## 3.1 Structural examination

Initially, structure is built by putting the atomic positions. After building the structure optimizes and obtained the lattice parameter. The atomic configurations for the elements being examined are as follows: The electron configurations of the given elements are as follows: Sodium (Na): 1s2 2s2 2p6 3s1, Lithium (Li): 1s²2s¹, Bismuth (Bi): 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 4f14 5s2 5p6 5d10 6s2 6p3, and Florine (F): 1s2 2s2 2p5. The lattice parameter was found to be NaBiF3 and LiBiF3 3.78Å and 4.11Å respectively. Concern physical characteristics were examined in terms of Murnaghan state equation which maintains the whole energy of concern materials. Overall results verified the volume of compounds are 54.01 (Å)3 and 69.42 (Å)3 NaBiF3 and LiBiF3 respectively. Unit cell of concern compounds are mentioned in Fig. 1. This study focuses on the optimization of the of NaBiF3 and LiBiF3 compounds. The energy computed at the equilibrium volume of the cell is denoted as eV, which is dependent on the volume. It is worth noting that the previous research lacks theoretical or experimental evidence for the comparative analysis of NaBiF3 and LiBiF3 compounds. Consequently, subsequent measurements corroborate the findings obtained from our initial measurements. The energies of formation were observed which are NaBiF3 and LiBiF3 are − 1.62 eV and − 1.57 eV, respectively. Moreover, the Goldschmidt tolerance factor, denoted as t, is employed in the context of the perovskite stability structure, and its definition is as follows:

$$t=\frac{{R}_{A}+{R}_{b}}{ \sqrt{2 } \left({R}_{B}+{R}_{b}\right)}$$

1

**3.2 Band Structure and DOS (density of States)**

The electronic profile is regarded as a distinguishing characteristic of materials. Additionally, these band structures illustrate the regions within the band gaps where electrons are localized or not. The valence band (VB) and the conduction band (CB) represent distinct energy bands within a material. The valence band (VB) is situated below the Fermi energy (EF), whereas the conduction band (CB) is located above this energy level. Given that all observations are conducted at absolute zero temperature (0 K). If the valence band maximum (VBM) coincides precisely with the conduction band minimum (CBM), the bandgap will exhibit a direct band gap while do not lie at the same point called indirect band gap. The valence band maximum (VBM) and conduction band minimum (CBM) of NaBiF3 exhibit perfect alignment, indicating a direct band gap in the ternary complex. On the other hand LiBiF3 shows indirect band gap. At absolute zero temperature (0 K), the material under consideration exhibit semiconductor properties. The band gap of LiBiF3 was calculated to be 1.71 eV and NaBiF3 possesses a bandgap of 2.14eV. Figures 2(a) and (b) illustrate the electrical band structure of NaBiF3 and LiBiF3 respectively. For further investigation the total density was considering the band structure mentioned in Fig. 3 .The highest peak is observed at a value of -19.805eV for LiBiCl3, while the secondary peak occurs at -9.85eV for NaBiCl3. Figures 4 (a) and (b) depicts illustrating the partial density of states (PDOS). The value of PDOS of NaBiCl3 and LiBiCl3 are found at d-state which means that the major contribution of electron is found in d-states tjat is present in valance band.

Table 1

Electronic band gap and lattice parameter

Lattice parameter (Bohr) NaBiF3 and LiBiF3 | Calculated (GGA) 5×10− 5 eV (presented work) NaBiF3 and LiBiF3 Volume | Other calculations [20] | The band gap (presented work) | Other studies [20] |

a = 3.78, b = 3.78, c = 3.78 | 54.01 (Å)3 | a = 4.78 Å, b = 4.78Å, c = 4.78 Å | 2.14 eV | 2.65 eV |

a = 4.11, b = 4.11 c = 4.11 | 69.42 (Å) 3 | a = 4.82 Å, b = 4.82Å, c = 4.82 Å | 1.71 eV | 3.12 eV |

## 3.2.1. Population Investigation

Before going to check the mechanical properties it is very important part to investigate the nature of the material like covalent or ionic bond. At this phenomenon the Mulliken population was used to calculate the bond nature of the materials. If value is nearly to zero of Mulliken population (MP) then bond nature of the material is to be considered ionic nature. If value is more positive then it also considers more ionic nature of the compounds. MP verifies that where the material is bonding and anti-bonding nature. If values is positive and negative it considered to be bonding and anti-bonding of the material respectively [21]. The value obtained from our calculation is 1.44 and 1.09 for NaBiF3 and LiBiF3 respectively. In calculation we noticed that NaBiF3 have more positive value which guaranties that it is more covalent bond as compare to the LiBiF3.

## 3.3. Mechanical properties

The arrangement of crystals is influenced by its elastic parameter values, which provide vital data on the mechanical features of the crystal nature. Here, the somatic features of materials, like that stiffness, and solidity are examined using the three elastic constant values, such as C44, C12, and C11. Table.2 shows the elastic parameters Cij. The value of Bulk modulus are determined by the relation of constant values of elastic parameters B= (C11 + 2C12)/3. All concern parameter satisfied that concern compounds are mechanical stable. Elastic constant values are mentioned in Table2. Table 3 displays the Pugh's index ratio (B/G), Poisson's ratio (v), and Young's modulus (E). Utilizing the B/G ratio, one may ascertain the brittleness and ductility of compounds [22]. The standardized value to check the brittle properties of the materials is 1.75 which means that if value is higher than ductile otherwise it considered to be brittle properties [18]. In calculation it is reported that NaBiF3 and LiBiF3 meet Pugh's requirement for ductility nature. Additionally, Poisson's ratio (σ) verified that material is ductile when 0.26 values are greater of concern compounds if less it consider brittle. In our reports it was observed that NaBiF3 and LiBiF3 are ductile. Table 2 demonstrates the summary of mechanical properties. Anisotropic factor A, are further evaluated by applying elastic constant. If values of anisotropic values are one then material is isotropic if deviate then the material is anisotropic properties. In calculation results it was reported that compounds shows the anisotropic properties [23].

Table 2

The calculated elastic constants (Cij) of NaBiF3 and LiBiF3 perovskites

Compounds | C11 | C12 | C44 |

NaBiF3 | 23.001 | 11.64 | 17.57 |

LiBiF3 | 6.866 | 4.592 | 3.740 |

Table 3

Calculated mechanical properties of NaBiF3 and LiBiF3

Compounds | BR | BV | B (GPa) | G (GPa) | Y (GPa) | σ | A | Pugh ratio |

NaBiF3 | 26.88 | 26.88 | 26.88 | 12.79 | 4.55 | 0.31 | 0.49 | 1.88 |

LiBiF3 | 5.72 | 5.72 | 5.72 | 4.43 | 6.18 | 0.29 | 0.46 | 1.94 |

$$\left\{\begin{array}{c}{C}_{11}-{C}_{12}> 0 \\ {C}_{11} > 0, {C}_{44}> 0\\ {C}_{11} + 2{C}_{12}> 0 \end{array}\right.$$

2

B = BH = \(\frac{1}{2}\) (BV+BR) ;G = GH = \(\frac{1}{2}\) (GV+GR) (3)

Values are evaluated by the following relationship.=

BV = \(\frac{{C}_{11} +{C}_{12}}{3}\)

GV = \(\frac{{C}_{11}-{C}_{12}+3{C}_{44}}{5}\)

GR = \(\frac{{(C}_{11}- {C}_{12 }){C}_{44}}{{4C}_{44}+3({C}_{11}-{C}_{12})}\)

Y = \(\frac{9BG}{3B+G}\)

ν =\(\frac{3B-2G}{2(3B+G)}\) (5)

## 3.4 Optical properties

Electromagnetic waves are being important rule in photoelectric properties. In optical properties we examine the conductivity, dielectric function ԑ(ω), Reflectivity *R*(ω), refractive index *n*(ω), absorption coefficient *I*(ω), and *L*(ω)energy function of our concern compounds. Wave-matter interactions, are responsible for all of these properties. To investigate optical qualities, one uses the dielectric functions ε(ω), which expressed as follows:

ԑ(ω) = ԑ1(ω) + iε2(ω) (6)

*n*(ω) = [ԑ1(ω)/2 + {ԑ12(ω) + ԑ12(ω)}/2]1/2 (7)

*L*(ω) = -Im (ԑ(ω)−1) = ԑ2(ω)/ ԑ1(ω) 2 + ԑ22 (8)

*I*(ω) = 21/2ω [{ԑ12(ω)+ ԑ12(ω)1/2 - ԑ1 (ω)}]1/2 (9)

*R*(ω) = (n + ik – 1)/ (n + ik + 1) (10)

The real and imaginary components of the dielectric equation are denoted as ε1(ω) and ε2(ω), respectively. Eq. (6) elegantly of the real and imaginary components. The real component of the quantity represents the manifestation of material polarization, while the imaginary part signifies the dissipation of energy, commonly referred to as the loss function.

The dielectric function ε(ω) was investigated in order to ascertain the response of the compounds to incident radiation. Whether imaginary or real, exhibits variations in its components as a consequence of the energy possessed by the incident photon. Upon careful observation, it has been determined that exhibiting slight fluctuations in response to variations in the frequency of electromagnetic waves. The reflectivity of NaBiF3 exhibits a maximum peak at 5.71 eV, where LiBiF3 demonstrates a maximum peak at 17.81 electron volts. At zero electron volts, the reflectance of NaBiF3 is observed to be 0.1237, whereas LiBiF3 exhibits a reflectance of 0.09223. The reflectivity exhibits a gradual increase, progressing from 0eV to 0.09223, subsequently reaching 0.958 and for LiBiF3 as illustrated in Fig. 6(a). The NaBiF3 and LiBiF3 composites are employed for the determination of the absorption coefficient I(ω) and dielectric function. In the compounds NaBiF3 and LiBiF3, the primary absorption occurs at energy levels of 6.91 electron volts (eV) and 4.58 eV, respectively. The maximum absorption peaks for NaBiF3 and LiBiF3 are observed at energy levels of 19.72 eV and 16.52 eV, respectively. The process of absorption commences at an energy level of 2.809 electron volts (eV) for the compound NaBiF3 and LiBiF3, which is 1.9 electron volt which is mentioned in Fig. 6 (b) [24]. Another fundamental characteristic is the refractive index, a quantity that quantifies the phenomenon of light ray deflection as it transitions from one denser to another denser medium. The incident light ray shall undergo refraction upon encountering these composites, specifically at the point of maximum refractive index (n). For NaBiF3, the refractive index is measured to be 2.11, corresponding to a peak energy of 5.19 eV and for LiBiF3 which is 3.1 at 2.12 eV. It was observed that the phenomenon of light ray bending exhibits a gradual increase when reaching the value of LiBiF3, from 2.5 to 3.1. On the other hand, NaiBiF3 demonstrates a refractive index of is slightly increases from 1.75 to 2.12. With an initial imaginary component k of NaBiF3 k at 1.4eV and for LiBiF3 which highest value at 1.92eV which is mention in Fig. 6 (c). Additionally, in the optical properties, one must consider the dielectric function, which is a fundamental factor. This function quantifies the relationship between the permittivity of a substance and the permittivity of free space. Properties of dielectric shall also elucidate the phenomenon of light polarization induced by charges, which the material is capable of accommodating. The real module of the dielectric parameters pertaining to NiBiF3 manifests at an energy of 4.079 eV, while for LiBiF3, it occurs at 2.952 electron volts. Imaginary parameter which is dominant peak is observed for NaBiF3 which is energy level of 5.92 eV, while the maximum peak with LiBiF3 occurs at 9.44 electron volts. In the case of NaBiF3 and LiBiF3, it is observed that the imaginary factor initiates at 0eV levels value is 2.753 and 6.682, respectively. Furthermore, it is noteworthy that this imaginary component exhibits a gradual increment as the energy increases at maximum 5.92 for NaBiF3 and for LiBiF3 which is 9.44 eV. The conductivity, which characterizes the material's ability to conduct electric charges, as depicted in Fig. 6 (e). The dominant component of the conductivity peak observed in NaBiF3 commences at an energy level of 10.19 electron volts (eV) and exhibits trend as the energy decreases up to 24.267 eV. Conversely, in the case of LiBiF3, the conductivity maximum peak occurs at an energy level of 8.48 eV and after this it was noted the peak trend slightly decrease with the energy electron volt up to 15.535 eV. The loss function values for NaBiF3 and LiBiF3 at 0 electron volts (eV) are zero. At this energy level, no dissipation of energy and the substance absorbs zero energy. The rises of energy, the loss of energy in the matter gradually increases. The highest energy loss occurs at 21.016 eV and 21.150 eV and for LiBiF3 which is 16.01eV. Beyond these energy levels, further increases in energy result in a decrease in energy loss within the matter. The aforementioned perovskite compounds are practical in electronic devices domains based on their optical characteristics.