(2.5.10) Köster : For decades, X α and Kohn-Sham DFT methods have served as a playground for the development of density-fitting methods. Commonly used approaches are the variational fitting of the Coulomb 225 and Fock 226 potentials. With these fitting approaches, the formal scaling of first-principles Hartree-Fock and Kohn-Sham calculations is reduced by one order of magnitude without lowering the accuracy of the underlying methodology. To avoid linear-algebra bottlenecks associated with variational density fitting, iterative Krylov subspace solvers are advocated. A further simplification of Kohn-Sham DFT implementations can be achieved by using the approximated density from the variational fitting of the Coulomb potential for the evaluation of the exchange-correlation energy and potential. The resulting energy expression remains variational and yields optimized structures and relative energies that are almost indistinguishable from standard Kohn-Sham approaches, but at a substantially reduced computational effort. The extension of this auxiliary DFT (ADFT) approach to perturbation theory permits first-principles molecular property calculations of systems with up to thousand atoms -for example, second-order analytic energy derivatives. Most recently, ADFT also serves as platform for the development of new DFAs.