  The phases of A$_2$Mn$_8$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 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. . . .
