From explicit to implicit density functionals

The concept of orbital- and eigenvalue-dependent exchange-correlation (xc) energy functionals is reviewed. We show how such functionals can be derived in a systematic fashion via a perturbation expansion, utilizing the Kohn–Sham system as a noninteracting reference system. We demonstrate that the second-order contribution to this expansion of the xc-energy functional includes the leading term of the van der Waals interaction. The optimized-potential method (OPM), which allows the calculation of the multiplicative xc-potential corresponding to an orbital- and eigenvalue-dependent xc-energy functional via an integral equation, is discussed in detail. We examine an approximate analytical solution of the OPM integral equation, pointing out that, for eigenvalue-dependent functionals, the three paths used in the literature for the derivation of this approximation yield different results. Finally, a number of illustrative results, both for the exchange-only limit and for the combination of the exact exchange with various correlation functionals, are given.