The crystal structure and charge and orbital order of magnetite below the Verwey temperature are calculated using a first-principles hybrid density functional theory (DFT) method. The initial atomic positions in the crystal-structure calculation are those recently refined from x-ray diffraction data for the $Cc$ space-group unit cell [Senn, Wright, and Attfield, Nature (London) 481, 173 (2012)]. Fermi contact and magnetic dipolar contributions to hyperfine fields at ${}^{57}\mathrm{Fe}$ nuclei obtained from hybrid DFT calculations are used to obtain NMR resonance frequencies for magnetite for a range of external magnetic field directions in a relatively weak field. NMR frequencies from hybrid density functional theory calculations are compared to NMR data [M. Mizoguchi, J. Phys. Soc. Jpn. 70, 2333 (2001)] for a range of applied magnetic field directions. NMR resonance frequencies of $B$-site Fe ions show large relative variations with applied field direction owing to anisotropic hyperfine fields from charge and orbital ordered Fe $3d$ minority-spin electrons at those sites. Good agreement between computed and measured NMR resonance frequencies confirms the pattern of charge and orbital order obtained from calculations. The charge and orbital order of magne-tite in its low-temperature phase obtained from hybrid DFT calculations is analyzed in terms of one-electron bonds between Fe ions. The Verwey transition in magnetite therefore resembles Mott-Peierls transitions in vanadium oxides which undergo symmetry-breaking transitions owing to electron-pair bond formation.

• Received 4 July 2014
• Revised 5 August 2014

DOI:https://doi.org/10.1103/PhysRevB.90.075134