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 TF 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.

- Kinetic energy
- Potential energies
- Total energy
- The Thomas–Fermi equation
- Inaccuracies and improvements
- See also
- Footnotes
- References

Working independently, Thomas and Fermi used this statistical model in 1927 to approximate the distribution of electrons in an atom. Although electrons are distributed nonuniformly in an atom, an approximation was made that the electrons are distributed uniformly in each small volume element *ΔV* (i.e. locally) but the electron density can still vary from one small volume element to the next.

For a small volume element *ΔV*, and for the atom in its ground state, we can fill out a spherical momentum space volume *V*_{F} up to the Fermi momentum *p*_{F} , and thus,^{ [4] }

where is the position vector of a point in *ΔV*.

The corresponding phase space volume is

The electrons in *ΔV*_{ph} are distributed uniformly with two electrons per *h ^{3}* of this phase space volume, where

The number of electrons in *ΔV* is

where is the electron number density.

Equating the number of electrons in *ΔV* to that in *ΔV*_{ph} gives,

The fraction of electrons at that have momentum between *p* and *p+dp* is,

Using the classical expression for the kinetic energy of an electron with mass *m _{e}*, the kinetic energy per unit volume at for the electrons of the atom is,

where a previous expression relating to has been used and,

Integrating the kinetic energy per unit volume over all space, results in the total kinetic energy of the electrons,^{ [6] }

This result shows that the total kinetic energy of the electrons can be expressed in terms of only the spatially varying electron density according to the Thomas–Fermi model. 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 is the potential energy of an electron at that is due to the electric field of the nucleus. For the case of a nucleus centered at 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 is nonzero.^{ [8] }^{ [9] } If we define the total potential by

then^{ [10] }

If the nucleus is assumed to be a point with charge *Ze* at the origin, then and will both be functions only of the radius , and we can define *φ(r)* by

where *a _{0}* is the Bohr radius.

For chemical potential *μ *= 0, this is a model of a neutral atom, with an infinite charge cloud where is everywhere nonzero and the overall charge is zero, while for *μ* < 0, it is a model of a positive ion, with a finite charge cloud and positive overall charge. The edge of the cloud is where *φ(r)*=0.^{ [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. In this case the atom ends at the radius *r* where d*φ*/d*r* = *φ*/*r*.^{ [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 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.

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.^{ [16] }^{ [17] }^{ [18] }^{ [19] } This can be overcome by improving the expression for the kinetic energy.^{ [20] }

One notable historical improvement to the Thomas–Fermi kinetic energy is the Weizsäcker (1935) correction,^{ [21] }

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.

- ↑ Thomas, L. H. (1927). "The calculation of atomic fields".
*Proc. Camb. Phil. Soc*.**23**(5): 542–548. Bibcode:1927PCPS...23..542T. doi:10.1017/S0305004100011683. - ↑ Fermi, Enrico (1927). "Un Metodo Statistico per la Determinazione di alcune Prioprietà dell'Atomo".
*Rend. Accad. Naz. Lincei*.**6**: 602–607. - ↑ Schrödinger, Erwin (December 1926). "An Undulatory Theory of the Mechanics of Atoms and Molecules" (PDF).
*Phys. Rev*.**28**(6): 1049–1070. Bibcode:1926PhRv...28.1049S. doi:10.1103/PhysRev.28.1049. - ↑ March 1992, p.24
- ↑ Parr and Yang 1989, p.47
- ↑ March 1983, p. 5, Eq. 11
- ↑ March 1983, p. 6, Eq. 15
- ↑ March 1983, p. 6, Eq. 18
- ↑ A Brief Review of Thomas-Fermi Theory, Elliott H. Lieb, http://physics.nyu.edu/LarrySpruch/Lieb.pdf, (2.2)
- ↑ March 1983, p. 7, Eq. 20
- ↑ March 1983, p. 8, Eq. 22, 23
- ↑ March 1983, p. 8
- ↑ March 1983, pp. 9-12.
- ↑ March 1983, p. 10, Figure 1.
- ↑ p. 1562, Feynman, Metropolis, and Teller 1949.
- ↑ Teller, E. (1962). "On the Stability of molecules in the Thomas–Fermi theory".
*Rev. Mod. Phys*.**34**(4): 627–631. Bibcode:1962RvMP...34..627T. doi:10.1103/RevModPhys.34.627. - ↑ Balàzs, N. (1967). "Formation of stable molecules within the statistical theory of atoms".
*Phys. Rev*.**156**(1): 42–47. Bibcode:1967PhRv..156...42B. doi:10.1103/PhysRev.156.42. - ↑ Lieb, Elliott H.; Simon, Barry (1977). "The Thomas–Fermi theory of atoms, molecules and solids".
*Adv. Math*.**23**(1): 22–116. doi:10.1016/0001-8708(77)90108-6. - ↑ Parr and Yang 1989, pp.114–115
- ↑ Parr and Yang 1989, p.127
- ↑ Weizsäcker, C. F. v. (1935). "Zur Theorie der Kernmassen".
*Zeitschrift für Physik*.**96**(7–8): 431–458. Bibcode:1935ZPhy...96..431W. doi:10.1007/BF01337700.

In physics, the **kinetic energy** (**KE**) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called **energy levels**. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

In physics, the **mean free path** is the average distance travelled by a moving particle between successive impacts (collisions), which modify its direction or energy or other particle properties.

An ideal **Fermi gas** is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statistics determine the energy distribution of fermions in a Fermi gas in thermal equilibrium, and is characterized by their number density, temperature, and the set of available energy states. The model is named after the Italian physicist Enrico Fermi.

In physics, **screening** is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity *ε*, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

**Density functional theory** (**DFT**) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent electron density. Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a functional to a change in a function on which the functional depends.

In thermodynamics and solid state physics, the **Debye model** is a method developed by Peter Debye in 1912 for estimating the phonon contribution to the specific heat in a solid. It treats the vibrations of the atomic lattice (heat) as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to – the **Debye T^{3} law**. Just like the Einstein model, it also recovers the Dulong–Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

In solid state physics and condensed matter physics, the **density of states** (**DOS**) of a system describes the number of states that are to be occupied by the system at each level of energy. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation.

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In solid-state physics, the **free electron model** is a simple model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quantum mechanical Fermi–Dirac statistics and hence it is also known as the **Drude–Sommerfeld model**.

**Jellium**, also known as the **uniform electron gas** (**UEG**) or **homogeneous electron gas** (**HEG**), is a quantum mechanical model of interacting electrons in a solid where the positive charges are assumed to be uniformly distributed in space whence the electron density is a uniform quantity as well in space. This model allows one to focus on the effects in solids that occur due to the quantum nature of electrons and their mutual repulsive interactions without explicit introduction of the atomic lattice and structure making up a real material. Jellium is often used in solid-state physics as a simple model of delocalized electrons in a metal, where it can qualitatively reproduce features of real metals such as screening, plasmons, Wigner crystallization and Friedel oscillations.

**Local-density approximations** (**LDA**) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT) that depend solely upon the value of the electronic density at each point in space. Many approaches can yield local approximations to the XC energy. However, overwhelmingly successful local approximations are those that have been derived from the homogeneous electron gas (HEG) model. In this regard, LDA is generally synonymous with functionals based on the HEG approximation, which are then applied to realistic systems.

In physics and quantum chemistry, specifically density functional theory, the **Kohn–Sham equation** is the one electron Schrödinger equation of a fictitious system of non-interacting particles that generate the same density as any given system of interacting particles. The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as *v _{s}*(

In computational chemistry, **orbital-free density functional theory** is a quantum mechanical approach to electronic structure determination which is based on functionals of the electronic density. It is most closely related to the Thomas–Fermi model. Orbital-free density functional theory is, at present, less accurate than Kohn–Sham density functional theory models, but has the advantage of being fast, so that it can be applied to large systems.

In quantum mechanics, **orbital magnetization**, **M**_{orb}, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, **M**_{spin}, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

**Heat transfer physics** describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

**Thomas–Fermi screening** is a theoretical approach to calculate the effects of electric field screening by electrons in a solid. It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector is much smaller than the fermi wavevector, i.e. the long-distance limit. It is named after Llewellyn Thomas and Enrico Fermi.

In solid state physics the **electron heat capacity** or **electronic specific heat** describes the contribution of electrons to the heat capacity. Heat is transported by phonons and by free electrons in solids. For pure metals, however, the electronic contributions dominate in the thermal conductivity. In impure metals, the electron mean free path is reduced by collisions with impurities, and the phonon contribution may be comparable with the electronic contribution.

- R. G. Parr and W. Yang (1989).
*Density-Functional Theory of Atoms and Molecules*. New York: Oxford University Press. ISBN 978-0-19-509276-9. - N. H. March (1992).
*Electron Density Theory of Atoms and Molecules*. Academic Press. ISBN 978-0-12-470525-8. - N. H. March (1983). "1. Origins – The Thomas–Fermi Theory". In S. Lundqvist and N. H. March (eds.).
*Theory of The Inhomogeneous Electron Gas*. Plenum Press. ISBN 978-0-306-41207-3.CS1 maint: uses editors parameter (link) - R. P. Feynman, N. Metropolis, and E. Teller. "Equations of State of Elements Based on the Generalized Thomas-Fermi Theory".
*Physical Review***75**, #10 (May 15, 1949), pp. 1561-1573.

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