The metallic bond and band theory

As we have just seen, the average potential theory cannot be used to determine the number of free electrons per atom of metal. In order to find it, we need to return to the average potential hypothesis and 

In the metal, each of the electrons is subject to the influence of all the nuclei and all the other electrons. In view of the periodic arrangement of the atoms, that potential is periodic, becoming infinite at each nucleus and minimal at the points furthest from the nuclei. We cannot hope to solve the Schrodinger equation for so complex a system. Certain calculations have been performed in specific cases - in particular, by Bloch, Brillouin, Wigner, Seitz and Slater, among others. 

We suppose that the wave function of the bond electrons is independent of the wave function of the ions comprising nuclei and inner-layer electrons. 

it is useful to solve the Schrodinger equation for the valence electrons, which is written as  

We construct the wave function of the valence electrons by linear combination of the atomic orbitals by adding the atoms in the line one after another, reasoning on the example of the s orbitals. 

---