We investigate temperature, pressure, and localization dependence of thermoelectric properties, phonon and de Haas–van Alphen (dHvA) frequencies of the anti-ferromagnetic (AFM) CeIn$$_3$$ 3 using density functional theory (DFT) and local, hybrid, and band correlated functionals. It is found that the maximum values of thermopower, power factor, and electronic figure of merit of this compound occur at low (high) temperatures provided that the 4f-Ce electrons are (not) localized enough. The maximum values of the thermopower, power factor, electronic figure of merit (conductivity parameters), and their related doping levels (do not) considerably depend on the localization degree and pressure. The effects of pressure on these parameters substantially depend on the degree of localization. The phonon frequencies are calculated to be real which shows that the crystal is dynamically stable. From the phonon band structure, the thermal conductivity is predicted to be homogeneous. This prediction is found consistent with the thermal conductivity components calculated along three Cartesian directions. In analogous to the thermoelectric properties, it is found that the dHvA frequencies also depend on both pressure and localization degree. To ensure that the phase transition at Néel temperature cannot remarkably affect the results, we verify the density of states (DOS) of the compound at the paramagnetic phase constructing a non-collinear magnetic structure where the angles of the spins are determined so that the resultant magnetic moment vanishes. The non-collinear results reveal that the DOS and whence the thermoelectric properties of the compound are not changed considerably by the phase transition. To validate the accuracy of the results, the total and partial DOSs are recalculated using DFT plus dynamical mean-field theory (DFT+DMFT). The DFT+DMFT DOSs, in agreement with the hybrid DOSs, predict the Kondo effect in this compound.

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