Reflections on the DFT success

The DFT method has a long history behind it, which began with Thomas, Dirac, Fermi, etc. At the beginning the successes were quite modest (the electron gas theory, known as the Xa method). Real success came after a publication by Jan Andzelm and Erich Wimmer. The DFT method, offering results at a correlated level for a wide spectrum of physical quantities, turned out to be roughly as inexpensive as the Hartree-Fock procedure - this is the most sensational feature of the method. 

We have a beacon - exact electron density distribution of harmonium 

Hohenberg and Kohn proved their famous theorem on the existenee of the energy functional, but nobody was able to give the functional for any system. All the 

DFT efforts are directed towards elaborating such a potential, and the only criterion of whether a model is any good, is comparison with experiment. However, it turned out that there is a system for which every detail of the DFT can be verified. Uniquely, the dragon may be driven out the hole and we may fearlessly and with impunity analyze all the details of its anatomy. The system is a bit artificial, it is the harmonic helium atom (harmonium) discussed on p. 185, in which the two electrons attract the nucleus by a harmonic force, while repelling each other by Coulombic interaction. For some selected force constants k, e.g., for A =, the Schrodinger equation can be solved analytically. The wave function is extremely simple, see p. 507. The electron density (normalized to 2) is computed as 

It turns out that in the case analyzed (and so far only in this case) we can calculate the exact total energy [eq. (11.15)], wonder potential vq that in the Kohn-Sham model gives the exact density distribution p [eq. (11.76)], exchange potential Vx and correlation [eqs. (11.67) and (11.68)]. Let us begin from the total energy. 

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