Weak charge disproportion and leading mechanisms in half-doped manganites.

The puzzling very weak charge disproportionation found in half- doped manganites such as La$_{1/2}$Ca$_{1/2}$MnO$_3$ is reconciled with the well-accepted Mn$^{3+}$/Mn$^{4+}$ picture of charge and orbital orders via our novel first-principles Wannier function analysis. The strong electron itinerancy is found to delocalize the ``Mn$^{3+} $'' Wannier states significantly, producing remarkable charge leaking into the ``Mn$^{4+}$'' sites. Thus, it is necessary to distinguish for this charge-transfer system actual charge from the occupation number. Finally, a realistic low-energy effective Hamiltonian is derived, revealing the interesting role of electron-electron interactions in the charge and orbital channels, which can be applied to other doping regions including the CMR phase.