Editing Density functional theory (section)

==Thomas–Fermi model==
The predecessor to density functional theory was the '''[[Thomas–Fermi model]]''', developed independently by both [[Llewellyn Thomas]] and [[Enrico Fermi]] in 1927. They used a statistical model to approximate the distribution of electrons in an atom. The mathematical basis postulated that electrons are distributed uniformly in phase space with two electrons in every <math>h^3</math> of volume.<ref name='ParrYang1989p47'>{{harvp|Parr|Yang|1989|p=47}}.</ref> For each element of coordinate space volume <math>\mathrm d^3 \mathbf r</math> we can fill out a sphere of momentum space up to the [[Fermi momentum]] <math>p_\text{F}</math><ref>{{cite book |last=March |first= N. H. |title=Electron Density Theory of Atoms and Molecules |publisher=Academic Press |year=1992 |isbn=978-0-12-470525-8 |page=24}}</ref>

: <math>\tfrac43 \pi p_\text{F}^3(\mathbf r).</math>

Equating the number of electrons in coordinate space to that in phase space gives

: <math>n(\mathbf r) = \frac{8\pi}{3h^3} p_\text{F}^3(\mathbf r).</math>

Solving for {{math|''p''<sub>F</sub>}} and substituting into the [[Classical mechanics|classical]] [[kinetic energy]] formula then leads directly to a kinetic energy represented as a [[functional (mathematics)|functional]] of the electron density:

: <math>\begin{align}
 t_\text{TF}[n] &= \frac{p^2}{2m_e} \propto \frac{(n^{1/3})^2}{2m_e} \propto n^{2/3}(\mathbf r), \\
 T_\text{TF}[n] &= C_\text{F} \int n(\mathbf r) n^{2/3}(\mathbf r) \,\mathrm d^3 \mathbf r = C_\text{F} \int n^{5/3}(\mathbf r) \,\mathrm d^3 \mathbf r,
\end{align}</math>

where

: <math>C_\text{F} = \frac{3h^2}{10m_e} \left(\frac{3}{8\pi}\right)^{2/3}.</math>

As such, they were able to calculate the [[energy]] of an atom using this kinetic-energy functional combined with the classical expressions for the nucleus–electron and electron–electron interactions (which can both also be represented in terms of the electron density).

Although this was an important first step, the Thomas–Fermi equation's accuracy is limited because the resulting kinetic-energy functional is only approximate, and because the method does not attempt to represent the [[exchange energy]] of an atom as a conclusion of the [[Pauli principle]]. An exchange-energy functional was added by [[Paul Dirac]] in 1928.

However, the Thomas–Fermi–Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect of [[electron correlation]].

[[Edward Teller]] (1962) showed that Thomas–Fermi theory cannot describe molecular bonding. This can be overcome by improving the kinetic-energy functional.

The kinetic-energy functional can be improved by adding the [[Carl Friedrich von Weizsäcker|von Weizsäcker]] (1935) correction:<ref name='Weizsäcker1935'>{{cite journal |title=Zur Theorie der Kernmassen |language=de |trans-title=On the theory of nuclear masses |journal=Zeitschrift für Physik |year=1935 |first=C. F. |last=von Weizsäcker |volume=96 |issue=7–8 |pages=431–458 |doi=10.1007/BF01337700 |bibcode=1935ZPhy...96..431W|s2cid=118231854 }}</ref><ref name='ParrYang1989p127'>{{harvp|Parr|Yang|1989|p=127}}.</ref>

: <math>T_\text{W}[n] = \frac{\hbar^2}{8m} \int \frac{|\nabla n(\mathbf r)|^2}{n(\mathbf r)} \,\mathrm d^3 \mathbf r.</math>