Self exchange-correlation function

To perform excited-state calculations, one has to approximate the exchange-correlation potential. Local self-interaction-free approximate exchange-correlation potentials have been proposed for this purpose [73]. We can try to construct these functionals as orbital-dependent functionals. There are different exchange-correlation functionals for the different excited states, and we suppose that the difference between the excited-state functionals can be adequately modeled through the occupation numbers (i.e., the electron configuration). Both the OPM and the KLI methods have been generalized for degenerate excited states [37,40]. 

Actually, the classical charge-cloud repulsion is somewhat inappropriate for electrons in that smearing an electron (a particle) out into a cloud forces it to repel itself, as any two regions of the cloud interact repulsively. One way to compensate for this physically incorrect electron self-interaction is with a good exchange-correlation functional (below). 

Such a distinction is made here for two reasons a) In the cases where the theoretical considerations lead to a given vxc directly, the parent functional is not known, b) In the process of developing approximations to the exchange-correlation functional, it is frequently the case that the functional is tested on electron densities obtained with a potential corresponding to another exchange-correlation potential i.e. not self-consistently. 

The core electrons of all atoms were treated via ultra-soft pseudo potentials [10,11] with a cut-off of 25 Ry for wave function, and 240 for electronic density. The PBE gradient-corrected exchange-correlation function was used in self-consistent DFT calculations. The geometry optimization was performed using a lxlxl /c-point mesh. Because of the natural paired electron occupancies of the adsorbates, spin polarization effects were not considered to be important and were not treated explicitly in this study. 

In the following we will discuss in more details methods for solving practically this self-consistent equation in order to find the ground state of a chemical system, however we first need to have tractable approximations for the exchange-correlation functional Exc-... 

The development that led to the emergence of DFT to its current status as the most widely used electronic structure computational method is that due to KS [274]. The ansatz used by KS replaces the interacting problem with an auxiliary independent particle one, with all the many-body effects included in an exchange-correlation functional. In practice, the KS scheme introduces an equivalent orbital picture (rigorously established), with the resulting KS equations solved self-consistently. 

Once the exchange-correlation functional is chosen, the N one-electron coupled, nonUnear Eqs. (13) can be solved self-consistently for the lAi s. To... 

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