Implicit density functionals

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. 

This chapter is devoted to orbital-dependent exchange-correlation (xc) functionals, a concept that has attracted more and more attention during the last ten years. After a few preliminary remarks, which clarify the scope of this review and introduce the basic notation, some motivation will be given why such implicit density functionals are of definite interest, in spite of the fact that one has to cope with additional complications (compared to the standard xc-functionals). The basic idea of orbital-dependent xc-functionals is then illustrated by the simplest and, at the same time, most important functional of this type, the exact exchange of density functional theory (DFT for a review see e.g. [1], or the chapter by J. Perdew and S. Kurth in this volume). 

Note, however, that this result is consistent with the basic statements of DPT As Uxc is a density functional itself, the right-hand side of (2.77) is an implicit density functional. 

This expression represents an implicit density functional for 7(ri,r2) in the same sense as (2.19) is an implicit density functional for the exchange energy. 

Group 2 complexes are formally electron deficient and conformationally floppy only small energies (often only 1-2 kcal mol-1) are required to alter their geometries by large amounts (e.g., bond angles by 20° or more). In such cases, the inclusion of electron-correlation effects becomes critical to an accurate description of the molecules structures. Both HF/MP2 and density functional theory (DFT) methods have been applied to organoalkaline earth compounds. DFT approaches, which implicitly incorporate electron correlation in a computationally efficient form, are generally the more widely used. Molecular orbital calculations that successfully reproduce bent... 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

The atom-centered multipole expansion used in the density formalisms described in chapter 3 implicitly assigns each density fragment to the nucleus at which it is centered. Since the shape of the density functions is fitted to the observed density in the least-squares minimalization, the partitioning is more flexible than that based on preconceived spherical atoms. 

Correlated Models. Models which take implicit or explicit account of the Correlation of electron motions. Moller-Plesset Models, Configuration Interaction Models and Density Functional Models... 

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