Basic Concept of Orbital-Dependent Functionals

This is the right point to clarify the meaning of the term orbital-dependent functional. The natural starting point for the discussion is the exact exchange Ex of DFT, which is the most simple functional of this type. The exact is defined as the Fock expression written in terms of KS orbitals [40,8], 

This is the most appropriate definition as it guarantees the exact cancellation of the self-interaction energy contained in Eu, which has been identified as the origin of the problem of the LDA/GGA with negative ions (and also seems to be relevant for the description of Mott insulators). It automatically induces a corresponding definition of the correlation functional of DFT, 

The step from explicitly density-dependent to orbital-dependent xc-func-tionals is in some sense analogous to the transition from the Thomas-Fermi variational equation to the KS equations In the latter transition the most important part of tot[ ], the kinetic energy, is recast in orbital-dependent form. The same concept is now applied to Exc- In that sense, one can call orbital-dependent functionals a third generation of density functionals. 

At this point, the idea of implicit xc-functionals might appear as a purely formal concept. In the following sections, however, it will be shown that implicit functionals can be used in practice. 

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