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''Fermi–Thomas theory''
 
  
Sometimes called the
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"statistical theory" ,
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it was invented by
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''Fermi–Thomas theory''
  
L.H. Thomas
+
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]]).
[[#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,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200601.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200602.png" />,
 
 
 
and the
 
 
 
ground state energy,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200603.png" />
 
 
 
for a large atom or molecule with a large number,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200604.png" />,
 
 
 
of electrons. Schrödinger's
 
 
 
equation, which would give the exact density and energy, cannot be
 
 
 
easily handled when
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200605.png" />
 
 
 
is large (cf. also
 
 
 
[[Schrödinger equation|Schrödinger equation]]).
 
 
 
A starting point for the theory is the
 
 
 
Thomas–Fermi energy functional.
 
 
 
For a molecule with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200606.png" />
 
 
 
nuclei of charges
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200607.png" />
 
 
 
and locations
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200608.png" />
 
 
 
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200609.png" />),
 
 
 
it is
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006011.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006012.png" /></td> </tr></table>
 
  
 +
A starting point for the theory is the Thomas–Fermi energy functional.
 +
For a molecule with $K$ nuclei of charges $Z_i > 0$ and locations $R_{i} \in \mathbf{R} ^ { 3 }$ ($i = 1 , \ldots , K$), it is
 +
\begin{equation} \tag{a1}
 +
\mathcal{E} ( \rho ) : =
 +
\frac { 3 } { 5 } \gamma \int _ { \mathbf R ^ { 3 } } \rho ( x ) ^ { 5 / 3 } d x - \int _ { \mathbf R ^ { 3 } } V ( x ) \rho ( x ) d x
 +
+\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,
 
in suitable units. Here,
 
+
\begin{equation*} V ( x ) = \sum _ { j = 1 } ^ { K } Z _ { j } | x - r _ { j } | ^ { - 1 }, \end{equation*}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006013.png" /></td> </tr></table>
+
\begin{equation*} U = \sum _ { 1 \leq i < j \leq K } Z _ { i } Z _ { j } | R _ { i } - R _ { j } | ^ { - 1 }, \end{equation*}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006014.png" /></td> </tr></table>
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006015.png" />.
 
 
 
The constraint on
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006016.png" />
 
 
 
is
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006017.png" />
 
 
 
 
and
 
and
 +
$\gamma = ( 3 \pi ^ { 2 } ) ^ { 2 / 3 }$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006018.png" />.
+
The constraint on $\rho$ is $\rho ( x ) \geq 0$ and $\int _ { \mathbf{R} ^ { 3 } } \rho = N$.
 
 
The functional
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006019.png" />
 
 
 
is convex (cf. also
 
  
[[Convex function (of a real variable)|Convex function (of a real variable)]]).
+
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 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 first term is roughly the minimum quantum-mechanical
+
The second term is the attractive interaction of the $N$ electrons with the $K$ nuclei, via the Coulomb potential $V$.
 
 
kinetic energy of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006020.png" />
 
 
 
electrons needed to produce an electron density
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006021.png" />.
 
 
 
The second term is the attractive interaction of the
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006022.png" />
 
 
 
electrons with the
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006023.png" />
 
 
 
nuclei, via the
 
 
 
Coulomb potential
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006024.png" />.
 
  
 
The third is approximately the electron-electron repulsive
 
The third is approximately the electron-electron repulsive
 
 
energy.
 
energy.
 +
$U$ is the nuclear-nuclear repulsion and is an important constant.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006025.png" />
+
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 $\int \rho = N$.
  
is the nuclear-nuclear repulsion and is an important constant.
+
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:
  
The
+
\begin{equation} \tag{a3} \Phi ( x ) = V ( x ) - \int _ { \mathbf{R} ^ { 3 } } | x - y | ^ { - 1 } \rho ( y ) d y. \end{equation}
  
Thomas–Fermi energy
+
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]]]):
  
is defined to be
+
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 $N > Z$, then $E ^ { \text{TF} } ( N ) = E ^ { \text{TF} } ( Z )$ and any minimizing sequence converges weakly in $L ^ { 5 / 3 } ( \mathbf{R} ^ { 3 } )$ to $\rho ^ { \operatorname {TF} } _{ Z }$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006026.png" /></td> </tr></table>
+
2) $\Phi ( x ) \geq 0$ for all $x$. (This need not be so for the real Schrödinger $\rho$.)
  
i.e., the Thomas–Fermi energy and density are obtained by minimizing
+
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 $N < Z$, $\rho _ { N } ^ { \operatorname {TF} }$ 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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006027.png" />
+
\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*}
  
with
+
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
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006028.png" />
+
\begin{equation} \tag{a4} E ^ { \operatorname{TF} } ( N ) > \sum _ { j = 1 } ^ { K } E _ { \operatorname{atom} } ^ { \operatorname{TF} } ( N _ { j } , Z _ { j } ), \end{equation}
  
and
+
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 ($R _ { j } \rightarrow \text{l}R _ { j }$, $\text{l} > 1$) 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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006029.png" />.
+
\begin{equation*} H = - \sum _ { i = 1 } ^ { N } [ \Delta _ { i } + V ( x _ { i } ) ] + \sum _ { 1 \leq i < j \leq N } | x _ { i } - x _ { j } | ^ { - 1 } + U, \end{equation*}
  
The
+
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 $0 \leq \lambda < 1$, then
  
[[Euler–Lagrange equation|Euler–Lagrange equation]],
+
\begin{equation} \tag{a5} \operatorname { lim } _ { Z \rightarrow \infty } \frac { E ^ { \text{TF} } ( \lambda Z ) } { E ^ { \text{Q} } ( \lambda Z ) } = 1. \end{equation}
  
in this case called the
+
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 $Z ^ { 7 / 3 }$.
  
Thomas–Fermi equation,
+
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
  
is
+
\begin{equation*} \rho _ { \text { atom } } ^ { \text {TF} } ( x ; N = \lambda Z , Z ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation*} = Z ^ { 2 } \rho _ { \text { atom } } ^ { \operatorname{TF} } ( Z ^ { 1 / 3 } x ; N = \lambda , Z = 1 ). \end{equation*}
  
where
+
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 $Z \rightarrow \infty$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006031.png" />,
+
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 $Z$.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006032.png" />
+
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.
  
is some constant
+
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 $E ^ { \text{Q} } ( N )$.
  
(a
+
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 $Z$.  
Lagrange multiplier; cf.
+
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.  
[[Lagrange multipliers|Lagrange multipliers]])
+
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.  
and
+
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]]]).
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006033.png" />
 
 
 
is the
 
 
 
Thomas–Fermi potential:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006035.png" />
 
 
 
that minimizes
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006036.png" />
 
 
 
if and only if
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006037.png" />.
 
 
 
This
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006038.png" />
 
 
 
is unique and it satisfies the Thomas–Fermi equation
 
 
 
(a2)
 
 
 
for some
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006039.png" />.
 
 
 
Every positive solution,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006040.png" />,
 
 
 
of
 
 
 
(a2)
 
 
 
is a minimizer of
 
 
 
(a1)
 
 
 
for
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006041.png" />.
 
 
 
If
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006042.png" />,
 
 
 
then
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006043.png" />
 
 
 
and any minimizing sequence converges weakly in
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006044.png" />
 
 
 
to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006045.png" />.
 
 
 
2)
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006046.png" />
 
 
 
for all
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006047.png" />.
 
 
 
(This need not be so for the real Schrödinger
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006048.png" />.)
 
 
 
3)
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006049.png" />
 
 
 
is a strictly monotonically decreasing function of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006050.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006051.png" />
 
 
 
(the
 
 
 
neutral case).
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006052.png" />
 
 
 
is the
 
 
 
chemical potential,
 
 
 
namely
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006053.png" /></td> </tr></table>
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006054.png" />
 
 
 
is a strictly convex, decreasing function of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006055.png" />
 
 
 
for
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006056.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006057.png" />
 
 
 
for
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006058.png" />.
 
 
 
If
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006059.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006060.png" />
 
 
 
has compact support.
 
 
 
When
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006061.png" />,
 
 
 
(a2)
 
 
 
becomes
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006062.png" />.
 
 
 
By applying the
 
 
 
[[Laplace operator|Laplace operator]]
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006063.png" />
 
 
 
to both sides, one obtains
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006064.png" /></td> </tr></table>
 
 
 
which is the form in which the Thomas–Fermi
 
 
 
equation is usually stated (but it
 
 
 
is valid only for
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006065.png" />).
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006066.png" />
 
 
 
there are
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006067.png" />
 
 
 
numbers
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006068.png" />
 
 
 
with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006069.png" />
 
 
 
such that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006070.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
 
 
 
where
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006071.png" />
 
 
 
is the Thomas–Fermi
 
 
 
energy with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006072.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006073.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006074.png" />.
 
 
 
The presence of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006075.png" />
 
 
 
in
 
 
 
(a1)
 
 
 
is crucial for this result. The inequality is strict. Not only does
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006076.png" />
 
 
 
decrease when the nuclei are pulled infinitely far apart (which is
 
 
 
what
 
 
 
(a4)
 
 
 
says) but any dilation of the nuclear coordinates
 
 
 
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006077.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006078.png" />)
 
 
 
will decrease
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006079.png" />
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006080.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006081.png" />,
 
 
 
the
 
 
 
ground state energy
 
 
 
(i.e., the infimum of the spectrum) of the
 
 
 
Schrödinger operator,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006082.png" />,
 
 
 
it was meant to approximate.
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006083.png" /></td> </tr></table>
 
 
 
which acts on the
 
 
 
anti-symmetric functions
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006084.png" />
 
 
 
(i.e., functions of space and spin). It used to be believed that
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006085.png" />
 
 
 
is asymptotically exact as
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006086.png" />,
 
 
 
but this is not quite right;
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006087.png" />
 
 
 
is also needed.
 
 
 
Lieb
 
 
 
and
 
 
 
Simon
 
 
 
[[#References|[a7]]]
 
 
 
proved that if one fixes
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006088.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006089.png" />
 
 
 
and sets
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006090.png" />,
 
 
 
with fixed
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006091.png" />,
 
 
 
and sets
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006092.png" />,
 
 
 
with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006093.png" />,
 
 
 
then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006094.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
 
 
 
In particular, a simple change of variables shows that
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006095.png" />
 
 
 
and hence the true energy of a large atom is asymptotically
 
 
 
proportional to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006096.png" />.
 
 
 
Likewise, there is a well-defined sense in which the
 
 
 
quantum-mechanical density converges to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006097.png" />
 
 
 
(cf.
 
 
 
[[#References|[a7]]]).
 
 
 
The Thomas–Fermi density for an atom located at
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006098.png" />,
 
 
 
which is spherically symmetric, scales as
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006099.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060100.png" /></td> </tr></table>
 
 
 
Thus, a large atom (i.e., large
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060101.png" />)
 
 
 
is smaller than a
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060102.png" />
 
 
 
atom by a factor
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060103.png" />
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060104.png" />.
 
 
 
Another important fact is the
 
 
 
large-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060105.png" />
 
 
 
asymptotics of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060106.png" />
 
 
 
for a neutral atom. As
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060107.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060108.png" /></td> </tr></table>
 
 
 
independent of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060109.png" />.
 
 
 
Again, this behaviour agrees with quantum mechanics — on a
 
 
 
length scale
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060110.png" />,
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060111.png" />,
 
 
 
the quantum-mechanical binding energy of a molecule is of lower order
 
 
 
in
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060112.png" />
 
 
 
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
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060113.png" />
 
 
 
one can show, using the
 
 
 
[[Lieb–Thirring inequalities|Lieb–Thirring inequalities]]
 
 
 
[[#References|[a12]]]
 
 
 
and the
 
 
 
Lieb–Oxford inequality
 
 
 
[[#References|[a6]]],
 
 
 
that
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060114.png" />,
 
 
 
with a modified
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060115.png" />,
 
 
 
gives a lower bound to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060116.png" />.
 
 
 
Several
 
 
 
"improvements"  
 
 
 
to Thomas–Fermi theory have been proposed, but none have a
 
 
 
fundamental significance in the sense of being
 
 
 
"exact"  
 
 
 
in the
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060117.png" />
 
 
 
limit. The
 
 
 
von Weizsäcker correction
 
 
 
consists in adding a term
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060118.png" /></td> </tr></table>
 
 
 
to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060119.png" />.
 
 
 
This preserves the convexity of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060120.png" />
 
 
 
and adds
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060121.png" />
 
 
 
to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060122.png" />
 
 
 
when
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060123.png" />
 
 
 
is large. It also has the effect that the range of
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060124.png" />
 
 
 
for which there is a minimizing
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060125.png" />
 
 
 
is extend from
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060126.png" />
 
 
 
to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060127.png" />.
 
 
 
Another correction, the
 
 
 
Dirac exchange energy,
 
 
 
is to add
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060128.png" /></td> </tr></table>
 
 
 
to
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060129.png" />.
 
 
 
This spoils the convexity but not the range
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060130.png" />
 
 
 
for which a
 
 
 
minimizing
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060131.png" />
 
 
 
exists, cf.
 
 
 
[[#References|[a7]]]
 
 
 
for both of these corrections.
 
 
 
When a uniform external magnetic field
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060132.png" />
 
 
 
is present, the operator
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060133.png" />
 
 
 
in
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060134.png" />
 
 
 
is replaced by
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060135.png" /></td> </tr></table>
 
 
 
with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060136.png" />
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060137.png" />
 
 
 
denoting the Pauli spin matrices (cf. also
 
 
 
[[Pauli matrices|Pauli matrices]]).
 
 
 
This leads to a modified Thomas–Fermi theory
 
 
 
that is asymptotically exact as
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060138.png" />,
 
 
 
but the theory depends on the manner in which
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060139.png" />
 
 
 
varies with
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060140.png" />.
 
 
 
There are five distinct regimes and theories:
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060141.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060142.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060143.png" />,
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060144.png" />,
 
 
 
and
 
 
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060145.png" />.
 
 
 
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
+
<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>
R. Benguria,  
+
<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>
E.H. Lieb,  
+
<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>
"The positivity of the pressure in Thomas–Fermi theory"
+
<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>
''Comm. Math. Phys.''
+
<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>'
, '''63'''
+
<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>
(1978)
+
<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>
pp. 193–218
+
<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>
((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''
 
''Elliott H. Lieb''
  
 
Copyright to this article is held by Elliott Lieb.
 
Copyright to this article is held by Elliott Lieb.

Latest revision as of 23:54, 14 February 2024

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, $\rho ( x )$, $x \in \mathbf{R} ^ { 3 }$, 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).

A starting point for the theory is the Thomas–Fermi energy functional. For a molecule with $K$ nuclei of charges $Z_i > 0$ and locations $R_{i} \in \mathbf{R} ^ { 3 }$ ($i = 1 , \ldots , K$), it is \begin{equation} \tag{a1} \mathcal{E} ( \rho ) : = \frac { 3 } { 5 } \gamma \int _ { \mathbf R ^ { 3 } } \rho ( x ) ^ { 5 / 3 } d x - \int _ { \mathbf R ^ { 3 } } V ( x ) \rho ( x ) d x +\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 < 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)).

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, 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) 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 [a7] (cf. also [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 $N > Z$, 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 $N < Z$, $\rho _ { N } ^ { \operatorname {TF} }$ has compact support. When $N = Z$, (a2) becomes $\gamma \rho ^ { 2 / 3 } = \Phi$. By applying the 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 [a4] (proved rigorously in [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 ) > \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 ($R _ { j } \rightarrow \text{l}R _ { j }$, $\text{l} > 1$) will decrease $E ^ { \text{TF} }$ 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 $E ^ { \text{TF} } ( N )$ 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 < 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 [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 $0 \leq \lambda < 1$, 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. [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 [a12] and the Lieb–Oxford inequality [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. [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). 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 [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.

How to Cite This Entry:
Thomas-Fermi theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thomas-Fermi_theory&oldid=15593