Thomas-Fermi theory

Fermi–Thomas theory

Sometimes called the

"statistical theory" ,

it was invented by

L.H. Thomas

[a13]

and

E. Fermi

[a2],

shortly after

E. Schrödinger

invented his

quantum-mechanical wave equation, in order to approximately

describe the

electron density,

,

,

and the

ground state energy,

for a large atom or molecule with a large number,

,

of electrons. Schrödinger's

equation, which would give the exact density and energy, cannot be

easily handled when

is large (cf. also

Schrödinger equation).

A starting point for the theory is the

Thomas–Fermi energy functional.

For a molecule with

nuclei of charges

and locations

(),

it is

(a1)

in suitable units. Here,

and

.

The constraint on

is

and

.

The functional

is convex (cf. also

Convex function (of a real variable)).

The justification for this functional is this:

The first term is roughly the minimum quantum-mechanical

kinetic energy of

electrons needed to produce an electron density

.

The second term is the attractive interaction of the

electrons with the

nuclei, via the

Coulomb potential

.

The third is approximately the electron-electron repulsive

energy.

is the nuclear-nuclear repulsion and is an important constant.

The

Thomas–Fermi energy

is defined to be

i.e., the Thomas–Fermi energy and density are obtained by minimizing

with

and

.

The

Euler–Lagrange equation,

in this case called the

Thomas–Fermi equation,

is

(a2)

where

,

is some constant

(a

Lagrange multiplier; cf.

Lagrange multipliers)

and

is the

Thomas–Fermi potential:

(a3)

The following essential mathematical facts about the

Thomas–Fermi equation were

established by

E.H. Lieb

and

B. Simon

[a7]

(cf. also

[a3]):

1)

There is a density

that minimizes

if and only if

.

This

is unique and it satisfies the Thomas–Fermi equation

(a2)

for some

.

Every positive solution,

,

of

(a2)

is a minimizer of

(a1)

for

.

If

,

then

and any minimizing sequence converges weakly in

to

.

2)

for all

.

(This need not be so for the real Schrödinger

.)

3)

is a strictly monotonically decreasing function of

and

(the

neutral case).

is the

chemical potential,

namely

is a strictly convex, decreasing function of

for

and

for

.

If

,

has compact support.

When

,

(a2)

becomes

.

By applying the

Laplace operator

to both sides, one obtains

which is the form in which the Thomas–Fermi

equation is usually stated (but it

is valid only for

).

An important property of the solution is

Teller's theorem

[a4]

(proved rigorously in

[a7]),

which implies that the

Thomas–Fermi molecule

is always unstable, i.e., for each

there are

numbers

with

such that

(a4)

where

is the Thomas–Fermi

energy with

,

and

.

The presence of

in

(a1)

is crucial for this result. The inequality is strict. Not only does

decrease when the nuclei are pulled infinitely far apart (which is

what

(a4)

says) but any dilation of the nuclear coordinates

(,

)

will decrease

in the neutral case

(positivity of the pressure)

[a3],

[a1].

This theorem plays an important role in the

stability of matter.

An important question concerns the connection between

and

,

the

ground state energy

(i.e., the infimum of the spectrum) of the

Schrödinger operator,

,

it was meant to approximate.

which acts on the

anti-symmetric functions

(i.e., functions of space and spin). It used to be believed that

is asymptotically exact as

,

but this is not quite right;

is also needed.

Lieb

and

Simon

[a7]

proved that if one fixes

and

and sets

,

with fixed

,

and sets

,

with

,

then

(a5)

In particular, a simple change of variables shows that

and hence the true energy of a large atom is asymptotically

proportional to

.

Likewise, there is a well-defined sense in which the

quantum-mechanical density converges to

(cf.

[a7]).

The Thomas–Fermi density for an atom located at

,

which is spherically symmetric, scales as

Thus, a large atom (i.e., large

)

is smaller than a

atom by a factor

in radius. Despite this seeming paradox, Thomas–Fermi

theory gives the correct

electron density in a real atom (so far as the bulk of the

electrons is concerned) as

.

Another important fact is the

large-

asymptotics of

for a neutral atom. As

,

independent of

.

Again, this behaviour agrees with quantum mechanics — on a

length scale

,

which is where the bulk of the electrons is to be found.

In light of the limit theorem

(a5),

Teller's theorem

can be understood as saying that, as

,

the quantum-mechanical binding energy of a molecule is of lower order

in

than the total ground state energy. Thus, Teller's theorem is

not a defect of Thomas–Fermi

theory (although it is sometimes interpreted that

way) but an important statement about the true quantum-mechanical

situation.

For finite

one can show, using the

Lieb–Thirring inequalities

[a12]

and the

Lieb–Oxford inequality

[a6],

that

,

with a modified

,

gives a lower bound to

.

Several

"improvements"

to Thomas–Fermi theory have been proposed, but none have a

fundamental significance in the sense of being

"exact"

in the

limit. The

von Weizsäcker correction

consists in adding a term

to

.

This preserves the convexity of

and adds

to

when

is large. It also has the effect that the range of

for which there is a minimizing

is extend from

to

.

Another correction, the

Dirac exchange energy,

is to add

to

.

This spoils the convexity but not the range

for which a

minimizing

exists, cf.

[a7]

for both of these corrections.

When a uniform external magnetic field

is present, the operator

in

is replaced by

with

and

denoting the Pauli spin matrices (cf. also

Pauli matrices).

This leads to a modified Thomas–Fermi theory

that is asymptotically exact as

,

but the theory depends on the manner in which

varies with

.

There are five distinct regimes and theories:

,

,

,

,

and

.

These

theories

[a8],

[a9]

are relevant for

neutron stars.

Another class of Thomas–Fermi theories with

magnetic fields is relevant for electrons confined to

two-dimensional geometries

(quantum dots)

[a10].

In this case there are three regimes. A convenient review

is

[a11].

Still another modification of Thomas–Fermi theory

is its extension from a

theory of the ground states of atoms and molecules (which corresponds

to zero temperature) to a theory of positive temperature states of

large systems such as stars

(cf.

[a5],

[a14]).

References

[a1]	R. Benguria, E.H. Lieb, "The positivity of the pressure in Thomas–Fermi theory" Comm. Math. Phys. , 63 (1978) pp. 193–218 ((Errata: 71 (1980), 94))
[a2]	E. Fermi, "Un metodo statistico per la determinazione di alcune priorieta dell'atome" Rend. Accad. Naz. Lincei , 6 (1927) pp. 602–607
[a3]	E.H. Lieb, "Thomas–Fermi and related theories of atoms and molecules" Rev. Mod. Phys. , 53 (1981) pp. 603–641 ((Errata: 54 (1982), 311))
[a4]	E. Teller, "On the stability of molecules in Thomas–Fermi theory" Rev. Mod. Phys. , 34 (1962) pp. 627–631
[a5]	J. Messer, "Temperature dependent Thomas–Fermi theory" , Lecture Notes Physics , 147 , Springer (1981)
[a6]	E.H. Lieb, S. Oxford, "An improved lower bound on the indirect Coulomb energy" Internat. J. Quant. Chem. , 19 (1981) pp. 427–439
[a7]	E.H. Lieb, B. Simon, "The Thomas–Fermi theory of atoms, molecules and solids" Adv. Math. , 23 (1977) pp. 22–116
[a8]	E.H. Lieb, J.P. Solovej, J. Yngvason, "Asymptotics of heavy atoms in high magnetic fields: I. lowest Landau band region" Commun. Pure Appl. Math. , 47 (1994) pp. 513–591
[a9]	E.H. Lieb, J.P. Solovej, J. Yngvason, "Asymptotics of heavy atoms in high magnetic fields: II. semiclassical regions" Comm. Math. Phys. , 161 (1994) pp. 77–124
[a10]	E.H. Lieb, J.P. Solovej, J. Yngvason, "Ground states of large quantum dots in magnetic fields" Phys. Rev. B , 51 (1995) pp. 10646–10665
[a11]	E.H. Lieb, J.P. Solovej, J. Yngvason, "Asymptotics of natural and artificial atoms in strong magnetic fields" W. Thirring (ed.) , The stability of matter: from atoms to stars, selecta of E.H. Lieb , Springer (1997) pp. 145–167 (Edition: Second)
[a12]	E.H. Lieb, W. Thirring, "Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities" E. Lieb (ed.) B. Simon (ed.) A. Wightman (ed.) , Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann , Princeton Univ. Press (1976) pp. 269–303 ((See also: W. Thirring (ed.), The stability of matter: from the atoms to stars, Selecta of E.H. Lieb, Springer, 1977))
[a13]	L.H. Thomas, "The calculation of atomic fields" Proc. Cambridge Philos. Soc. , 23 (1927) pp. 542–548
[a14]	W. Thirring, "A course in mathematical physics" , 4 , Springer (1983) pp. 209–277

Elliott H. Lieb

Copyright to this article is held by Elliott Lieb.