First-principles description of van der Waals-bonded spin-polarized systems using vdW-DF$+U$ method---application to solid oxygen at low pressure

The description of the molecular solid phase of O$_2$, especially its ground-state antiferromagnetic insulating phase, is known to be quite unsatisfactory within the conventional local and semilocal density functional approximations used in the Kohn-Sham formalism of density functional theory. The recently-developed van der Waals functionals that take into account nonlocal correlations have also shown subpar performance in this regard. The difficulty lies in the subtle balance between the van der Waals interactions and the exchange coupling between the antiferromagnetic and ferromagnetic molecule pairs in the molecular crystal. Here, we report that the DFT$+U$ approach used in combination with the vdW-DF functional performs surprisingly well in this regard, and discuss the reasoning behind this behavior. We also apply this approach to study the recently-reported magnetic field-induced $\theta$ phase of solid O$_2$.