“Thomas–Fermi model”的意思、由来-开放百科全书

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.

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.

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 V_F up to the Fermi momentum p_F , and thus,^[4]

The electrons in ΔV_ph are distributed uniformly with two electrons per h³ of this phase space volume, where h is Planck's constant.^[5] Then the number of electrons in ΔV_ph 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,

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).

Potential energies

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,

Total energy

The total energy of the electrons is the sum of their kinetic and potential energies,^[7]

The Thomas–Fermi equation

In order to minimize the energy E while keeping the number of electrons constant, we add a Lagrange multiplier term of the form

which must hold wherever

is nonzero.^[8]^[9] If we define the total potential

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 a₀ is the Bohr radius.^[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

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 dφ/dr = φ/r.^[14]^[15]

Inaccuracies and improvements

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.

See also

Footnotes

1. ^{{cite journal|title=The calculation of atomic fields|journal=Proc. Camb. Phil. Soc.|year=1927|first=L. H.|last=Thomas|volume=23|issue=5|pages=542–548|doi=10.1017/S0305004100011683|bibcode = 1927PCPS...23..542T }}
2. ^{{cite journal|title=Un Metodo Statistico per la Determinazione di alcune Prioprietà dell'Atomo|journal=Rend. Accad. Naz. Lincei|year=1927|first=Enrico|last=Fermi|volume=6|issue=|pages=602–607|url=http://babel.hathitrust.org/cgi/pt?seq=339&view=image&size=100&id=mdp.39015001321200&u=1&num=278}}
3. ^{{cite journal| last = Schrödinger| first = Erwin| authorlink = Erwin Schrödinger| title = An Undulatory Theory of the Mechanics of Atoms and Molecules| journal = Phys. Rev.| volume = 28| issue = 6| pages = 1049–1070 |date=December 1926| url = http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf| format = PDF| doi = 10.1103/PhysRev.28.1049|bibcode = 1926PhRv...28.1049S }}
4. ^March 1992, p.24
5. ^Parr and Yang 1989, p.47
6. ^March 1983, p. 5, Eq. 11
7. ^March 1983, p. 6, Eq. 15
8. ^March 1983, p. 6, Eq. 18
9. ^A Brief Review of Thomas-Fermi Theory, Elliott H. Lieb, http://physics.nyu.edu/LarrySpruch/Lieb.pdf, (2.2)
10. ^March 1983, p. 7, Eq. 20
11. ^March 1983, p. 8, Eq. 22, 23
12. ^March 1983, p. 8
13. ^March 1983, pp. 9-12.
14. ^March 1983, p. 10, Figure 1.
15. ^p. 1562, Feynman, Metropolis, and Teller 1949.
16. ^{{cite journal|last=Teller|first=E.|year=1962|title=On the Stability of molecules in the Thomas–Fermi theory|journal=Rev. Mod. Phys.|volume=34|issue=4|pages=627–631|doi=10.1103/RevModPhys.34.627|bibcode = 1962RvMP...34..627T }}
17. ^{{cite journal|last=Balàzs|first=N.|year=1967|title=Formation of stable molecules within the statistical theory of atoms|journal=Phys. Rev.|volume=156|issue=1|pages=42–47|doi=10.1103/PhysRev.156.42|bibcode = 1967PhRv..156...42B }}
18. ^{{cite journal|last=Lieb|first=Elliott H.|author2=Simon, Barry |year=1977|title=The Thomas–Fermi theory of atoms, molecules and solids |journal=Adv. Math.|volume=23|issue=1|pages=22–116|doi=10.1016/0001-8708(77)90108-6}}
19. ^Parr and Yang 1989, pp.114–115
20. ^Parr and Yang 1989, p.127
21. ^{{cite journal|title=Zur Theorie der Kernmassen|journal=Zeitschrift für Physik|year=1935|first=C. F. v.|last=Weizsäcker|volume=96|issue=7-8|pages=431–458|doi=10.1007/BF01337700|bibcode = 1935ZPhy...96..431W }}