Editing Density functional theory (section)

==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 [[screening effect|screen]] the nucleus, thus forming with the [[nucleus (atomic structure)|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 1950s.

[[File:Pseudopotential.png|thumb|Pseudopotential representing the effective core charge. The physical image of the system with the accurate wavefunction and potential is replaced by a pseudo-wavefunction and a pseudopotential up to a cutoff value. In the image on the right, core electrons and atomic core are considered as the effective core in DFT calculations]]

===''Ab initio'' pseudo-potentials===

A crucial step toward more realistic pseudo-potentials was given by William C. Topp and [[John Hopfield]],<ref>{{Cite journal |last1=Topp |first1=William C. |last2=Hopfield |first2=John J. |date=1973-02-15 |title=Chemically Motivated Pseudopotential for Sodium |journal=Physical Review B |volume=7 |issue=4 |pages=1295–1303 |doi=10.1103/PhysRevB.7.1295 |bibcode=1973PhRvB...7.1295T}}</ref> 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-wavefunctions to coincide with the true valence wavefunctions beyond a certain distance {{mvar|r<sub>l</sub>}}. The pseudo-wavefunctions are also forced to have the same norm (i.e., the so-called norm-conserving condition) as the true valence wavefunctions and can be written as

: <math>\begin{align}
 R_l^\text{PP}(r) &= R_{nl}^\text{AE}(r), \text{ for } r > r_l,\\
 \int_0^{r_l} \big|R_l^\text{PP}(r)\big|^2 r^2 \,\mathrm{d}r &= \int_0^{r_l} \big|R_{nl}^\text{AE}(r)\big|^2 r^2 \,\mathrm{d}r,
\end{align}</math>

where {{math|''R<sub>l</sub>''(''r'')}} is the radial part of the [[wavefunction]] with [[angular momentum]] {{mvar|l}}, and PP and AE denote the pseudo-wavefunction and the true (all-electron) wavefunction respectively. The index {{mvar|n}} in the true wavefunctions denotes the [[valence (chemistry)|valence]] level. The distance {{mvar|r<sub>l</sub>}} beyond which the true and the pseudo-wavefunctions are equal is also dependent on {{mvar|l}}.