically symmetrical. This suggests that all electronic charges are localized around anions-a typical ionic bonding. The same trend is also displayed for the second compound with some small difference in the distribution of due to the charge transfer, which differs in the two titled compounds (see Figure 7C). We also present in Figure 7B,D, the molecular graph of our two perovskites.
These plots were done following the character of Wyckoff's, bond CP and ring CP family [28]. The form of the atomic basin generated by the CCP point suggests that both SrRhO 3 and SrZrO 3 are belonging to the R11 family (see Figure 7B,D) [28]. Here, we can define a topological charge of each atomic basin. As results, Q(Sr) = 1.59 electron, Q(Rh) = 1.47 electron and Q(O) = −1.05 electron for SrRhO 3 , and Q(Sr) = 1.61 electron, Q(Zr) = 2.49 electron and Q(O) = −1.37 electron for the SrZrO 3 compound. When we compare these charges with the oxidation number corresponding to each of the atoms, we find that the charge transfer is mainly due to the strontium one, with an equal percentage of 80% in both perovskites. However, we note that replacing the cation Rh with electro-negativity of 2.2 by the Zr one with low electro-negativity of 1.4 has a negligible effect on the nature of the bonding. The SrZrO 3 has a more ionic trend. However, even the constituent cations show different charge transfer (Zr and Rh are transferred by 62% and 37%) and each of them varies as strongly as the oxide ions. The latter adapt their charges according to electro-neutrality requirements. There is a second topological index that has been proposed as a global measurement of the degree of metallicity, the electron density flatness defined as f = ρ min where ρ min c is the minimal electron density found on the unit cell (it necessarily corresponds to a critical point) and ρ max b is the maximal electron density among bond critical points. This flatness has values close to one on common metallic compounds and close to zero on localized bonding compounds [29]. The delocalization of electrons (flatness) in the SrRhO 3 and SrZrO 3 compounds are respectively equal to 80.61% and 5.62%. This suggests that the global chemical behavior of SrRhO 3 is metallic, whereas SrZrO 3 is nonmetallic. To go further in the characterization of the bonds nature of the titled compounds we have employed another tool named the electron localization function (ELF) [30], which is based on a measure of the likelihood of finding an electron in the neighborhood space of a reference electron located at a given point and with the same spin. Taking into account that the homogeneous electron gas has an ELF value of 0.5, valence electrons of metals should deviate very slightly from this quantity. The ELF isosurfaces, as well as the one-dimensional projections of Sr-O-Rh and Sr-O-Zr bond paths of the investigated perovskites are depicted in Figure 8. It is strikingly clear from the plot that O-Zr and O-Rh are different. The integration of population on the ELF attractor of the SrZrO 3 compound shows two types of Zr-O bonds, the first one has ELF maxima equal to 0.78 and the second to 0.83; the former has the negligible population, but the latter has 0.95 electrons. The integration gives a number of the delocalized attractor as lone pair bonds around the Sr and Zr cations; the population in this attractor varies from 1.4 to 1.1 electrons. Regarding the second compound (SrRhO 3 ), no O-Rh bonds have been found, only delocalized lone pair ones with ELF maxima near to 0.5 are reported. Here, we should emphasize that the delocalization of these attractors in the SrRhO 3 compound provides a reliable measure of the delocalization of wave function and its localization in the SrZrO 3 compound.

Elastic Properties
In this subsection, we turn our attention to study the mechanical properties of SrRhO3 and SrZrO3 via calculating their elastic constants. These constants define the properties of material that undergo stress, deform and then recover, returning to its original shape after stress ceases. They

Elastic Properties
In this subsection, we turn our attention to study the mechanical properties of SrRhO3 and SrZrO3 via calculating their elastic constants. These constants define the properties of material that undergo stress, deform and then recover, returning to its original shape after stress ceases. They

Elastic Properties
In this subsection, we turn our attention to study the mechanical properties of SrRhO 3 and SrZrO 3 via calculating their elastic constants. These constants define the properties of material that undergo stress, deform and then recover, returning to its original shape after stress ceases. They have a significant role in finding information about the brittleness, ductility, stiffness and the mechanical stability of the material [31]. The elastic constants require knowledge of the derivative of the energy as a function of the lattice strain. In the case of the cubic system, this strain is chosen in such a way that the volume of the unit cell is preserved. Thus, for the calculation of elastic constants C 11 , C 12 and C 44 , for these compounds we have used the method developed by Morteza Jamal [32] and integrated in Wien2k code as the IRelast package (Cubic-elastic_13.2). The calculated C ij constants are listed in Table 3. Our computed C ij data are in reasonable agreement with previous theoretical results. In view of Table 3, it can be noticed that the calculated values of the elastic modulus C 11 , which are related to the unidirectional compression along the principal crystallographic directions, are much higher than that of C 44 , which represent the resistance to the shear deformation, indicating the weak resistance to the shear deformation compared to the resistance to the unidirectional compression. The mechanical stability of a cubic material requires that its independent elastic constants should satisfy the following Born's stability criteria [33,34]: From Table 3, we can see that all required conditions given in the above Equation (1) are simultaneously satisfied, which clearly indicates that the SrRhO 3 and SrZrO 3 are mechanically stable.
The three elastic constants C 11 , C 12 and C 44 are estimated from first-principles calculations for SrRhO 3 and SrZrO 3 single-crystals. However, the prepared materials are in general polycrystalline, and therefore it is important to evaluate the corresponding moduli for the polycrystalline species using the Hill's approach [35]. In this approach, the effective modulus for polycrystals could be approximated by the arithmetic mean of the two well-known bounds for monocrystals according to Voigt [36] and Reuss [37]. Then, for the cubic system, the shear modulus S in the mentioned approximations: Voigt (V), Reuss (R) and Hill (H) are calculated from the elastic constants of the single crystal, in the following form: To compute the Young's modulus (Y), Poisson's ratio (ν), and the anisotropic factor (A), the following equations are used, respectively: The computed bulk modulus, shear modulus, Young's modulus, Poisson's ratio and anisotropic factor are listed in Table 3 along with the theoretical results [14,38]. The bulk modulus and shear modulus can be used to measure the material hardness [39]. In general, when S H values increase, the materials become stiffer. From the obtained values of shear modulus S H one can remark that SrZrO 3 is stiffer than SrRhO 3 . To categorize the brittle and ductility behaviors of a material, the ratio of the bulk modulus to the shear modulus, B/S, an empirical relationship related to the plastic and elastic properties of the material, is used. According to Pugh [40], a high B/S value is associated with ductility, while a low value is consistent with brittleness. The critical value that separates the two behaviors has been determined to be 1.75. The results listed in Table 3 clearly indicate that the SrRhO 3 (SrZrO 3 ) compound has B/S H ratio higher (smaller) than the critical value of 1.75, which classifies SrRhO 3 (SrZrO 3 ) compound as ductile (brittle) material. In addition, to identify the materials as ductile or brittle, we also applied the Cauchy's pressure rule defined as the difference between the elastic constants C 12 -C 44 [41]. According to this rule, if the Cauchy's pressure is positive (negative), the material will be ductile (brittle) in nature. As shown in Table 3, It is seen that the Cauchy's pressure is positive for SrRhO 3 and negative for SrZrO 3 , confirming the ductile nature for SrRhO 3 and the brittle behavior for the SrZrO 3 compound. We may also refer to Frantsevich et al. [42] who distinguishes the ductility and brittleness of materials in terms of Poisson's ratio (ν). According to this rule, if the Poisson's ratio is less than 0.26, the material will be brittle in nature; otherwise, the material will be ductile. As shown, the computed Poisson's ratiosare 0.30 for SrRhO 3 and 0.23 for SrZrO 3 , categorizing SrZrO 3 as brittle compounds and SrRhO 3 as ductile compounds. These results exactly agree with the results of the B/S ratio and Cauchy's pressure.
The anisotropy factor is an important parameter to measure the degree of materials anisotropy; also, it has a significant usage in engineering science to inspect the potential of micro-cracks in the material [43,44]. For completely isotropic materials, the anisotropy factor A takes the value of the unity and the deviation from unity measures the degree of elastic anisotropy. The calculated values of the anisotropic factor A are found to be equal to 1.85 for SrRhO 3 and 0.71 for SrZrO 3 , suggesting that both compounds are anisotropic in nature and SrRhO 3 is characterized by a profound anisotropy.

Optical Properties
Since the investigated compounds have cubic symmetry, we need to calculate only one dielectric tensor component to completely characterize their linear optical properties. The frequency-dependent complex dielectric function ε(ω) = ε 1 (ω) + iε 2 (ω); where ε 1 (ω) and ε 2 (ω) are the real and imaginary components of the dielectric function, respectively; is known to describe the optical response of the medium at all photon energies E = ω, using the formalism of Ehrenreich and Cohen [45].
Complex dielectric function can be derived from the definition given by Hedin [46] r, t, r , t = δ r − r δ t − t − P r, t, r , t v r − r dr (6) where P is the polarization propagator and is the Coulomb interaction; P can be given by the following form: where V is the unit cell volume, f 0 is the Fermi distribution function and ε k is the single particle energy. The matrix element can be given by: M n,n' (k,q) = u nk |e −iq,r |u n'k , q is the wave vector of light and it is much smaller than the wave vector of electrons in the system; the matrix elements M n,n' (k,q) with small q can be given by: The sum over n' and n must be split into two terms, one with n' = n corresponding to intra-band electronic transitions, and the second with n' = n, corresponding to inter-band transitions, the intra-band part of the dielectric function (ω, 0) can be given by: while the inter band part can be written as where n' = n in Equation (10). The imaginary part of the ε(ω) in the long wavelength limit has been obtained directly from the electronic structure calculation, using the joint density of states (JDOS) and the transition moments elements M n,n' (k,q): The integral is over the first Brillouin zone. The real part of ε(ω) can be derived from the imaginary part using the Kramers-Kronig relations.
where P implies the principal value of the integral. The knowledge of the real and imaginary parts of the dielectric function allows the calculation of other important optical functions such as the refractive index n(ω), reflectivity R(ω), extinction coefficient k(ω), energy loss function L(ω) and absorption coefficient α(ω) by using the following expressions [47][48][49]: Real and imaginary parts of the dielectric constant are displayed in Figure 9a-d for the SrRhO 3 compound. The value of the static real part of the dielectric function ε 1 (0) for the SrRhO 3 compound within intra and inter band transition in Figure 9a is negative, while the imaginary part of the static dielectric function ε 2 (0) within intra and inter band transition Figure 9c is positive; this implies two important facts: Firstly, the SrRhO 3 compound has considerable metallic behavior, which agrees with the energy band structure calculations. Secondly, the negative value of (ε 1 ) especially in the energy range 0-1.15 eV in Figure 9a and the highly positive value of ε 2 (ω) at the early beginning of Figure 9c, reveal the loss of light transit. Real dielectric constant within intra band transition for SrRhO 3 in Figure 9b has small peaks; the first two of them are centered at 1.42 eV and 3.73 eV. We see that (ε 1 ) for the SrRhO 3 compound in Figure 9a, it has roots in the 1.15 eV, 1.55 eV, 5.0 eV, 8.45 eV and 13.5 eV. When these roots occur, (ε 1 ) = 0, the compound does not respond to incident light, this fact is mainly due to the Plasmon oscillation. The sharp increase in ε 1 (ω) with intra and inter band transition, in Figure 9a, at an energy range 0-1.15 eV, indicates that the compound does not interact with the incident photons at this energy range. The static dielectric constant ε 1 (0) of SrRhO 3 with inter + intra in Figure 9a and intra band transition Figure 9b are −180 and 32, respectively. Negative value ε 1 (0) of with inter + intra band transition ensures the metallic behavior of SrRhO 3 compound. By taking only the intra band transition into account as seen in Figure 9b,d, both the real and imaginary parts of dielectric constant have positive values, which indicates a mix of metallic and semiconducting behavior for the SrRhO 3 compound. This implies that for metallic compounds inter and intra band transitions must be taken into account.  Figure 9b has small peaks; the first two of them are centered at 1.42 eV and 3.73 eV. We see that (ε1) for the SrRhO3 compound in Figure 9a, it has roots in the 1.15 eV, 1.55 eV, 5.0 eV, 8.45 eV and 13.5 eV. When these roots occur, (ε1) = 0, the compound does not respond to incident light, this fact is mainly due to the Plasmon oscillation. The sharp increase in ε1(ω) with intra and inter band transition, in Figure 9a, at an energy range 0-1.15 eV, indicates that the compound does not interact with the incident photons at this energy range. The static dielectric constant ε1(0) of SrRhO3 with inter + intra in Figure 9a and intra band transition Figure 9b are −180 and 32, respectively. Negative value ε1(0) of with inter + intra band transition ensures the metallic behavior of SrRhO3 compound. By taking only the intra band transition into account as seen in Figure 9b,d, both the real and imaginary parts of dielectric constant have positive values, which indicates a mix of metallic and semiconducting behavior for the SrRhO3 compound. This implies that for metallic compounds inter and intra band transitions must be taken into account. Optical conductivity is a quantity depending on the inter band and intra band transitions. In Figure 10a,b, the real and imaginary parts of conductivity are illustrated for the SrRhO3 compound, by taking the inter and intra band transition into account; the static real conductivity is high, while it is zero when only intra band transition is taken into account. As the incident light energy increases, the real and imaginary parts of conductivity with the two approaches; intra + inter and intra band transition; both almost have the same behavior.  Optical conductivity is a quantity depending on the inter band and intra band transitions. In Figure 10a,b, the real and imaginary parts of conductivity are illustrated for the SrRhO 3 compound, by taking the inter and intra band transition into account; the static real conductivity is high, while it is zero when only intra band transition is taken into account. As the incident light energy increases, the real and imaginary parts of conductivity with the two approaches; intra + inter and intra band transition; both almost have the same behavior. The refractive index n(ω) and extinct factor k(ω) for the SrRhO3 compound are illustrated in Figures 11a,b and 12a,b, respectively. The SrRhO3 has a high n(0) and k(0) with intra and inter band transition, which indicates metallic behavior for the real and imaginary parts of the dielectric. As the incident light energy increases the n(ω); and k(ω) goes to a lower values. Many peaks are shown in n(ω) and k(ω) spectra, these peaks originate from intra-band transition. The extinction coefficient depends on the amount of absorption of the photon when it propagates in the material, while the refractive index indicates the phase velocity of the electromagnetic wave. The reflectivity spectra of SrRhO3 as a function of energy are shown in Figure 13a. The static reflectivity of SrRhO3 within intra and inter transition is 0.95, while it is 0.5 within intra transition. The reflectivity of SrRhO3 in the low energy region goes down as the incident light energy increases, The refractive index n(ω) and extinct factor k(ω) for the SrRhO 3 compound are illustrated in Figure 11a,b and Figure 12a,b, respectively. The SrRhO 3 has a high n(0) and k(0) with intra and inter band transition, which indicates metallic behavior for the real and imaginary parts of the dielectric. As the incident light energy increases the n(ω); and k(ω) goes to a lower values. Many peaks are shown in n(ω) and k(ω) spectra, these peaks originate from intra-band transition. The extinction coefficient depends on the amount of absorption of the photon when it propagates in the material, while the refractive index indicates the phase velocity of the electromagnetic wave. The refractive index n(ω) and extinct factor k(ω) for the SrRhO3 compound are illustrated in Figures 11a,b and 12a,b, respectively. The SrRhO3 has a high n(0) and k(0) with intra and inter band transition, which indicates metallic behavior for the real and imaginary parts of the dielectric. As the incident light energy increases the n(ω); and k(ω) goes to a lower values. Many peaks are shown in n(ω) and k(ω) spectra, these peaks originate from intra-band transition. The extinction coefficient depends on the amount of absorption of the photon when it propagates in the material, while the refractive index indicates the phase velocity of the electromagnetic wave. The reflectivity spectra of SrRhO3 as a function of energy are shown in Figure 13a. The static reflectivity of SrRhO3 within intra and inter transition is 0.95, while it is 0.5 within intra transition. The reflectivity of SrRhO3 in the low energy region goes down as the incident light energy increases, The refractive index n(ω) and extinct factor k(ω) for the SrRhO3 compound are illustrated in Figures 11a,b and 12a,b, respectively. The SrRhO3 has a high n(0) and k(0) with intra and inter band transition, which indicates metallic behavior for the real and imaginary parts of the dielectric. As the incident light energy increases the n(ω); and k(ω) goes to a lower values. Many peaks are shown in n(ω) and k(ω) spectra, these peaks originate from intra-band transition. The extinction coefficient depends on the amount of absorption of the photon when it propagates in the material, while the refractive index indicates the phase velocity of the electromagnetic wave. The reflectivity spectra of SrRhO3 as a function of energy are shown in Figure 13a. The static reflectivity of SrRhO3 within intra and inter transition is 0.95, while it is 0.5 within intra transition. The reflectivity of SrRhO3 in the low energy region goes down as the incident light energy increases, The reflectivity spectra of SrRhO 3 as a function of energy are shown in Figure 13a. The static reflectivity of SrRhO 3 within intra and inter transition is 0.95, while it is 0.5 within intra transition. The reflectivity of SrRhO 3 in the low energy region goes down as the incident light energy increases, while it increases in the high energy region-the far ultraviolet region (FUV). The absorption coefficient spectra of SrRhO 3 is plotted in Figure 13b, absorption spectra, as shown, begins at the early beginning and increases as the incident photon energy increases with some peaks along the spectrum. The observed peaks in the spectra related to electron transitions from conduction to valence bands, sharp peaks in the absorption spectrum may be accordance with transitions between valance and conduction band (inter band transitions) that can be considerably far from each other. The SrRhO 3 compound is a good absorbent compound because it is a metal compound and it is identical with intra + inter and intra band transition. Peaks in the spectrum of the absorption coefficient are proportional to the peaks in the (ε 2 ) spectrum. The energy loss function L(ω) is describing the energy loss of the fast electrons that propagate inside the material. The energy loss spectrum of SrRhO 3 is depicted in Figure 13c. Energy loss for SrRhO 3 is high along the whole spectrum and it is identical with intra + inter and intra band transition in the high energy region. We observe some peaks, the highest peaks are related to the plasma frequency [50]. From these Figures, the plasma frequency of SrRhO 3 occurs at 10.9 eV and 13.2 eV. while it increases in the high energy region-the far ultraviolet region (FUV). The absorption coefficient spectra of SrRhO3 is plotted in Figure 13b, absorption spectra, as shown, begins at the early beginning and increases as the incident photon energy increases with some peaks along the spectrum. The observed peaks in the spectra related to electron transitions from conduction to valence bands, sharp peaks in the absorption spectrum may be accordance with transitions between valance and conduction band (inter band transitions) that can be considerably far from each other. The SrRhO3 compound is a good absorbent compound because it is a metal compound and it is identical with intra + inter and intra band transition. Peaks in the spectrum of the absorption coefficient are proportional to the peaks in the (ε2) spectrum. The energy loss function L(ω) is describing the energy loss of the fast electrons that propagate inside the material. The energy loss spectrum of SrRhO3 is depicted in Figure 13c. Energy loss for SrRhO3 is high along the whole spectrum and it is identical with intra + inter and intra band transition in the high energy region. We observe some peaks, the highest peaks are related to the plasma frequency [50].  Figure 14a,b displays the calculated real and imaginary parts of the dielectric function for the SrZrO3 compound for a radiation up to 14 eV. It is seen that the calculated linear optical components ε1(ω) and ε2(ω) spectra for SrZrO3 are different from SrRhO3. The calculated ε2(ω) spectra show that the first critical point (threshold energy) of the dielectric function occurs at about 4.52 eV within the mBJ approach (GGA: 3.6 eV) for SrZrO3; the first critical points are comparable with the energy band gap computed from the energy band structure. We can see from Figure 14b, that ε2(ω) displaced to the high energy region within the mBJ approach, but the first peak in ε2(ω) within the GGA approach has a higher value. The threshold's energy is followed by some peaks that originate from direct optical transition between the valence band and conduction bands. The main peak in the absorptive spectra is positioned at about 7.2 eV within mBJ (GGA: 6.2 eV). The real part ε1(ω) gives us information about the material's polarizability; the static dielectric constant ε1(0) of SrZrO3 is 3  Figure 14a,b displays the calculated real and imaginary parts of the dielectric function for the SrZrO 3 compound for a radiation up to 14 eV. It is seen that the calculated linear optical components ε 1 (ω) and ε 2 (ω) spectra for SrZrO 3 are different from SrRhO 3 . The calculated ε 2 (ω) spectra show that the first critical point (threshold energy) of the dielectric function occurs at about 4.52 eV within the mBJ approach (GGA: 3.6 eV) for SrZrO 3 ; the first critical points are comparable with the energy band gap computed from the energy band structure. We can see from Figure 14b, that ε 2 (ω) displaced to the high energy region within the mBJ approach, but the first peak in ε 2 (ω) within the GGA approach has a higher value. The threshold's energy is followed by some peaks that originate from direct optical transition between the valence band and conduction bands. The main peak in the absorptive spectra is positioned at about 7.2 eV within mBJ (GGA: 6.2 eV). The real part ε 1 (ω) gives us information about the material's polarizability; the static dielectric constant ε 1 (0) of SrZrO 3 is 3 within the mBJ approach (GGA: 3.92), as we can see that the mBJ approach has a lower value of ε 1 (0). The value of ε 1 (ω) within mBJ displaced to the higher energy region along the spectrum. The behavior with the high dielectric constant makes SrZrO 3 a possible useful candidate for manufacturing high capacitors [51]. The negative value in the spectra of ε 1 (ω) for the SrZrO 3 compound is located only in the ultraviolet spectrum; and shows the metallic behavior of this compound in the mentioned region. within the mBJ approach (GGA: 3.92), as we can see that the mBJ approach has a lower value of ε1(0). The value of ε1(ω) within mBJ displaced to the higher energy region along the spectrum. The behavior with the high dielectric constant makes SrZrO3 a possible useful candidate for manufacturing high capacitors [51]. The negative value in the spectra of ε1(ω) for the SrZrO3 compound is located only in the ultraviolet spectrum; and shows the metallic behavior of this compound in the mentioned region.
(a) (b) In Figure 15a,b, the real and imaginary parts of conductivity are illustrated for SrZrO3, respectively. It can be seen from Figure 15a, that the real part of conductivity starts to have considerable values at about 4.75 eV within the mBJ approach (GGA: 3.6 eV). From the conductivity and imaginary part of the dielectric function spectrums (Figure 14b), it is shown that both of them start to have a considerable value from approximately the same value. We have also observed some peaks in the conductivity spectra as also observed in the ε2(ω) spectra. The conductivity increases as the material becomes more photon (energy) absorbent.  Figure 16a,b displaying the extinction coefficient and refractive index n(ω). One can remark that the extinction coefficient of SrZrO3 starts to have considerable value at 4.75 eV within mBJ (GGA: 3.6 eV). We can clearly see that k(ω), conductivity and ε2(ω) start to have considerable values at the same point. Some peaks are presented along the spectrum, these peaks are related to the electrons transitions from valence to conduction bands. It is clearly seen that the extinction coefficient and the imaginary part of the epsilon vary in the same way. The static value of the refractive index n(0) in Figure 16b for the SrZrO3 compound is 1.75 within the mBJ approach (GGA: 1.9), which is a small value compared to n(0) for the SrRhO3 compound. In Figure 15a,b, the real and imaginary parts of conductivity are illustrated for SrZrO 3 , respectively. It can be seen from Figure 15a, that the real part of conductivity starts to have considerable values at about 4.75 eV within the mBJ approach (GGA: 3.6 eV). From the conductivity and imaginary part of the dielectric function spectrums (Figure 14b), it is shown that both of them start to have a considerable value from approximately the same value. We have also observed some peaks in the conductivity spectra as also observed in the ε 2 (ω) spectra. The conductivity increases as the material becomes more photon (energy) absorbent. within the mBJ approach (GGA: 3.92), as we can see that the mBJ approach has a lower value of ε1(0). The value of ε1(ω) within mBJ displaced to the higher energy region along the spectrum. The behavior with the high dielectric constant makes SrZrO3 a possible useful candidate for manufacturing high capacitors [51]. The negative value in the spectra of ε1(ω) for the SrZrO3 compound is located only in the ultraviolet spectrum; and shows the metallic behavior of this compound in the mentioned region.
(a) (b) In Figure 15a,b, the real and imaginary parts of conductivity are illustrated for SrZrO3, respectively. It can be seen from Figure 15a, that the real part of conductivity starts to have considerable values at about 4.75 eV within the mBJ approach (GGA: 3.6 eV). From the conductivity and imaginary part of the dielectric function spectrums (Figure 14b), it is shown that both of them start to have a considerable value from approximately the same value. We have also observed some peaks in the conductivity spectra as also observed in the ε2(ω) spectra. The conductivity increases as the material becomes more photon (energy) absorbent.  Figure 16a,b displaying the extinction coefficient and refractive index n(ω). One can remark that the extinction coefficient of SrZrO3 starts to have considerable value at 4.75 eV within mBJ (GGA: 3.6 eV). We can clearly see that k(ω), conductivity and ε2(ω) start to have considerable values at the same point. Some peaks are presented along the spectrum, these peaks are related to the electrons transitions from valence to conduction bands. It is clearly seen that the extinction coefficient and the imaginary part of the epsilon vary in the same way. The static value of the refractive index n(0) in Figure 16b for the SrZrO3 compound is 1.75 within the mBJ approach (GGA: 1.9), which is a small value compared to n(0) for the SrRhO3 compound.  Figure 16a,b displaying the extinction coefficient and refractive index n(ω). One can remark that the extinction coefficient of SrZrO 3 starts to have considerable value at 4.75 eV within mBJ (GGA: 3.6 eV). We can clearly see that k(ω), conductivity and ε 2 (ω) start to have considerable values at the same point. Some peaks are presented along the spectrum, these peaks are related to the electrons transitions from valence to conduction bands. It is clearly seen that the extinction coefficient and the imaginary part of the epsilon vary in the same way. The static value of the refractive index n(0) in Figure 16b for the SrZrO 3 compound is 1.75 within the mBJ approach (GGA: 1.9), which is a small value compared to n(0) for the SrRhO 3 compound. Reflectivity, energy loss and absorption functions are illustrated in Figure 17a-c, respectively. The static reflectivity of SrZrO3 is about 0.07 within the mBJ approach (GGA: 0.11). The static reflectivity of SrRhO3 is more than nine times greater than the reflectivity of SrZrO3. The reflectivity value increases rapidly in the high energy region; far ultraviolet region (FUV); ab = nd for the SrZrO3 compound, indicating that SrZrO3 is suitable as a wave reflectance compound in the far ultraviolet region (FUV). Energy loss and absorption functions are seen in Figure 17b,c; both of them quietly behave in the same way. Both absorption and energy loss for SrZrO3 begin at about 4.8 eV within the mBJ approach (GGA: 3.8 eV). The SrZrO3 is a good absorbent compound, but not in the low energy region; it is good absorbent in the far ultraviolet region, as it shows metallic behavior in the high energy region.  Reflectivity, energy loss and absorption functions are illustrated in Figure 17a-c, respectively. The static reflectivity of SrZrO 3 is about 0.07 within the mBJ approach (GGA: 0.11). The static reflectivity of SrRhO 3 is more than nine times greater than the reflectivity of SrZrO 3 . The reflectivity value increases rapidly in the high energy region; far ultraviolet region (FUV); ab = nd for the SrZrO 3 compound, indicating that SrZrO 3 is suitable as a wave reflectance compound in the far ultraviolet region (FUV). Energy loss and absorption functions are seen in Figure 17b,c; both of them quietly behave in the same way. Both absorption and energy loss for SrZrO3 begin at about 4.8 eV within the mBJ approach (GGA: 3.8 eV). The SrZrO 3 is a good absorbent compound, but not in the low energy region; it is good absorbent in the far ultraviolet region, as it shows metallic behavior in the high energy region. Reflectivity, energy loss and absorption functions are illustrated in Figure 17a-c, respectively. The static reflectivity of SrZrO3 is about 0.07 within the mBJ approach (GGA: 0.11). The static reflectivity of SrRhO3 is more than nine times greater than the reflectivity of SrZrO3. The reflectivity value increases rapidly in the high energy region; far ultraviolet region (FUV); ab = nd for the SrZrO3 compound, indicating that SrZrO3 is suitable as a wave reflectance compound in the far ultraviolet region (FUV). Energy loss and absorption functions are seen in Figure 17b,c; both of them quietly behave in the same way. Both absorption and energy loss for SrZrO3 begin at about 4.8 eV within the mBJ approach (GGA: 3.8 eV). The SrZrO3 is a good absorbent compound, but not in the low energy region; it is good absorbent in the far ultraviolet region, as it shows metallic behavior in the high energy region.

Conclusions
In summary, we have studied the structural, electronic, elastic and optical properties of SrTMO 3 (TM = Rh, Zr) using the FP-LAPW method within the PBE-GGA and mBJ-GGA approaches in the framework of density functional theory. The lattice parameter is found to be in good agreement with experimental and theoretical results. From the band structure, we found that the SrRhO 3 compound has a metallic nature within both BPE-GGA and mBJ-GGA approaches, while SrZrO 3 is found to be an insulator within mBJ-GGA, while it is a wide energy band gap semiconductor within PBE-GGA. The elastic constants (C 11 , C 12 , C 44 ), bulk modulus, shear modulus and Young's modulus are also calculated and discussed. By analyzing the B/S ratio and Poisson's ratio, we found that SrRhO 3 has a ductile nature, while SrZrO 3 has brittle nature. SrZrO 3 has a more ionic trend compared to the SrRhO 3 . The optical properties such as dielectric constant, absorption coefficient, reflectivity coefficient, refractive index, optical conductivity and energy loss function were investigated in the energy range (0-14) eV. According to the dielectric constant, SrZrO 3 has a large static dielectric constant which may make it promising as a good dielectric material; while SrRhO 3 has a negative real dielectric constant which indicates a metallic behavior. It would be interesting to make detailed measurements of the transport and thermodynamic properties of SrRhO 3 under applied fields looking for metamagnetic behavior and non-Fermi-liquid scalings. In conclusion, our theoretical calculations on SrTMO 3 (TM = Rh, Zr) will help to use these materials for practical applications.