Abstract: Based on density functional theory, a systematic investigation of the geometric structure, electronic structure, and optical properties of Ba₃(PO₄)₂, Pb₃(PO₄)₂, BiPO₄, and YPO₄, which are composed of P—O coordination tetrahedra with Y, Ba, Pb, and Bi, has been conducted. The research indicates that the substitution of metal atoms can alter thestructural framework of compound, thereby regulating its band gap and optical properties, providing an effective approach for designing materials with excellent comprehensive performance. All four compounds are indirect band gap materials with relatively wide band gaps, with band gaps of 5.188, 3.879, 3.870, and 4.886 eV for Ba₃(PO₄)₂, Pb₃(PO₄)₂, BiPO₄, and YPO₄, respectively. According to Mulliken population analysis, the cations Ba, Pb, Bi, and Y form O—X (X = Ba, Pb, Bi, and Y) bonds with oxygen, which have similar bond lengths and exhibit strong ionic bond characteristics. The bottom of the conduction band in the four compounds is occupied by the outermost orbitals of the metal cation, and the main contributor to the top of the valence band is the O-2p orbital. The 2p orbitals of the oxygen atoms also exhibit strong localization near the Fermi level, and the P-3p orbitals bond with the O-2p orbitals, showing strong covalent P—O bonds. The birefringence of the four crystals Ba₃(PO₄)₂, Pb₃(PO₄)₂, BiPO₄, and YPO₄ are 0.003 7, 0.027 0, 0.059 0 and 0.149 0, respectively, with BiPO₄ and YPO₄ showing the highest birefringence and anisotropy among the four systems, which is due to the asymmetric crystal structural framework caused by different cations.

Key words: phosphate crystal, first-principle, electronic structure, birefringence, band gap