  The functional properties of transition metal perovskite oxides are known to
result from a complex interplay of magnetism, polarization, strain, and
stoichiometry. Here, we show that for materials with a cooperative Jahn-Teller
distortion, such as LaMnO$_3$ (LMO), the orbital order can also couple to the
defect chemistry and induce novel material properties. At low temperatures, LMO
exhibits a strong Jahn-Teller distortion that splits the $e_g$ orbitals of the
high-spin Mn$^{3+}$ ions and leads to alternating long, short, and intermediate
Mn--O bonds. Our DFT+$U$ calculations show that, as a result of this orbital
order, the charge localization in LMO upon oxygen vacancy formation differs
from other manganites, like SrMnO$_3$, where the two extra electrons reduce the
two Mn sites adjacent to the vacancy. In LMO, relaxations around the defect
depend on which type of Mn--O bond is broken, affecting the $d$-orbital
energies and leading to asymmetric and hence polar excess-electron localization
with respect to the vacancy. Moreover, we show that the Mn--O bond lengths,
orbital order and consequently the charge localization and polarity are tunable
via strain.
