Partial molecular orbitals in face-sharing 3$d$ manganese trimer: Comparative studies on Ba${}_4$TaMn${}_3$O${}_{12}$ and Ba${}_4$NbMn${}_3$O${}_{12}$

We present a molecular orbital candidate Ba${}_4$TaMn${}_3$O${}_{12}$ with a face-sharing octahedra trimer, by comparing it with a related compound Ba${}_4$NbMn${}_3$O${}_{12}$. The synthesis of the polycrystalline powder is optimized by suppressing the secondary impurity phase via x-ray diffraction. Magnetic susceptibility measurements on the optimized samples reveal a weak magnetic hysteresis with magnetic transitions consistent with heat capacity results. The effective magnetic moments from susceptibility indicate a strongly coupled $S=2$ antiferromagnetic trimer at around room temperature, whereas the estimated magnetic entropy from heat capacity suggests the localized $S=3\mathrm{/}2$ timer. These results can be explainable by a partial molecular orbital state, in which three $t_{2g}$ electrons are localized in each Mn ion and one $e_g$ electron is delocalized over two-end Mn ions of the trimer based on density functional theory calculations. This unconventional 3$d$ orbital state is comprehended as a consequence of competition between the hybrid interatomic orbitals within the Mn trimer and the local moment formation by on-site Coulomb correlations.