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Detailed first-principle calculations of properties in zinc blende quaternary alloy ${\mathrm{\text{B}}}_x{\mathrm{\text{Al}}}_y{\mathrm{\text{In}}}_{1-x-y}\mathrm{\text{N}}$ at various concentrations are investigated using density functional theory (DFT) within virtual crystal approximation (VCA) implemented in alchemical mixing approximation. The calculated bandgaps show direct transitions at $\mathrm{\Gamma }$–$\mathrm{\Gamma }$ and indirect transitions at $\mathrm{\Gamma }$–$X$, which are opened by increasing boron concentration. The density of state (DOS) revealed upper valence band (VB1) domination by $p$-states atoms, while $s$-states dominate the lower valence band (VB2); also, the DOS shows the contribution of $d$-states to the conduction band. The first critical point in the dielectric constant ranges between 0.07–4.47 eV and is due to the first threshold optical transitions in the energy bandgap. Calculated static dielectric function (DF) ${𝜖}_1(0)$ is between 5.15 and 10.35, an indication that small energy bandgaps yield large static DFs. The present results indicate $\mathrm{\text{ZB}}$-${\mathrm{\text{B}}}_x{\mathrm{\text{Al}}}_y{\mathrm{\text{In}}}_{1-x-y}\mathrm{\text{N}}$ alloys are suitable candidates of deep ultraviolet light emitting diodes (LEDs), laser diodes (LDs) and modern solar cell since the concentrations $x$ and $y$ make the bandgap and lattice constant of $\mathrm{\text{ZB}}$-${\mathrm{\text{B}}}_x{\mathrm{\text{Al}}}_y{\mathrm{\text{In}}}_{1-x-y}\mathrm{\text{N}}$ quaternary alloys tunable to desirable values.

The quaternary alloy (QA) is simulated on the Bethe lattice (BL) in the form of AB_pC_qD_r and its phase diagrams are calculated by using the exact recursion relations (ERR) for the coordination number z = 3. The QA is designed on the BL by placing A atoms (spin-1/2) on the odd shells and randomly placing B (spin-3/2), C (spin-5/2) or D (spin-1) atoms with probabilities p, q and r, respectively, on the even shells. A compact form of formulation for the QA is obtained in the standard-random approach which can easily be reduced to ternary alloy (TA) and mixed-spin models by the appropriate values of the random variables p, q and r. The phase diagrams are calculated on the temperature and ratio of bilinear interaction parameter planes for given values of probabilities.

In the present work, we have performed first principles calculations to study the structural and electronic properties of the MgFBr_xCl_1-x quaternary alloys using the pseudo-potential plane wave approach within the framework of density functional theory. By using the optimized initial parameters, we have obtained the physical quantities such as equilibrium lattice constants a and c, cohesive energy and band gap and then fitted the results by a quadratic expression for all x compositions. The results of bulk modulus exhibit nearly linear concentration dependence (LCD) but other quantities show nonlinear dependence. Finally, we have calculated the total and angular momentum decomposed (partial) density of states and determined the contributions of different orbitals of each atoms.