Managing the Many-Particle Hamiltonian

Density Functional Theory and the Local Density Approximation Even in light of the insights afforded by the Born-Oppenheimer approximation, our problem remains hopelessly complex. The true wave function of the system may be written as i/f(ri, T2, T3,. .., Vf ), where we must bear in mind, N can be a number of Avogadrian proportions. Furthermore, if we attempt the separation of variables ansatz, what is found is that the equation for the i electron depends in a nonlinear way upon the single particle wave functions of all of the other electrons. Though there is a colorful history of attempts to cope with these difficulties, we skip forth to the major conceptual breakthrough that made possible a systematic approach to these problems. 

In the language of the beginning of the chapter, the basic idea used in the density functional setting is to effect a replacement 

In particular, Hohenberg and Kohn advanced the view that the total energy of the many-electron system may be written as a unique functional of the electron density of the form, 

This results in a set of single particle equations, known as the Kohn-Sham 

Note that at this point we have turned the original (hopeless) many-body problem into a series of effective single particle Schrodinger equations. 

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