Exchange correlation functionals, local theory

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Until now we only considered the formal framework of density functional theory. However, the theory would be of little use if we would not be able to construct good approximate functionals for the exchange-correlation energy and exchange-correlation functional. Historically the first approximation for the exchange-correlation functional to be used was the local density approximation. In this approximation the exchange-correlation functional is taken to be... 

Density functional calculations have become an increasingly popular method of computational molecular quantum chemistry during the last two decades. While density functional theory had been an established tool in solid state physics much earlier, its way into mainstream molecular quantum chemistry was paved by the advent of gradient-corrected exchange-correlation functionals. These functionals alleviate the problems the earlier local-density schemes have especially at the border of a finite molecular system and thus allowed... 

It is impossible to develop a current-dependent relativistic exchange-correlation functional, which is computationally tractable and reduces to spin-density functional theory in the weakly relativistic limit. One reason is that there is no local approximation to such a functional since j vanishes for any homogeneous system. This means that the relativistic electron gas cannot serve as a starting point. More insight is gained from a Gordon decomposition of the current density j (see e.g. Refs. [7, 24]), which shows that j consists of an orbital part and the curl of a magnetisation density fh. 

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