  Na$_{0.5}$(Bi$_3$/4RE$_{1/4}$)$_{0.5}$TiO$_3$ (RENBT, RE = Nd, Gd, Dy, and
Ho) compounds were investigated in the framework of first-principles
calculations using the full potential linearized augmented plane wave (FP-LAPW)
method based on the spin-polarized density functional theory implemented in the
WIEN2k code. Combined charge density distribution and Ti K-edge x-ray
absorption spectra revealed that the RENBT compositions with high polarization
values were accompanied by a higher TiO$_6$ distortion, DyNBT, and NdNBT
compounds. The effect of the rare-earth elements on the polarization were
confirmed experimentally with the collection of the hysteresis loops. The
investigation of the electronic properties of the compounds highlighted the
emergence of a magnetization owing to the 4f orbital effect of the rare-earth
elements. Besides, the investigation of the chemical ordering showed a
short-range chemical ordering for the pure composition and an increased A-site
disorder for dysprosium doped NBT system. The increased disorder may speak for
increased relaxor properties in the RE doped compositions.
