  The equations of state, formation energy and migration energy barrier of the
oxygen vacancy in SrFeO${_3}$ and LaFeO${_3}$ were calculated with the
diffusion quantum Monte Carlo (DMC) method. Calculations were also performed
with various Density Functional Theory (DFT) approximations for comparison. DMC
reproduces the measured cohesive energies of these materials with errors below
0.23(5) eV and the structural properties within 1% of the experimental values.
The DMC formation energies of the oxygen vacancy in SrFeO${_3}$ and LaFeO${_3}$
under oxygen-rich conditions are 1.3(1) and 6.24(7) eV, respectively. Similar
calculations with semi-local DFT approximations for LaFeO$_3$ yielded vacancy
formation energies 1.5 eV lower. Comparison of charge density evaluated with
DMC and DFT approximations shows that DFT tends to overdelocalize the electrons
in defected SrFeO${_3}$ and LaFeO${_3}$. Calculations with DMC and LDA yield
similar vacancy migration energy barriers, indicating that steric/electrostatic
effects mainly determine migration barriers in these materials.
