  Functionals that strive to correct for such self-interaction errors, such as
those obtained by imposing the Perdew-Zunger self-interaction correction or the
generalized Koopmans' condition, become orbital dependent or orbital-density
dependent, and provide a very promising avenue to go beyond density-functional
theory, especially when studying electronic, optical and dielectric properties,
charge-transfer excitations, and molecular dissociations. Unlike conventional
density functionals, these functionals are not invariant under unitary
transformations of occupied electronic states, which leave the total charge
density intact, and this added complexity has greatly inhibited both their
development and their practical applicability. Here, we first recast the
minimization problem for non-unitary invariant energy functionals into the
language of ensemble density-functional theory, decoupling the variational
search into an inner loop of unitary transformations that minimize the energy
at fixed orbital subspace, and an outer-loop evolution of the orbitals in the
space orthogonal to the occupied manifold. Then, we show that the potential
energy surface in the inner loop is far from convex parabolic in the early
stages of the minimization and hence minimization schemes based on these
assumptions are unstable, and present an approach to overcome such difficulty.
The overall formulation allows for a stable, robust, and efficient variational
minimization of non-unitary-invariant functionals, essential to study complex
materials and molecules, and to investigate the bulk thermodynamic limit, where
orbitals converge typically to localized Wannier functions. In particular,
using maximally localized Wannier functions as an initial guess can greatly
reduce the computational costs needed to reach the energy minimum while not
affecting or improving the convergence efficiency.
