The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The Thomas–Fermi model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more sophisticated density approximation to the kinetic energy within modern orbital-free density functional theory. Working independently, Thomas and Fermi used this model in 1927 to approximate the distribution of electrons in an atom. Although electrons are distributed nonuniformly in an atom, the approximation was made that in each small volume element ΔV (i.e. locally), the electrons are distributed uniformly. The electron density n ( r ) {\displaystyle n(\mathbf {r} )} can still vary from one volume element to the next.
Kinetic energy For a small volume element ΔV, and for the atom in its ground state, we can fill out a spherical momentum-space volume VF up to the Fermi momentum pF, and thus
V F = 4 3 π p F 3 ( r ) , {\displaystyle V_{\text{F}}={\frac {4}{3}}\pi p_{\text{F}}^{3}(\mathbf {r} ),}
where r {\displaystyle \mathbf {r} } is the position vector of a point in ΔV. The corresponding phase-space volume is
Δ V ph = V F Δ V = 4 3 π p F 3 ( r ) Δ V . {\displaystyle \Delta V_{\text{ph}}=V_{\text{F}}\,\Delta V={\frac {4}{3}}\pi p_{\text{F}}^{3}(\mathbf {r} )\,\Delta V.}
In the phase-space volume ΔVph, the electrons are distributed uniformly with density 2/h3 where h is the Planck constant. The number of electrons in ΔVph is
Δ N ph = 2 h 3 Δ V ph = 8 π 3 h 3 p F 3 ( r ) Δ V . {\displaystyle \Delta N_{\text{ph}}={\frac {2}{h^{3}}}\,\Delta V_{\text{ph}}={\frac {8\pi }{3h^{3}}}p_{\text{F}}^{3}(\mathbf {r} )\,\Delta V.}
The electron number density in real space is this number per volume ΔV, and hence
n ( r ) = Δ N ph Δ V = 8 π 3 h 3 p F 3 ( r ) . {\displaystyle n(\mathbf {r} )={\frac {\Delta N_{\text{ph}}}{\Delta V}}={\frac {8\pi }{3h^{3}}}p_{\text{F}}^{3}(\mathbf {r} ).}
The fraction of electrons at r {\displaystyle \mathbf {r} } that have momentum between p and p + dp is
… excerpt ends here. Continue reading the full article.
