Thomas–Fermi model

The predecessor to density functional theory was the Thomas-Fermi model, developed by Thomas and 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 h³ of volume. For each element of coordinate space volume d³r we can fill out a sphere of momentum space up to the Fermi momentum p_f

$(4/3)\pi p_{f}^{3}({\vec {r}})$

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

$n({\vec {r}})={\frac {8\pi }{3h^{3}}}p_{f}^{3}({\vec {r}})$

Solving for p_f and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:

$T_{TF}[n]=C_{F}\int n^{5/3}({\vec {r}})d^{3}r$

where   $C_{F}={\frac {3h^{2}}{10m}}\left({\frac {3}{8\pi }}\right)^{2/3}$

As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nuclear-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 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.

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 Weizsäcker (1935) correction:

$T_{W}[n]={\frac {1}{8}}{\frac {\hbar ^{2}}{m}}\int {\frac {|\nabla n({\vec {r}})|^{2}}{n({\vec {r}})}}dr$

Related Books

• R. G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989), ISBN 0-19-504279-4, ISBN 0-19-509276-7 (pbk.).
• N.H. March, Electron Density Theory of Atoms and Molecules (Academic Press, 1992), ISBN 0-12-470525-1.