Motivation for Orbital-Dependent Functionals

The first question to be addressed is Why would one think about using orbital-dependent functionals, given the tremendous success of the GGA The answer to this question necessarily consists of a list of situations in which the GGA, which is by now the standard workhorse of DFT, fails. 

It seems worthwhile to emphasize that this deficiency of the GGA can not be explained by relativistic effects The inclusion of relativistic corrections in the GGA [32] does not improve the results (see Table 2.1 - the fully relativistic Tg and thus the fully relativistic KS equations have been applied in all calculations). This observation suggests that the GGA has some difficulties with the treatment of higher angular momentum d and /), similarly to the LDA [33]. 

Negative Ions. In contrast to the loss of accuracy for heavy elements, the second problem of the GGA, its failure for negative ions, is of qualitative nature. It originates from the (semi-)local density-dependence of the LDA and GGA exchange potential. The situation is most easily analyzed in the case of the LDA, for which one has 

In the asymptotic regime of finite systems, in which the density decays exponentially, one thus finds an exponential decay of 

The same is true for the LDA correlation potential. Moreover, for neutral atoms the electrostatic potential of the nucleus cancels with the monopole term in vh, (2-3). Consequently, the total Ug also decays faster than 1/r. This implies that, within the framework of the LDA, a neutral atom does not exhibit a Rydberg series of excited states and thus is not able to bind an additional electron, i.e. to form a negative ion. 

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