  By means of density-functional simulations for half-doped manganites, such as
pseudocubic Pr0.5Ca0.5MnO3 and bilayer PrCa2Mn2O7, we discuss the occurrence of
ferroelectricity and we explore its crucial relation to the crystal structure
and to peculiar charge/spin/orbital ordering effects. In pseudocubic
Pr0.5Ca0.5MnO3, ferroelectricity is induced in the Zener polaron type
structure, where Mn ions are dimerized. In marked contrast, in bilayer
PrCa2Mn2O7, it is the displacements of apical oxygens bonded to either Mn3+ or
Mn4+ ions that play a key role in the rising of ferroelectricity. Importantly,
local dipoles due to apical oxygens are also intimately linked to charge and
orbital ordering patterns in MnO2 planes, which in turn contribute to
polarization. Finally, an important outcome of our work consists in proposing
Born effective charges as a valid mean to quantify charge disproportionation
effects, in terms of anisotropy and size of electronic clouds around Mn ions.
