Fully nonlocal

Recently, a new kind of analytic pseudo-potentials, directly in a fully nonlocal form, has been proposed [129,160]. Coefficients are fitted to minimize penalty functions, like atomic properties, to ensure that these properties are well reproduced for the pseudoatom. It has been further generalized for use in the context of QM/MM situations [161] or to include semi-empirically the long-range van der Waals attraction [161,162]. 

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. 

We only mention another fully nonlocal type of density functional for which a generalization to RDFT has been put forward [36], the weighted density ap-... 

In this respect three problems seem to be worthwhile mentioning. The GGA, which has become the standard xc-functional in the nonrelativistic context by now, can neither describe negative ions nor dispersion forces and also fails to reproduce the ground state of highly correlated systems. The first aspect reflects the fact that the single-particle spectrum produced by the GGA is far from the exact KS spectrum, due to the exponential decay of the GGA potential. The spectrum, however, is not only pertinent for the existence of negative ions, but is also particularly important for the study of excitation or ionization processes. The second problem of the GGA points at its semi-local character Only a fully nonlocal functional, which can build up an attractive force even in regions where the density vanishes, is able to reproduce dispersion forces. 

Q -t- 5p. Two paths can be followed towards the construction of actual current density functionals On the one hand, one can rewrite (325) as fully nonlocal density functional utilizing either that f x) - f[y) — 8p x) — 8p y) [23,21] or that Vp x) = V5y (x) [209]. On the other hand, one can restrict oneself to a long-wavelength expansion of the response kernels in (325), assuming 5p q) to be strongly localized, i.e. 8p x) to be rather delocalized. This approach leads to gradient corrections and has been pursued extensively in the case of the nonrelativistic Exc[d -... 

To make easier the connection with expressions in a finite spin-orbital basis, we systematically use four-point indexes for all the two-electron quantities. The starting point is therefore a fully nonlocal time-dependent Hamiltonian... 

Figure 4 Schematic of sampling choices for various QC formulations, (a) Local QC The repatom and its crystallite are completely contained inside one element (X indicates the quadrature point), (b) Nonlocal QC in the energy formulation The repatom is still contained inside the element, but its crystallite is not. (c) Fully nonlocal QC The repatoms coincide with the FEM nodes, and the atoms comprising their crystallites are individually considered in the energy calculation.

Fully Nonlocal QC The nonlocal formulation of the QC method was developed for modeling inhomogeneous structural features. A first formulation was presented in the original QC studies this method was later expanded (see, e.g.. Ref. 89) and, finally, the fully nonlocal QC (FNL-QC) method was developed by Knap and Ortiz.The key point of the nonlocal formulation is that each atom within the representative crystallite is displaced according to the actual continuum displacement field at its position. Thus, the displacement field considered when computing the energy (or force) of a repatom can be nonuniform. In the original formulation, the repatoms are... 

To achieve higher numerical efficiency, it is common practice to transform the semi-local pseudopotential (O Eq. 7.72) to a fully nonlocal form. 

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