  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.
