  We present a dynamical mean-field theory (DMFT) study of the charge and
orbital correlations in finite-size La$_{0.5}$Ca$_{0.5}$MnO$_3$ (LCMO)
nanoclusters. Upon nanostructuring LCMO to clusters of 3 nm diameter, the size
reduction induces an insulator-to-metal transition in the high-temperature
paramagnetic phase. This is ascribed to the reduction in charge
disproportionation between Mn sites with different nominal valence [Das et al.,
Phys. Rev. Lett. 107, 197202 (2011)]. Here we show that upon further reducing
the system size to a few-atom nanoclusters, quantum confinement effects come
into play. These lead to the opposite effect: the nanocluster turns insulating
again and the charge disproportionation between Mn sites, as well as the
orbital polarization, are enhanced. Electron doping by means of external gate
voltage on few-atom nanoclusters is found to trigger a site- and
orbital-selective Mott transition. Our results suggest that LCMO nanoclusters
could be employed for the realization of technological devices, exploiting the
proximity to the Mott transition and its control by size and gate voltage.
