Functional semi-local

Kristyan, S., Pulay, P., 1994, Can (Semi)Local Density Functional Theory Account for the London Dispersion Forces , Chem. Phys. Lett., 229, 175. 

The description of bonding at transition metal surfaces presented here has been based on a combination of detailed experiments and quantitative theoretical treatments. Adsorption of simple molecules on transition metal surfaces has been extremely well characterized experimentally both in terms of geometrical structure, vibrational properties, electronic structure, kinetics, and thermo-chemistry [1-3]. The wealth of high-quality experimental data forms a unique basis for the testing of theoretical methods, and it has become clear that density functional theory calculations, using a semi-local description of exchange and correlation effects, can provide a semi-quantitative description of surface adsorption phenomena [4-6]. Given that the DFT calculations describe reality semi-quantitatively, we can use them as a basis for the analysis of catalytic processes at surfaces. 

The complexity of the functional form for Cc0Te increases from equation (67) to equation (71). To remove the necessity of accepting any particular predetermined functional form Kahn and Goddard29 evaluated a semi-local potential by making use of... 

There are exceptions from this trend. A noted example is the polarizability of sodium clusters significantly underestimated by semi-local functionals.73... 

The term non-local is used sometimes in die literature in association with gradient-dependent (GGA) functionals. This nomenclature is not applied in this work. The LDA-, GGA-, and meta-GGA functionals are referred to as semi-local as they do not account for any long-range non-locality of the exchange-correlation energy density excseim local(r) = exc(p(r), Vp(r), V2p(r), r(r)) whereas... 

Kristyan S, Pulay P (1994) Can (semi)local density-functional theory account for London dispersion forces, Chem Phys Lett, 229 175-180... 

For semi-local xc functionals such as the local density approximation, general gradient approximations or meta functionals using the kinetic energy density, the xc energy can, using (18) and (19), be written as... 

There are different kinds of exchange-correlation functionals. The local and semi-local ones are the most efficient from the computational cost standpoint. They were the first ones to be used in actual calculations and they give quite accurate results for a number of properties, such as equilibrium geometries, vibration frequencies, and crystal compressibilities. 

Table 1.1 MAEs (in kcal/mol) in atomization energies, barrier heights, and noncovalent binding energies of the global hybrids based on various local or semi-local functionals. The value of the mixing parameter ao for each case is also reported. The MAEs of the original local or semi-local functionals are given in parenthesis...

The semi-classical equations of motion obtained above involve only the transverse adiabatic vector potential which is, by definition, independent of the choice of gauge functions/(q) and g(q). The (Aj -f A2)/2M term in the potential is also independent of those two arbitrary functions. The locally quadratic approach to Gaussian dynamics therefore gives physically equivalent results for any choice of /(q) and g(q). The finding that the locally quadratic Hamiltonian approach developed here is strictly invariant with respect to choice of phases of the adiabatic electronic eigenstates supersedes the approximate discussion of gauge invariance given earlier by Romero-Rochin and Cina [25] (see also [40]). 

Another spin spiral with q = [0,0, rx] with rx = 0.5 — 0.6 depending on volume, was found theoretically very early [138, 139, 140, 141, 142], but it was only recently that a spin spiral with a wave vector near the experimental value was found theoretically [143, 144], and then at much lower volumes. In all these calculations one has assumed bulk geometry, which seems reasonable since the number of Fe atoms is quite big 105 — 107. The other common approximation is the use of local or semi-local exchange-correlation functionals, and this might be one of the reasons for the failure of reproducing experimental results. 

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. 

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