Optimization of Functionals of Orthonormal Functions in the Absence of Unitary Invariance

We discuss the optimization of a functional with respect to sets of orthonormal functions where unitary invariance does not apply. This problem arises, for example, when density functionals with explicit self-interaction correction are used for systems of electrons. There, unitary invariance cannot be used to reformulate the minimization of the energy with respect to each of the functions as an eigenvalue problem as can be done for the commonly used GGA-DFT and Hartree-Fock theory. By including optimization with respect to unitary transformations as an explicit step in the iterative minimization procedure, fast convergence can, nevertheless, be obtained. Furthermore, by working with two sets of orthonormal functions, the optimal functions and a set of eigenfunctions, the implementation of the extended functional form in existing software becomes easier. The additional computations arising from the lack of unitary invariance can largely be carried out in parallel.