Structural, electronic, mechanical and thermodynamic properties of Rh$_3$Zr$_x$V$_{1−x}$ and Rh$_3$Hf$_x$V$_{1−x} ($x = 0$, 0.125, 0.25, 0.75, 0.875 and 1) combinations are investigated by means of first-principles calculations based on the density functional theory within the generalized gradient approximation. Here, Rh$_3$V is chosen as the parent binary compound and the doping elements are zirconium and hafnium with the above-mentioned concentrations. The calculated lattice parameters and elastic modulus of binary Rh$_3$Hf, Rh$_3$V and Rh$_3$Zr are in good agreement with the available experimental and other theoretical results. In this study, the following ternary materials viz., Rh$_3$Zr$_{0.75}$V$_{0.25}$, Rh$_3$Hf$_{0.25}$V$_{0.75}$ and Rh$_3$Hf$_{0.75}$V$_{0.25}$ are found to be brittle/more brittle than the parent binary compound Rh$_3$V, whereas the other ternary combinations, namelyRh$_3$Zr$_{0.125}$V$_{0.875}$, Rh$_3$Zr$_{0.25}$V$_{0.75}$, Rh$_3$Zr$_{0.875}$V$_{0.125}$, Rh$_{3}Hf$_{0.125}$V$_{0.875}$ and Rh$_3$Hf$_{0.875}$V$_{0.125}$ are found to be more ductilethanRh3V. The more brittle ternary combination, namely Rh$_3$Hf$_{0.75}$V$_{0.25}$ ($B = 229.32$ GPa) has the maximum Young’s modulus,shear modulus and hardness values; whereas the more ductile ternary Rh$_3$Zr$_{0.25}$V$_{0.75}$ combination ($B = 243.54$ GPa) is found to have the least values of Young’s modulus, shear modulus and hardness. The band structure, density of stateshistograms and charge density plots are drawn and discussed. Computed Debye temperature (θD), Grüneisen parameter ($\zeta$) and melting temperature ($T_{\rm m}$) of the parent binary compound Rh$_3$V, the more brittle Rh$_3$Hf$_{0.75}V$_{0.25}$ combination and themore ductile Rh$_3$Zr$_{0.25}$V$_{0.75}$ combination are given by (895 K, 1.3491, 2788 K), (790 K, 1.2701, 2736K) and (698 K, 1.7972, 2529 K), respectively.

A tin–bismuth–telluride material was confirmed as an efficient and harmless material in thermoelectric applications by the results obtained from the density functional theory. The calculations were carried out using the FPLAPW method with Wien2k code. This is the classic thermoelectric material used in refrigeration and thermoelectric generators due to its high Seebeck coefficient value and low thermal conductivity. Here Tin is replaced in parent SnTe by bismuth as various doping concentrations. The spin-orbit coupling was used in both electronic and thermoelectric properties calculations. Also we discussed mechanical properties of Sn$_{(1–x)}$Bi$_x$Te (x = 0, 0.125, 0.25, 0.5, 0.75, 0.875 and 1) materials. The result confirms that all the doped materials are ductile in nature and parent SnTe is of brittle nature. Here we have discussed the results on both spin-orbit coupling (SOC) and non-SOC calculations in thermoelectric properties. Changes occurred in band structure, and density of states according to SOC and non-SOC calculations were clearly explained. Also, the calculations of Seebeck coefficient, electrical conductivity with relaxation time, power factor,electronic thermal conductivity and figure of merit were made with SOC and without SOC over the temperature range of 300–1000 K. From the results, it was obtained that, within the SOC calculations, thermoelectric properties of the studied materials were enhanced at high temperature.