The Kohn-Sham Auxiliary System of Equations

The Hohenberg-Kohn theorem says nothing specific about the form of Thk[w], and therefore the utility of the density functional theory depends on the choice of sufficiently accurate approximations for it. In order to do this, the unknown functional, E[n], is rewritten as a sum of the Hartree total energy and another but presumably smaller unknown functional called the exchange-correlation functional, Exc[n]. 

Kohn and Shahm published the second excellent work in 1965 [13] whose formulation of density functional theory has become the basis of much of present-day methods for treating electrons in atoms, molecules, and solids. The approach proposed by Kohn and Sham replaced the original many-body problem by an auxiliary independent-particle problem. 

Kohn and Sham assumed that the ground-state density of the original interacting system is equal to that of another yet unknown noninteracting system. They showed how it is possible to replace the many-body problem by an equivalent set of self consistent one-electron equations. 

This leads to independent-particle equations for the noninteracting system that can be considered exactly soluble (in practice by numerical means) with all the difficult many-body terms incorporated into an exchange-correlation functional of the electron density. The Kohn-Sham approach has indeed led to very useful approximations that are now the basis of most calculations that attempt to make ab initio predictions for the properties of solids and large molecular system. The approach is remarkably accurate, most notably for wide-band systems, such as the group IV and 11-V semiconductors, the sp bonded metals like sodium and aluminum, insulators like diamond, sodium chloride, and molecules with covalent and ionic bonds. It also appears to be successful in many cases in which the electrons have stronger effects of correlation, such as in transition metals. 

The Kohn-Sham construction of an auxiliary system rests upon two assumptions  

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