Density functional theory/Pseudo-potentials

Pseudo-potentials

The many electron Schrödinger 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; they also partially screen the nucleus, 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 an 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 and Hellmann in 1935. In spite of the simplification pseudo-potentials introduce in calculations, they remained forgotten until the late 50's.

Ab initio Pseudo-potentials

A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield and more recently Cronin, who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudo-potentials are obtained inverting the free atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo wave-functions to coincide with the true valence wave functions beyond a certain distance ${\displaystyle rl_{.}}$ . The pseudo wave-functions are also forced to have the same norm as the true valence wave-functions and can be written as

${\displaystyle R_{\rm {l}}^{\rm {pp}}(r)=R_{\rm {nl}}^{\rm {AE}}(r).}$
${\displaystyle \int _{0}^{rl}dr|R_{\rm {l}}^{\rm {PP}}(r)|^{2}r^{2}=\int _{0}^{rl}dr|R_{\rm {nl}}^{\rm {AE}}(r)|^{2}r^{2}.}$

where ${\displaystyle R_{\rm {l}}(r).}$  is the radial part of the wavefunction with angular momentum ${\displaystyle l_{.}}$ , and ${\displaystyle pp_{.}}$  and ${\displaystyle AE_{.}}$  denote, respectively, the pseudo wave-function and the true (all-electron) wave-function. The index n in the true wave-functions denotes the valence level. The distance beyond which the true and the pseudo wave-functions are equal, ${\displaystyle rl_{.}}$ , is also ${\displaystyle l_{.}}$ -dependent.