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''Fermi–Thomas theory''

Sometimes called the

"statistical theory" ,

it was invented by

L.H. Thomas

[[#References|[a13]]]

and

E. Fermi

[[#References|[a2]]],

shortly after

E. Schrödinger

invented his

quantum-mechanical wave equation, in order to approximately

describe the

electron density,

$\rho ( x )$,

$x \in \mathbf{R} ^ { 3 }$,

and the

ground state energy,

$E ( N )$

for a large atom or molecule with a large number,

$N$,

of electrons. Schrödinger's

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

easily handled when

$N$

is large (cf. also

[[Schrödinger equation|Schrödinger equation]]).

A starting point for the theory is the

Thomas–Fermi energy functional.

For a molecule with

$K$

nuclei of charges

$Z_i &gt; 0$

and locations

$R_{i} \in \mathbf{R} ^ { 3 }$

($i = 1 , \ldots , K$),

it is

\begin{equation} \tag{a1} \mathcal{E} ( \rho ) : = \end{equation}

\begin{equation*} := \frac { 3 } { 5 } \gamma \int _ { \mathbf R ^ { 3 } } \rho ( x ) ^ { 5 / 3 } d x - \int _ { \mathbf R ^ { 3 } } V ( x ) \rho ( x ) d x + \end{equation*}

\begin{equation*} +\frac { 1 } { 2 } \int _ { {\bf R} ^ { 3 } } \int _ { {\bf R} ^ { 3 } } \frac { \rho ( x ) \rho ( y ) } { | x - y | } d x d y + U \end{equation*}

in suitable units. Here,

\begin{equation*} V ( x ) = \sum _ { j = 1 } ^ { K } Z _ { j } | x - r _ { j } | ^ { - 1 }, \end{equation*}

\begin{equation*} U = \sum _ { 1 \leq i &lt; j \leq K } Z _ { i } Z _ { j } | R _ { i } - R _ { j } | ^ { - 1 }, \end{equation*}

and

$\gamma = ( 3 \pi ^ { 2 } ) ^ { 2 / 3 }$.

The constraint on

$\rho$

is

$\rho ( x ) \geq 0$

and

$\int _ { \mathbf{R} ^ { 3 } } \rho = N$.

The functional

$\rho \rightarrow \mathcal{E} ( \rho )$

is convex (cf. also

[[Convex function (of a real variable)|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

$N$

electrons needed to produce an electron density

$\rho$.

The second term is the attractive interaction of the

$N$

electrons with the

$K$

nuclei, via the

Coulomb potential

$V$.

The third is approximately the electron-electron repulsive

energy.

$U$

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

The

Thomas–Fermi energy

is defined to be

\begin{equation*} E ^ { \text{TF} } ( N ) = \operatorname { inf } \{ \mathcal{E} ( \rho ) : \rho \in L ^ { 5 / 3 } , \int \rho = N , \rho \geq 0 \}, \end{equation*}

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

${\cal E} ( \rho )$

with

$\rho \in L ^ { 5 / 3 } ( \mathbf{R} ^ { 3 } )$

and

$\int \rho = N$.

The

[[Euler–Lagrange equation|Euler–Lagrange equation]],

in this case called the

Thomas–Fermi equation,

is

\begin{equation} \tag{a2} \gamma \rho ( x ) ^ { 2 / 3 } = [ \Phi ( x ) - \mu ]_+ , \end{equation}

where

$[ a ] + = \operatorname { max } \{ 0 , a \}$,

$\mu$

is some constant

(a

Lagrange multiplier; cf.

[[Lagrange multipliers|Lagrange multipliers]])

and

$\Phi$

is the

Thomas–Fermi potential:

\begin{equation} \tag{a3} \Phi ( x ) = V ( x ) - \int _ { \mathbf{R} ^ { 3 } } | x - y | ^ { - 1 } \rho ( y ) d y. \end{equation}

The following essential mathematical facts about the

Thomas–Fermi equation were

established by

E.H. Lieb

and

B. Simon

[[#References|[a7]]]

(cf. also

[[#References|[a3]]]):

1)

There is a density

$\rho _ { N } ^ { \operatorname {TF} }$

that minimizes

${\cal E} ( \rho )$

if and only if

$N \leq Z : = \sum _ { j = 1 } ^ { K } Z _ { j }$.

This

$\rho _ { N } ^ { \operatorname {TF} }$

is unique and it satisfies the Thomas–Fermi equation

(a2)

for some

$\mu \geq 0$.

Every positive solution,

$\rho$,

of

(a2)

is a minimizer of

(a1)

for

$N = \int \rho$.

If

$N &gt; Z$,

then

$E ^ { \text{TF} } ( N ) = E ^ { \text{TF} } ( Z )$

and any minimizing sequence converges weakly in

$L ^ { 5 / 3 } ( \mathbf{R} ^ { 3 } )$

to

$\rho ^ { \operatorname {TF} } _{ Z }$.

2)

$\Phi ( x ) \geq 0$

for all

$x$.

(This need not be so for the real Schrödinger

$\rho$.)

3)

$\mu = \mu ( N )$

is a strictly monotonically decreasing function of

$N$

and

$\mu ( Z ) = 0$

(the

neutral case).

$\mu$

is the

chemical potential,

namely

\begin{equation*} \mu ( N ) = - \frac { \partial E ^ { \text{TF} } ( N ) } { \partial N }. \end{equation*}

$E ^ { \text{TF} } ( N )$

is a strictly convex, decreasing function of

$N$

for

$N \leq Z$

and

$E ^ { \text{TF} } ( N ) = E ^ { \text{TF} } ( Z )$

for

$N \geq Z$.

If

$N &lt; Z$,

$\rho _ { N } ^ { \operatorname {TF} }$

has compact support.

When

$N = Z$,

(a2)

becomes

$\gamma \rho ^ { 2 / 3 } = \Phi$.

By applying the

[[Laplace operator|Laplace operator]]

$\Delta$

to both sides, one obtains

\begin{equation*} - \Delta \Phi ( x ) + 4 \pi \gamma ^ { - 3 / 2 } \Phi ( x ) ^ { 3 / 2 } = 4 \pi \sum _ { j = 1 } ^ { K } Z _ { j } \delta ( x - R _ { j } ), \end{equation*}

which is the form in which the Thomas–Fermi

equation is usually stated (but it

is valid only for

$N = Z$).

An important property of the solution is

Teller's theorem

[[#References|[a4]]]

(proved rigorously in

[[#References|[a7]]]),

which implies that the

Thomas–Fermi molecule

is always unstable, i.e., for each

$N \leq Z$

there are

$K$

numbers

$N _ { j } \in ( 0 , Z _ { j } )$

with

$\sum _ { j } N _ { j } = N$

such that

\begin{equation} \tag{a4} E ^ { \operatorname{TF} } ( N ) &gt; \sum _ { j = 1 } ^ { K } E _ { \operatorname{atom} } ^ { \operatorname{TF} } ( N _ { j } , Z _ { j } ), \end{equation}

where

$E _ { \operatorname{atom} } ^ { \operatorname{TF} } ( N _ { j } , Z _ { j } )$

is the Thomas–Fermi

energy with

$K = 1$,

$Z = Z_j$

and

$N = N_{j}$.

The presence of

$U$

in

(a1)

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

$E ^ { \text{TF} }$

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

what

(a4)

says) but any dilation of the nuclear coordinates

($R _ { j } \rightarrow \text{l}R _ { j }$,

$\text{l} &gt; 1$)

will decrease

$E ^ { \text{TF} }$

in the neutral case

(positivity of the pressure)

[[#References|[a3]]],

[[#References|[a1]]].

This theorem plays an important role in the

stability of matter.

An important question concerns the connection between

$E ^ { \text{TF} } ( N )$

and

$E ^ { \text{Q} } ( N )$,

the

ground state energy

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

Schrödinger operator,

$H$,

it was meant to approximate.

\begin{equation*} H = - \sum _ { i = 1 } ^ { N } [ \Delta _ { i } + V ( x _ { i } ) ] + \sum _ { 1 \leq i &lt; j \leq N } | x _ { i } - x _ { j } | ^ { - 1 } + U, \end{equation*}

which acts on the

anti-symmetric functions

$\wedge ^ { N } L ^ { 2 } ( \mathbf{R} ^ { 3 } ; \mathbf{C} ^ { 2 } )$

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

$E ^ { \text{TF} }$

is asymptotically exact as

$N \rightarrow \infty$,

but this is not quite right;

$Z \rightarrow \infty$

is also needed.

Lieb

and

Simon

[[#References|[a7]]]

proved that if one fixes

$K$

and

$Z _ { j } / Z$

and sets

$R _ { j } = Z ^ { - 1 / 3 } R _ { j } ^ { 0 }$,

with fixed

$R _ { j } ^ { 0 } \in \mathbf{R} ^ { 3 }$,

and sets

$N = \lambda Z$,

with

$0 \leq \lambda &lt; 1$,

then

\begin{equation} \tag{a5} \operatorname { lim } _ { Z \rightarrow \infty } \frac { E ^ { \text{TF} } ( \lambda Z ) } { E ^ { \text{Q} } ( \lambda Z ) } = 1. \end{equation}

In particular, a simple change of variables shows that

$E _ { \text{atom} } ^ { \text{TF} } ( \lambda , Z ) = Z ^ { 7 / 3 } E _ { \text{atom} } ^ { \text{TF} } ( \lambda , 1 )$

and hence the true energy of a large atom is asymptotically

proportional to

$Z ^ { 7 / 3 }$.

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

quantum-mechanical density converges to

$\rho _ { N } ^ { \operatorname {TF} }$

(cf.

[[#References|[a7]]]).

The Thomas–Fermi density for an atom located at

$R = 0$,

which is spherically symmetric, scales as

\begin{equation*} \rho _ { \text { atom } } ^ { \text {TF} } ( x ; N = \lambda Z , Z ) = \end{equation*}

\begin{equation*} = Z ^ { 2 } \rho _ { \text { atom } } ^ { \operatorname{TF} } ( Z ^ { 1 / 3 } x ; N = \lambda , Z = 1 ). \end{equation*}

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

$Z$)

is smaller than a

$Z = 1$

atom by a factor

$Z ^ { - 1 / 3 }$

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

$Z \rightarrow \infty$.

Another important fact is the

large-$| x |$

asymptotics of

$\rho _ { \text { atom } } ^ { \text{TF} }$

for a neutral atom. As

$| x | \rightarrow \infty$,

\begin{equation*} \rho _ { \text{atom} } ^ { \text{TF} } ( x , N = Z , Z ) \sim \gamma ^ { 3 } \left( \frac { 3 } { \pi } \right) ^ { 3 } | x | ^ { - 6 }, \end{equation*}

independent of

$Z$.

Again, this behaviour agrees with quantum mechanics — on a

length scale

$Z ^ { - 1 / 3 }$,

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

$Z \rightarrow \infty$,

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

in

$Z$

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

$Z$

one can show, using the

[[Lieb–Thirring inequalities|Lieb–Thirring inequalities]]

[[#References|[a12]]]

and the

Lieb–Oxford inequality

[[#References|[a6]]],

that

$E ^ { \text{TF} } ( N )$,

with a modified

$\gamma$,

gives a lower bound to

$E ^ { \text{Q} } ( N )$.

Several

"improvements" 

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

fundamental significance in the sense of being

"exact" 

in the

$Z \rightarrow \infty$

limit. The

von Weizsäcker correction

consists in adding a term

\begin{equation*} \text{(const)} \int _ { {\bf R} ^ { 3 } } | \nabla \sqrt { \rho ( x ) } | ^ { 2 } d x \end{equation*}

to

${\cal E} ( \rho )$.

This preserves the convexity of

${\cal E} ( \rho )$

and adds

$(\text{const})Z ^ { 2 }$

to

$E ^ { \text{TF} } ( N )$

when

$Z$

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

$N$

for which there is a minimizing

$\rho$

is extend from

$[ 0 , Z ]$

to

$[ 0 , Z + ( \text { const } ) K ]$.

Another correction, the

Dirac exchange energy,

is to add

\begin{equation*} - ( \text {const} ) \int _ { {\bf R} ^ { 3 } } \rho ( x ) ^ { 4 / 3 } d x \end{equation*}

to

${\cal E} ( \rho )$.

This spoils the convexity but not the range

$[ 0 , Z ]$

for which a

minimizing

$\rho$

exists, cf.

[[#References|[a7]]]

for both of these corrections.

When a uniform external magnetic field

$B$

is present, the operator

$- \Delta$

in

$H$

is replaced by

\begin{equation*} | i \nabla + A ( x ) | ^ { 2 } + \sigma . B ( x ), \end{equation*}

with

$\operatorname{curl}A = B$

and

$\sigma$

denoting the Pauli spin matrices (cf. also

[[Pauli matrices|Pauli matrices]]).

This leads to a modified Thomas–Fermi theory

that is asymptotically exact as

$Z \rightarrow \infty$,

but the theory depends on the manner in which

$B$

varies with

$Z$.

There are five distinct regimes and theories:

$B \ll Z ^ { 4 / 3 }$,

$B \sim Z ^ { 4 / 3 }$,

$Z ^ { 4 / 3 } \ll B \ll Z ^ { 3 }$,

$B \sim Z ^ { 3 }$,

and

$B \gg Z ^ { 3 }$.

These

theories

[[#References|[a8]]],

[[#References|[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)

[[#References|[a10]]].

In this case there are three regimes. A convenient review

is

[[#References|[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.

[[#References|[a5]]],

[[#References|[a14]]]).

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">

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))</td></tr><tr><td valign="top">[a2]</td> <td valign="top">

E. Fermi, 

"Un metodo statistico per la determinazione di alcune priorieta dell'atome"

''Rend. Accad. Naz. Lincei''

, '''6'''

(1927)

pp. 602–607</td></tr><tr><td valign="top">[a3]</td> <td valign="top">

E.H. Lieb, 

"Thomas–Fermi and related theories of atoms and molecules"

''Rev. Mod. Phys.''

, '''53'''

(1981)

pp. 603–641

((Errata: 54 (1982), 311))</td></tr><tr><td valign="top">[a4]</td> <td valign="top">

E. Teller, 

"On the stability of molecules in Thomas–Fermi theory"

''Rev. Mod. Phys.''

, '''34'''

(1962)

pp. 627–631</td></tr><tr><td valign="top">[a5]</td> <td valign="top">

J. Messer, 

"Temperature dependent Thomas–Fermi theory"

, ''Lecture Notes Physics''

, '''147'''

, Springer

(1981)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">

E.H. Lieb, 

S. Oxford, 

"An improved lower bound on the indirect Coulomb energy"

''Internat. J. Quant. Chem.''

, '''19'''

(1981)

pp. 427–439</td></tr><tr><td valign="top">[a7]</td> <td valign="top">

E.H. Lieb, 

B. Simon, 

"The Thomas–Fermi theory of atoms, molecules and solids"

''Adv. Math.''

, '''23'''

(1977)

pp. 22–116</td></tr><tr><td valign="top">[a8]</td> <td valign="top">

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</td></tr><tr><td valign="top">[a9]</td> <td valign="top">

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</td></tr><tr><td valign="top">[a10]</td> <td valign="top">

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</td></tr><tr><td valign="top">[a11]</td> <td valign="top">

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)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">

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))</td></tr><tr><td valign="top">[a13]</td> <td valign="top">

L.H. Thomas, 

"The calculation of atomic fields"

''Proc. Cambridge Philos. Soc.''

, '''23'''

(1927)

pp. 542–548</td></tr><tr><td valign="top">[a14]</td> <td valign="top">

W. Thirring, 

"A course in mathematical physics"

, '''4'''

, Springer

(1983)

pp. 209–277</td></tr></table>

''Elliott H. Lieb''

Copyright to this article is held by Elliott Lieb.