The Pseudo-potential Concept

The many-electron Schrodinger equation can be very much simplified if electrons are divided in two groups valence electrons and inner core electrons. The electrons in the inner shells are strongly bound and do not play a significant role in the chemical binding of atoms, thus forming with the nucleus an (almost) inert core. Binding properties are almost completely due to the valence electrons, especially in metals and semiconductors. 

This separation suggests that inner electrons can be ignored in a large number of cases, thereby reducing the atom to a ionic core that interacts with the valence electrons. The use of an effective interaction, a pseudopotential, that approximates the potential felt by the valence electrons, was first proposed by Fermi in 1934 [28]. Hellmann in 1935 [29] suggested that the form 

Let the exact solutions of the Schrodinger equation for the inner electrons be denoted by ipc), and ipv) those for the valence electrons. Then 

This equation indicates that states p ) satisfy a Schrbdinger-like equation with an energy-dependent pseudo-Hamiltonian 

A consequence of the cancellation between the two terms of (6.47) is the surprisingly good description of the electronic structure of solids given by the nearly-free electron approximation. The fact that many metal and semiconductor band structures are a small distortion of the free electron gas band structure suggests that the valence electrons do indeed feel a weak potential. The Phillips and Kleinman potential explains the reason for this cancellation. 

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