  In this study, we have reported electronic structure, lattice dynamics, and
thermoelectric (TE) transport properties of a new family of pentanary
substituted TiNiSn systems using the 18 valence electron count (VEC) rule. From
our calculated band structures and density of states, we show that by
preserving the 18 VEC through aliovalent substitutions at the Ti site of TiNiSn
semiconducting behavior can be achieved, and hence one can tune the band
structure and band gap to maximize the thermoelectric figure of merit (ZT)
value. Two approaches have been used for calculating the lattice thermal
conductivity ($\kappa_{L}$), one by fully solving the linearized phonon
Boltzmann transport (LBTE) equation from first$-$principles anharmonic lattice
dynamics calculations implemented in Phono3py code and other using Slack's
equation with calculated Debye temperature and Gr\"{u}neisen parameter using
the calculated elastic constant values. The calculated $\kappa_{L}$ values
decrease from parent TiNiSn to pentanary substituted TiNiSn systems as expected
due to fluctuation in atomic mass. The calculated $\kappa_{L}$ for Hf
containing systems La$_{0.25}$Hf$_{0.5}$V$_{0.25}$NiSn and non Hf containing
system La$_{0.25}$Zr$_{0.5}$V$_{0.25}$NiSn calculated from Phono3py (Slack's
equation) are found to be 0.37 (1.04) and 0.16 (0.95) W/mK, at 550\,K,
respectively and the corresponding ZT value are found to be 0.54 (0.4) and 0.77
(0.53). Among the considered systems, the calculated phonon spectra and heat
capacity show that La$_{0.25}$Hf$_{0.5}$V$_{0.25}$NiSn has more
optical$-$acoustic band mixing which creates more phonon$-$phonon scattering
and hence lower the $\kappa_{L}$ value and maximizing the ZT. Based on the
calculated results we conclude that one can design high-efficiency
thermoelectric materials by considering the 18 VEC rule with aliovalent
substitution.
