The rhodium doping in the LaCo$_{1-x}$Rh$_{x}$O$_3$ perovskite series ($x=0.02-0.5$) has been studied by X-ray diffraction, electric transport and magnetization measurements, complemented by electronic structure GGA+U calculations in supercell for different concentration regimes. No charge transfer between Co$^{3+}$ and Rh$^{3+}$ is evidenced. The diamagnetic ground state of LaCoO$_3$, based on Co$^{3+}$ in low-spin (LS) state, is disturbed even by a small doping of Rh. The driving force is the elastic energy connected with incorporation of a large Rh$^{3+}$ cation into the matrix of small LS Co$^{3+}$ cations, which is relaxed by formation of large Co$^{3+}$ in high-spin (HS) state in the next-nearest sites to the inserted Rh atom. With increasing temperature, the population of Co$^{3+}$ in HS state increases through thermal excitation, and a saturated phase is obtained close to room temperature, consisting of a nearest-neighbor correlation of small (LS Co$^{3+}$) and large (HS Co$^{3+}$ and LS Rh$^{3+}$) cations in a kind of double perovskite structure. The stabilizing role of elastic and electronic energy contributions is demonstrated in supercell calculations for dilute Rh concentration compared to other dopants with various trivalent ionic radius.
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