  The apparent contradiction between the recently observed weak charge
disproportion and the traditional Mn$^{3+}$/Mn$^{4+}$ picture of the
charge-orbital orders in half-doped manganites is resolved by a novel Wannier
states analysis of the LDA$+U$ electronic structure. Strong electron itinerancy
in this charge-transfer system significantly delocalizes the occupied
low-energy "Mn$^{3+}$" Wannier states such that charge leaks into the
"Mn$^{4+}$"-sites. Furthermore, the leading mechanisms of the charge order are
quantified via our first-principles derivation of the low-energy effective
Hamiltonian. The electron-electron interaction is found to play a role as
important as the electron-lattice interaction. \ignore{A general picture of
doped holes in strongly correlated charge-transfer systems is presented and
applied to the study of charge order in half-doped manganites, using a novel
Wannier states analysis of the LDA$+U$ electronic structure. While residing
primarily in the oxygen atoms, the doped holes form additional effective $e_g$
orbitals at the low-energy scale, leading to an effective Mn$^{3+}$/Mn$^{4+}$
valence picture that enables weak charge disproportion, resolving the current
serious contradictions between the recent experimental observations of charge
distribution and traditional models. Furthermore, the leading mechanisms of the
observed charge order are quantified via our first-principles derivation of the
low-energy effective Hamiltonian
