Total Energies in the Local Density Approximation

Once the effective single particle equations have been derived, there are almost limitless possibilities in terms of the different schemes that have been invented to solve them. By way of contrast, the assertion is often made that if the calculations in these different realizations of the first-principles methodology are made correctly, then the predictions of these different implementations should be the same. 

Interactions Between Nuclei and Electrons Note that in our treatment of the Kohn-Sham equations, we assume the existence of an external potential Vexj(r) which characterizes the interactions of the nuclei and the electrons. Given our interest in modeling the properties of solid materials. 

Vext will arise as a superposition of the potentials associated with each of the nuclei making up the solid. In the most direct of pictures, one can imagine treating the solid by superposing the bare nuclear potentials for each atom, and subsequently solving for the disposition of each and every electron. This approach is the basis of the various all-electron methods. 

In our statement of the Kohn-Sham equations, nothing has been said about the way in which i/ (r) itself will be determined. Many methods share the common feature that the wave function is presumed to exist as a linear combination of some set of basis functions which might be written generically as 

Part of the variety associated with the many different incarnations of density functional codes is associated with this choice of basis functions. For example, in the spirit of the earlier tight-binding discussion, one can imagine a basis constructed out of functions localized about each atomic site. Alternatively, we can 

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