Approximations of exchange and correlation energy Exc (LDA)

Model of Hedin and Lundqvist

It's one of the approwimations used to express the  exchange and correlation energy. In rhis model, the exchange and correlations terms are dtermined separetly. The exchange term is defined by:

where  rs is the parameter of electrons gaz verifying the condition:

and the exchange potential term become:

The correlation term is defined by:

Model of Ceperly and Adler

This model is based upon the quantic Monte Carlo technique applied for electrons gaz. The correlation energy was adapted by Vosko, Wik and Nusair as follows:

In the form od VWN, the choice of rs is: 10, 30 and 50. The calculation is precise with a margin of error of 0.05 mRy.

The approach of Wigner which is imporoved by Pine gives the following formula:

Parametrizaton of the model of Ceperley and Adler

The model of Ceperley and Adler is not used for calculation under its primitive form. The parametrization of Perdew and Zunger for the exhange and correlation energy of this model allows to its almost direct exploitation to calculate the total energy within the LDA approach. This parametrization is established as follows:

                                                         Exc = Ex + Ec

and the exchange and correlation energies are :

Advantages of the model of Ceperley and Adler parametrized by Perdew and Zunger

- The results of Ceperley and Adler are based upon a stochastic sampling of an exact solution of the interaction of an electrons gaz.

- Their results interpolated by Perdew and Zunger are acceptable within the limit of the low and high densities.

- Une extenstion to include the spin polarization is possible.

Reference: The thesis of Pr Ali Zaoui working in France

ÉrUos DES PROPRIETES STRUCTURALES, ÉT ELECTRONIQUES ET OPTIQUES DES SEMICONDUCTEURS DE TYPE AnB8-n ET DE LEURS ALLIAGES

https://drive.google.com/open?id=0B-QmTukdEFfQOFE5UzVRdEZmQnVRbW9ERTJUYzFRNkRYbmhz