The phases of $A_2{\mathrm{Mn}}_8{\mathrm{O}}_{16}$ hollandite group oxides emerge from the competition between ionic interactions, Jahn-Teller effects, charge ordering, and magnetic interactions. Their balanced treatment with feasible computational approaches can be challenging for commonly used approximations in density functional theory. Three examples ($A$ = Ag, Li, and K) are studied with a sequence of different approximate exchange-correlation functionals. Starting from a generalized gradient approximation (GGA), an extension to include van der Waals interactions and a recently proposed meta-GGA are considered. Then local Coulomb interactions for the Mn $3d$ electrons are more explicitly considered with the DFT + $U$ approach. Finally, selected results from a hybrid functional approach provide a reference. Results for the binding energy of the A species in the parent oxide highlight the role of van der Waals interactions. Relatively accurate results for insertion energies can be achieved with a low-$U$ and a high-$U$ approach. In the low-$U$ case, the materials are described as band metals with a high-symmetry, tetragonal crystal structure. In the high-$U$ case, the electrons donated by A result in formation of local ${\mathrm{Mn}}^{3+}$ centers and corresponding Jahn-Teller distortions characterized by a local order parameter. The resulting degree of monoclinic distortion depends on charge ordering and magnetic interactions in the phase formed. The reference hybrid functional results show charge localization and ordering. Comparison to low-temperature experiments of related compounds suggests that charge localization is the physically correct result for the hollandite group oxides studied here. Finally, while competing effects in the local magnetic coupling are subtle, the fully anisotropic implementation of DFT + $U$ gives the best overall agreement with results from the hybrid functional.

• Received 1 September 2017

DOI:https://doi.org/10.1103/PhysRevMaterials.1.075401