The Thomas–Fermi (TF) model,[1][2] 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.
[3] 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 statistical model in 1927 to approximate the distribution of electrons in an atom.
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,[4] where
The corresponding phase space volume is The electrons in ΔVph are distributed uniformly with two electrons per h3 of this phase space volume, where h is the Planck constant.
As such, they were able to calculate the energy of an atom using this expression for the kinetic energy 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).
The potential energy of an atom's electrons, due to the electric attraction of the positively charged nucleus is where
with charge Ze, where Z is a positive integer and e is the elementary charge, The potential energy of the electrons due to their mutual electric repulsion is, The total energy of the electrons is the sum of their kinetic and potential energies,[7] In order to minimize the energy E while keeping the number of electrons constant, we add a Lagrange multiplier term of the form to E. Letting the variation with respect to n vanish then gives the equation which must hold wherever
by then[10] If the nucleus is assumed to be a point with charge Ze at the origin, then
[11] From using the above equations together with Gauss's law, φ(r) can be seen to satisfy the Thomas–Fermi equation[12] For chemical potential μ = 0, this is a model of a neutral atom, with an infinite charge cloud where
[13] For μ > 0, it can be interpreted as a model of a compressed atom, so that negative charge is squeezed into a smaller space.
[14][15] Although this was an important first step, the Thomas–Fermi equation's accuracy is limited because the resulting expression for the kinetic energy is only approximate, and because the method does not attempt to represent the exchange energy of an atom as a conclusion of the Pauli exclusion principle.
A term for the exchange energy was added by Dirac in 1930,[16] which significantly improved its accuracy.
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.
In 1962, Edward Teller showed that Thomas–Fermi theory cannot describe molecular bonding – the energy of any molecule calculated with TF theory is higher than the sum of the energies of the constituent atoms.
More generally, the total energy of a molecule decreases when the bond lengths are uniformly increased.
[22] One notable historical improvement to the Thomas–Fermi kinetic energy is the Weizsäcker (1935) correction,[23] which is the other notable building block of orbital-free density functional theory.
The problem with the inaccurate modelling of the kinetic energy in the Thomas–Fermi model, as well as other orbital-free density functionals, is circumvented in Kohn–Sham density functional theory with a fictitious system of non-interacting electrons whose kinetic energy expression is known.