Accuracy of the KLI Approximation

Before addressing the issue of correlation in more detail, it is instructive to study the x-only limit from a quantitative point of view. This analysis serves two purposes The first is to assert the accuracy of the KLI approximation. As is clear from the discussion of Sect. 2.2.5, any large-scale application of orbital-dependent functionals will have to rely on the efficiency of the KLI approximation. One thus has to make sure that this approach yields reasonable results at least for the simplest orbital-dependent functional, the exact Ex- The second aim of this section is to demonstrate that orbital-dependent functionals are worth the increased computational effort, i.e. that they in fact yield improvements over the standard functionals. 

The next comparison to be made is that of OPM and Hartree-Fock results. In Sect. 2.2.2 it has been emphasized that the x-only OPM represents a restricted Hartree-Fock energy minimization One minimizes the same energy expression, but under the subsidiary condition of having a multiplicative exchange potential. How important is this subsidiary condition As Table 2.2 shows, the differences are rather small. For He the OPM energy is identical with the HF value, as in this case the HF equation can be trivially recast as a KS equation with the OPM exchange potential (2.52). Moreover, even for the heaviest elements the differences between OPM and HF energies are 

Until now total energies have been considered. However, the physical and chemical properties usually depend on energy differences. In Table 2.4 the most simple energy difference, namely the ionization potential (IP), is studied for atoms. Again the KLI results are extremely close to the OPM data, which agree very well with the Hartree-Fock IPs. On the other hand, one finds the (well-known) errors in the case of the LDA and the GGA. 

At this point a side remark seems appropriate. All potentials shown in Fig. 2.3 originate from self-consistent calculations within the corresponding schemes. One might then ask how these curves change if the same density (and thus the same orbitals) are used for the evaluation of the different functionals This issue is addressed in Fig. 2.4 in which the solution of the OPM integral equation on the basis of three different sets of orbitals is plotted. 

In addition to the exact x-only orbitals used for Fig. 2.3 also the exact KS orbitals [57] and the LDA orbitals are inserted into (2.22)-(2.28). It turns out that the three solutions are almost indistinguishable. The origin of the orbitals (and thus of the density) is much less important for the structure of atomic Wxc than the functional form of E c- In other words Fig. 2.3 would look very similar if all functionals were evaluated with the same density. 

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Table 4.2 also demonstrates the accuracy of the KLI approximation in the relativistic situation. In fact, for heavy elements the differences between the KLI and the full OPM energies are smaller than those resulting from a perturbative treatment of the transverse interaction. 

In its most general form the ROPM requires the solution of a set of four integral equations in order to determine the xc-components of v. As a consequence the ROPM selfconsistency procedure is much more demanding than standard RKS-calculations. Even in the nonrelativistic case most applications thus either addressed spherical systems [63-66] or utilized the atomic sphere approximation [67,68], Only few applications are available in which a spherical approximation is not exploited [69-71]. However, the computational demands of implicit functionals can be substantially reduced by resorting to a very efficient and accurate semi-analytical approximation to the 0PM which has been introduced by Krieger, Li and lafrate (KLI) [72], This scheme is easily extended to the ROPM [56,54]. Applications of the KLI approximation within RDFT confirm the level of accuracy found in the nonrelativistic limit [73]. With this technique the use of implicit functionals represents a real alternative to the application of the RGGA. 

The KLI approximation preserves both the KLI identity (2.48) and the asymptotic behavior of v, (2.49) (for finite systems). It is exact for spin-saturated two-electron systems, i.e. it also satisfies (2.52). Moreover, all applications available so far point at the rather high accuracy of this approximation, at least in the case of the exact exchange (see Sect. 2.3). 

The high accuracy attained by complex orbital functionals implemented via the OEP, and the fact that it is easier to devise orbital functionals than explicit density functionals, makes the OEP concept attractive, but the computational cost of solving the OEP integral equation is a major drawback. However, this computational cost is significantly reduced by the KLI approximation and other recently proposed simplifications. " In the context of the EXX method (i.e. using the Fock exchange term as an orbital functional) the OEP is a viable way to proceed. For more complex orbital functionals, additional simplifications may be necessary. " ... 

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