Kohn-Sham Non-interacting System

Although we can search over all antisymmetric N-electron wavefunctions in (1.59), the minimizing wavefunction for a given density will be a noninteracting wavefunction (a single Slater determinant or a linear combination of a few) for some external potential such that 

Thus we have derived the Kohn-Sham method [6] of Sect. 1.1.2. 

The Kohn-Sham method treats Ts[ ] exactly, leaving only to be 

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