First-Principles Implicit Correlation Functionals

Some of the examples considered in the previous section already indicated that the exact exchange, while providing obvious progress compared with the LDA and the GGA, has to be combined with an appropriate orbital-dependent correlation functional in order to be useful in practice. Given the first-principles nature of the exact E, it is natural to derive such a correlation functional in a systematic fashion. The first task is to establish a suitable expression for the exact E c which can serve as starting point for the subsequent discussion of different approximations. This exact formula for E c at the same time resolves the discrepancy which was found between the original OPM equation (2.27) and the Sham-Schliiter equation (2.45). 

1 Many-Body Theory on the Basis of the Kohn Sham System Exact Expression for E c 

Let us assume for a moment that Vg, the total Kohn-Sham potential, is known [16,17,62]. This allows the definition of a noninteracting A -particle 

Hamiltonian Hg, which is the sum of the kinetic energy and an external potential term based on 

The ground-state o) (assumed to be nondegenerate) corresponding to Hg is obtained by solution of the Schrodinger equation, 

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Given the complexity of the first-principles implicit correlation functionals of Sect. 2.4, one is automatically led to look for simpler and thus more efficient semi-empirical alternatives. Two functionals of this type have been suggested for use within the 0PM. 

Finally, the performance of the presently available implicit correlation functionals is studied in Sect. 2.6. In particular, the success of the first-principles perturbative correlation functional with the description of dispersion forces is demonstrated [24]. On the other hand, this functional leads to a divergent correlation potential in the case of finite systems [25]. This failure prompts an approximate handling of the associated OPM integral equation in the spirit of the KLI approximation, which avoids the asymptotic divergence and produces comparatively accurate atomic correlation potentials. 

The systematic derivation of implicit correlation functionals is discussed in Sect. 2.4. In particular, perturbation theory based on the Kohn-Sham (KS) Hamiltonian [16,17,18] is used to derive an exact relation for l xc- This expression is then expanded to second order in the electron-electron coupling constant in order to obtain the simplest first-principles correlation functional [18]. The corresponding OPM integral equation as well as extensions like the random phase approximation (RPA) [19,20] and the interaction strength interpolation (ISI) [21] are also introduced. 

Given the motivation for implicit correlation functionals the first question to be addressed is that of dispersion forces. As none of the semi-empirical functionals of Sect. 2.5 can deal with these long-range forces, the present discussion focuses on the second order correlation functional (2.82) as the simplest first-principles functional. 

As a fully nonlocal alternative to these explicit density functionals orbital-dependent (implicit) density functionals have been suggested. In addition to the exact exchange [51,52] some approximate correlation functionals are available, both empirical [166] and first-principles forms [57,59,60], As is already clear from Section 3.4 this concept can also be used in the relativistic situation. The status of relativistic implicit functionals [54] will be reviewed in Section 4.1. In particular, the various ingredients of the exact exchange will be analyzed. Subsequently the results obtained with the exact exchange will then serve as reference data for the analysis of the RLDA and RGGA. 

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