  Modern extensions of density functional theory such as the density functional
theory plus U and the density functional theory plus dynamical mean-field
theory require choices, including selection of variable (charge vs spin
density) for the density functional and specification of the correlated
subspace. This paper examines these issues in the context of the "plus U"
extensions of density functional theory, in which additional correlations on
specified correlated orbitals are treated using a Hartree-Fock approximation.
Differences between using charge-only or spin-density-dependent
exchange-correlation functionals and between Wannier and projector-based
definitions of the correlated orbitals are considered on the formal level and
in the context of the structural energetics of the rare earth nickelates. It is
demonstrated that theories based on spin-dependent exchange-correlation
functionals can lead to large and in some cases unphysical effective on-site
exchange couplings. Wannier and projector-based definitions of the correlated
orbitals lead to similar behavior near ambient pressure, but substantial
differences are observed at large pressures. Implications for other beyond
density functional methods such as the combination of density functional and
dynamical mean field theory are discussed.
