  Iron oxide magnetic nanoparticles (NPs) are stimuli-responsive materials at
the forefront of nanomedicine. Their realistic finite temperature simulations
are a formidable challenge for first-principles methods. Here, we use density
functional tight binding to open up the required time and length scales and
obtain global minimum structures of Fe3O4 NPs of realistic size (1400 atoms,
2.5 nm) and of different shapes, which we then refine with hybrid density
functional theory methods to accomplish proper electronic and magnetic
properties, which have never been accurately described in simulations. On this
basis, we develop a general empirical formula and prove its predictive power
for the evaluation of the total magnetic moment of Fe3O4 NPs. By converting the
total magnetic moment into the macroscopic saturation magnetization, we
rationalize the experimentally observed dependence with shape. We also reveal
interesting reconstruction mechanisms and unexpected patterns of charge
ordering.
