Electronic density - the superstar

John Slater showed that the Hartree-Fock method applied to metals gives the exchange 

In the DFT method we begin from the Bom-Oppenheimer approximation, that allows us to obtain the electronic wave function corresponding to fixed positions of the nuclei. We will be interested in the ground-state of the system. 

we need only the electron density distribution defined as  

It is seen that we obtain p by carrying out the integration of over the coordinates space and spin) of all the electrons except one (in our case electron 1 with coordinates r, cri) and in addition the summation over its spin coordinate ( xi). Thus we obtain a function of the position of electron 1 in space p(r). The wave function is antisymmetric with respect to the exchange of the coordinates of any two electrons, and, therefore, is symmetric with respect to such an exchange. 

the definition of p is independent of the label of the electron we do not integrate over. According to this definition, p represents nothing else but the density of the electron cloud carrying N electrons, hence (integration over the whole 3D space)  

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