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''Fermi–Thomas theory''
 
''Fermi–Thomas theory''
  
Line 29: Line 37:
 
electron density,
 
electron density,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200601.png" />,
+
$\rho ( x )$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200602.png" />,
+
$x \in \mathbf{R} ^ { 3 }$,
  
 
and the
 
and the
Line 37: Line 45:
 
ground state energy,
 
ground state energy,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200603.png" />
+
$E ( N )$
  
 
for a large atom or molecule with a large number,
 
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" />,
+
$N$,
  
 
of electrons. Schrödinger's
 
of electrons. Schrödinger's
Line 49: Line 57:
 
easily handled when
 
easily handled when
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200605.png" />
+
$N$
  
 
is large (cf. also
 
is large (cf. also
Line 61: Line 69:
 
For a molecule with
 
For a molecule with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200606.png" />
+
$K$
  
 
nuclei of charges
 
nuclei of charges
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200607.png" />
+
$Z_i &gt; 0$
  
 
and locations
 
and locations
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200608.png" />
+
$R_{i} \in \mathbf{R} ^ { 3 }$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t1200609.png" />),
+
($i = 1 , \ldots , K$),
  
 
it is
 
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>
+
\begin{equation} \tag{a1} \mathcal{E} ( \rho ) : = \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/t12006011.png" /></td> </tr></table>
+
\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*}
  
<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>
+
\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,
 
in suitable units. Here,
  
<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*} 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/t12006014.png" /></td> </tr></table>
+
\begin{equation*} U = \sum _ { 1 \leq i &lt; j \leq K } Z _ { i } Z _ { j } | R _ { i } - R _ { j } | ^ { - 1 }, \end{equation*}
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006015.png" />.
+
$\gamma = ( 3 \pi ^ { 2 } ) ^ { 2 / 3 }$.
  
 
The constraint on
 
The constraint on
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006016.png" />
+
$\rho$
  
 
is
 
is
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006017.png" />
+
$\rho ( x ) \geq 0$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006018.png" />.
+
$\int _ { \mathbf{R} ^ { 3 } } \rho = N$.
  
 
The functional
 
The functional
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006019.png" />
+
$\rho \rightarrow \mathcal{E} ( \rho )$
  
 
is convex (cf. also
 
is convex (cf. also
Line 117: Line 125:
 
kinetic energy of
 
kinetic energy of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006020.png" />
+
$N$
  
 
electrons needed to produce an electron density
 
electrons needed to produce an electron density
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006021.png" />.
+
$\rho$.
  
 
The second term is the attractive interaction of the
 
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" />
+
$N$
  
 
electrons with the
 
electrons with the
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006023.png" />
+
$K$
  
 
nuclei, via the
 
nuclei, via the
Line 135: Line 143:
 
Coulomb potential
 
Coulomb potential
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006024.png" />.
+
$V$.
  
 
The third is approximately the electron-electron repulsive
 
The third is approximately the electron-electron repulsive
Line 141: Line 149:
 
energy.
 
energy.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006025.png" />
+
$U$
  
 
is the nuclear-nuclear repulsion and is an important constant.
 
is the nuclear-nuclear repulsion and is an important constant.
Line 151: Line 159:
 
is defined to be
 
is defined to be
  
<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>
+
\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
 
i.e., the Thomas–Fermi energy and density are obtained by minimizing
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006027.png" />
+
${\cal E} ( \rho )$
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006028.png" />
+
$\rho \in L ^ { 5 / 3 } ( \mathbf{R} ^ { 3 } )$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006029.png" />.
+
$\int \rho = N$.
  
 
The
 
The
Line 175: Line 183:
 
is
 
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/t12006030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \gamma \rho ( x ) ^ { 2 / 3 } = [ \Phi ( x ) - \mu ]_+ , \end{equation}
  
 
where
 
where
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006031.png" />,
+
$[ a ] + = \operatorname { max } \{ 0 , a \}$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006032.png" />
+
$\mu$
  
 
is some constant
 
is some constant
Line 193: Line 201:
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006033.png" />
+
$\Phi$
  
 
is the
 
is the
Line 199: Line 207:
 
Thomas–Fermi potential:
 
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>
+
\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
 
The following essential mathematical facts about the
Line 223: Line 231:
 
There is a density
 
There is a density
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006035.png" />
+
$\rho _ { N } ^ { \operatorname {TF} }$
  
 
that minimizes
 
that minimizes
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006036.png" />
+
${\cal E} ( \rho )$
  
 
if and only if
 
if and only if
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006037.png" />.
+
$N \leq Z : = \sum _ { j = 1 } ^ { K } Z _ { j }$.
  
 
This
 
This
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006038.png" />
+
$\rho _ { N } ^ { \operatorname {TF} }$
  
 
is unique and it satisfies the Thomas–Fermi equation
 
is unique and it satisfies the Thomas–Fermi equation
Line 243: Line 251:
 
for some
 
for some
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006039.png" />.
+
$\mu \geq 0$.
  
 
Every positive solution,
 
Every positive solution,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006040.png" />,
+
$\rho$,
  
 
of
 
of
Line 259: Line 267:
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006041.png" />.
+
$N = \int \rho$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006042.png" />,
+
$N &gt; Z$,
  
 
then
 
then
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006043.png" />
+
$E ^ { \text{TF} } ( N ) = E ^ { \text{TF} } ( Z )$
  
 
and any minimizing sequence converges weakly in
 
and any minimizing sequence converges weakly in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006044.png" />
+
$L ^ { 5 / 3 } ( \mathbf{R} ^ { 3 } )$
  
 
to
 
to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006045.png" />.
+
$\rho ^ { \operatorname {TF} } _{ Z }$.
  
 
2)
 
2)
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006046.png" />
+
$\Phi ( x ) \geq 0$
  
 
for all
 
for all
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006047.png" />.
+
$x$.
  
 
(This need not be so for the real Schrödinger
 
(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" />.)
+
$\rho$.)
  
 
3)
 
3)
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006049.png" />
+
$\mu = \mu ( N )$
  
 
is a strictly monotonically decreasing function of
 
is a strictly monotonically decreasing function of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006050.png" />
+
$N$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006051.png" />
+
$\mu ( Z ) = 0$
  
 
(the
 
(the
Line 305: Line 313:
 
neutral case).
 
neutral case).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006052.png" />
+
$\mu$
  
 
is the
 
is the
Line 313: Line 321:
 
namely
 
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>
+
\begin{equation*} \mu ( N ) = - \frac { \partial E ^ { \text{TF} } ( N ) } { \partial N }. \end{equation*}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006054.png" />
+
$E ^ { \text{TF} } ( N )$
  
 
is a strictly convex, decreasing function of
 
is a strictly convex, decreasing function of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006055.png" />
+
$N$
  
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006056.png" />
+
$N \leq Z$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006057.png" />
+
$E ^ { \text{TF} } ( N ) = E ^ { \text{TF} } ( Z )$
  
 
for
 
for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006058.png" />.
+
$N \geq Z$.
  
 
If
 
If
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006059.png" />,
+
$N &lt; Z$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006060.png" />
+
$\rho _ { N } ^ { \operatorname {TF} }$
  
 
has compact support.
 
has compact support.
Line 343: Line 351:
 
When
 
When
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006061.png" />,
+
$N = Z$,
  
 
(a2)
 
(a2)
Line 349: Line 357:
 
becomes
 
becomes
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006062.png" />.
+
$\gamma \rho ^ { 2 / 3 } = \Phi$.
  
 
By applying the
 
By applying the
Line 355: Line 363:
 
[[Laplace operator|Laplace operator]]
 
[[Laplace operator|Laplace operator]]
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006063.png" />
+
$\Delta$
  
 
to both sides, one obtains
 
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>
+
\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
 
which is the form in which the Thomas–Fermi
Line 367: Line 375:
 
is valid only for
 
is valid only for
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006065.png" />).
+
$N = Z$).
  
 
An important property of the solution is
 
An important property of the solution is
Line 385: Line 393:
 
is always unstable, i.e., for each
 
is always unstable, i.e., for each
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006066.png" />
+
$N \leq Z$
  
 
there are
 
there are
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006067.png" />
+
$K$
  
 
numbers
 
numbers
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006068.png" />
+
$N _ { j } \in ( 0 , Z _ { j } )$
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006069.png" />
+
$\sum _ { j } N _ { j } = N$
  
 
such that
 
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>
+
\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
 
where
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006071.png" />
+
$E _ { \operatorname{atom} } ^ { \operatorname{TF} } ( N _ { j } , Z _ { j } )$
  
 
is the Thomas–Fermi
 
is the Thomas–Fermi
Line 411: Line 419:
 
energy with
 
energy with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006072.png" />,
+
$K = 1$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006073.png" />
+
$Z = Z_j$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006074.png" />.
+
$N = N_{j}$.
  
 
The presence of
 
The presence of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006075.png" />
+
$U$
  
 
in
 
in
Line 429: Line 437:
 
is crucial for this result. The inequality is strict. Not only does
 
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" />
+
$E ^ { \text{TF} }$
  
 
decrease when the nuclei are pulled infinitely far apart (which is
 
decrease when the nuclei are pulled infinitely far apart (which is
Line 439: Line 447:
 
says) but any dilation of the nuclear coordinates
 
says) but any dilation of the nuclear coordinates
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006077.png" />,
+
($R _ { j } \rightarrow \text{l}R _ { j }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006078.png" />)
+
$\text{l} &gt; 1$)
  
 
will decrease
 
will decrease
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006079.png" />
+
$E ^ { \text{TF} }$
  
 
in the neutral case
 
in the neutral case
Line 461: Line 469:
 
An important question concerns the connection between
 
An important question concerns the connection between
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006080.png" />
+
$E ^ { \text{TF} } ( N )$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006081.png" />,
+
$E ^ { \text{Q} } ( N )$,
  
 
the
 
the
Line 475: Line 483:
 
Schrödinger operator,
 
Schrödinger operator,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006082.png" />,
+
$H$,
  
 
it was meant to approximate.
 
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>
+
\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
 
which acts on the
Line 485: Line 493:
 
anti-symmetric functions
 
anti-symmetric functions
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006084.png" />
+
$\wedge ^ { N } L ^ { 2 } ( \mathbf{R} ^ { 3 } ; \mathbf{C} ^ { 2 } )$
  
 
(i.e., functions of space and spin). It used to be believed that
 
(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" />
+
$E ^ { \text{TF} }$
  
 
is asymptotically exact as
 
is asymptotically exact as
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006086.png" />,
+
$N \rightarrow \infty$,
  
 
but this is not quite right;
 
but this is not quite right;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006087.png" />
+
$Z \rightarrow \infty$
  
 
is also needed.
 
is also needed.
Line 511: Line 519:
 
proved that if one fixes
 
proved that if one fixes
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006088.png" />
+
$K$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006089.png" />
+
$Z _ { j } / Z$
  
 
and sets
 
and sets
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006090.png" />,
+
$R _ { j } = Z ^ { - 1 / 3 } R _ { j } ^ { 0 }$,
  
 
with fixed
 
with fixed
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006091.png" />,
+
$R _ { j } ^ { 0 } \in \mathbf{R} ^ { 3 }$,
  
 
and sets
 
and sets
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006092.png" />,
+
$N = \lambda Z$,
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006093.png" />,
+
$0 \leq \lambda &lt; 1$,
  
 
then
 
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>
+
\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
 
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" />
+
$E _ { \text{atom} } ^ { \text{TF} } ( \lambda , Z ) = Z ^ { 7 / 3 } E _ { \text{atom} } ^ { \text{TF} } ( \lambda , 1 )$
  
 
and hence the true energy of a large atom is asymptotically
 
and hence the true energy of a large atom is asymptotically
Line 545: Line 553:
 
proportional to
 
proportional to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006096.png" />.
+
$Z ^ { 7 / 3 }$.
  
 
Likewise, there is a well-defined sense in which the
 
Likewise, there is a well-defined sense in which the
Line 551: Line 559:
 
quantum-mechanical density converges to
 
quantum-mechanical density converges to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t12006097.png" />
+
$\rho _ { N } ^ { \operatorname {TF} }$
  
 
(cf.
 
(cf.
Line 559: Line 567:
 
The Thomas–Fermi density for an atom located at
 
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" />,
+
$R = 0$,
  
 
which is spherically symmetric, scales as
 
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>
+
\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/t120060100.png" /></td> </tr></table>
+
\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
 
Thus, a large atom (i.e., large
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060101.png" />)
+
$Z$)
  
 
is smaller than a
 
is smaller than a
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060102.png" />
+
$Z = 1$
  
 
atom by a factor
 
atom by a factor
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060103.png" />
+
$Z ^ { - 1 / 3 }$
  
 
in radius. Despite this seeming paradox, Thomas–Fermi
 
in radius. Despite this seeming paradox, Thomas–Fermi
Line 587: Line 595:
 
electrons is concerned) as
 
electrons is concerned) as
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060104.png" />.
+
$Z \rightarrow \infty$.
  
 
Another important fact is the
 
Another important fact is the
  
large-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060105.png" />
+
large-$| x |$
  
 
asymptotics of
 
asymptotics of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060106.png" />
+
$\rho _ { \text { atom } } ^ { \text{TF} }$
  
 
for a neutral atom. As
 
for a neutral atom. As
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060107.png" />,
+
$| x | \rightarrow \infty$,
  
<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>
+
\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
 
independent of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060109.png" />.
+
$Z$.
  
 
Again, this behaviour agrees with quantum mechanics — on a
 
Again, this behaviour agrees with quantum mechanics — on a
Line 611: Line 619:
 
length scale
 
length scale
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060110.png" />,
+
$Z ^ { - 1 / 3 }$,
  
 
which is where the bulk of the electrons is to be found.
 
which is where the bulk of the electrons is to be found.
Line 623: Line 631:
 
can be understood as saying that, as
 
can be understood as saying that, as
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060111.png" />,
+
$Z \rightarrow \infty$,
  
 
the quantum-mechanical binding energy of a molecule is of lower order
 
the quantum-mechanical binding energy of a molecule is of lower order
Line 629: Line 637:
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060112.png" />
+
$Z$
  
 
than the total ground state energy. Thus, Teller's theorem is
 
than the total ground state energy. Thus, Teller's theorem is
Line 643: Line 651:
 
For finite
 
For finite
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060113.png" />
+
$Z$
  
 
one can show, using the
 
one can show, using the
Line 659: Line 667:
 
that
 
that
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060114.png" />,
+
$E ^ { \text{TF} } ( N )$,
  
 
with a modified
 
with a modified
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060115.png" />,
+
$\gamma$,
  
 
gives a lower bound to
 
gives a lower bound to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060116.png" />.
+
$E ^ { \text{Q} } ( N )$.
  
 
Several
 
Several
Line 681: Line 689:
 
in the
 
in the
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060117.png" />
+
$Z \rightarrow \infty$
  
 
limit. The
 
limit. The
Line 689: Line 697:
 
consists in adding a term
 
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>
+
\begin{equation*} \text{(const)} \int _ { {\bf R} ^ { 3 } } | \nabla \sqrt { \rho ( x ) } | ^ { 2 } d x \end{equation*}
  
 
to
 
to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060119.png" />.
+
${\cal E} ( \rho )$.
  
 
This preserves the convexity of
 
This preserves the convexity of
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060120.png" />
+
${\cal E} ( \rho )$
  
 
and adds
 
and adds
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060121.png" />
+
$(\text{const})Z ^ { 2 }$
  
 
to
 
to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060122.png" />
+
$E ^ { \text{TF} } ( N )$
  
 
when
 
when
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060123.png" />
+
$Z$
  
 
is large. It also has the effect that the range of
 
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" />
+
$N$
  
 
for which there is a minimizing
 
for which there is a minimizing
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060125.png" />
+
$\rho$
  
 
is extend from
 
is extend from
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060126.png" />
+
$[ 0 , Z ]$
  
 
to
 
to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060127.png" />.
+
$[ 0 , Z + ( \text { const } ) K ]$.
  
 
Another correction, the
 
Another correction, the
Line 733: Line 741:
 
is to add
 
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>
+
\begin{equation*} - ( \text {const} ) \int _ { {\bf R} ^ { 3 } } \rho ( x ) ^ { 4 / 3 } d x \end{equation*}
  
 
to
 
to
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060129.png" />.
+
${\cal E} ( \rho )$.
  
 
This spoils the convexity but not the range
 
This spoils the convexity but not the range
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060130.png" />
+
$[ 0 , Z ]$
  
 
for which a
 
for which a
Line 747: Line 755:
 
minimizing
 
minimizing
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060131.png" />
+
$\rho$
  
 
exists, cf.
 
exists, cf.
Line 757: Line 765:
 
When a uniform external magnetic field
 
When a uniform external magnetic field
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060132.png" />
+
$B$
  
 
is present, the operator
 
is present, the operator
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060133.png" />
+
$- \Delta$
  
 
in
 
in
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060134.png" />
+
$H$
  
 
is replaced by
 
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>
+
\begin{equation*} | i \nabla + A ( x ) | ^ { 2 } + \sigma . B ( x ), \end{equation*}
  
 
with
 
with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060136.png" />
+
$\operatorname{curl}A = B$
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060137.png" />
+
$\sigma$
  
 
denoting the Pauli spin matrices (cf. also
 
denoting the Pauli spin matrices (cf. also
Line 787: Line 795:
 
that is asymptotically exact as
 
that is asymptotically exact as
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060138.png" />,
+
$Z \rightarrow \infty$,
  
 
but the theory depends on the manner in which
 
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" />
+
$B$
  
 
varies with
 
varies with
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060140.png" />.
+
$Z$.
  
 
There are five distinct regimes and theories:
 
There are five distinct regimes and theories:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060141.png" />,
+
$B \ll Z ^ { 4 / 3 }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060142.png" />,
+
$B \sim Z ^ { 4 / 3 }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060143.png" />,
+
$Z ^ { 4 / 3 } \ll B \ll Z ^ { 3 }$,
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060144.png" />,
+
$B \sim Z ^ { 3 }$,
  
 
and
 
and
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120060/t120060145.png" />.
+
$B \gg Z ^ { 3 }$.
  
 
These
 
These
Line 856: Line 864:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">
+
<table><tr><td valign="top">[a1]</td> <td valign="top">
  
 
R. Benguria,  
 
R. Benguria,  
Line 872: Line 880:
 
pp. 193–218
 
pp. 193–218
  
((Errata: 71 (1980), 94))</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
+
((Errata: 71 (1980), 94))</td></tr><tr><td valign="top">[a2]</td> <td valign="top">
  
 
E. Fermi,  
 
E. Fermi,  
Line 884: Line 892:
 
(1927)
 
(1927)
  
pp. 602–607</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">
+
pp. 602–607</td></tr><tr><td valign="top">[a3]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 898: Line 906:
 
pp. 603–641
 
pp. 603–641
  
((Errata: 54 (1982), 311))</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">
+
((Errata: 54 (1982), 311))</td></tr><tr><td valign="top">[a4]</td> <td valign="top">
  
 
E. Teller,  
 
E. Teller,  
Line 910: Line 918:
 
(1962)
 
(1962)
  
pp. 627–631</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">
+
pp. 627–631</td></tr><tr><td valign="top">[a5]</td> <td valign="top">
  
 
J. Messer,  
 
J. Messer,  
Line 922: Line 930:
 
, Springer
 
, Springer
  
(1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">
+
(1981)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 936: Line 944:
 
(1981)
 
(1981)
  
pp. 427–439</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">
+
pp. 427–439</td></tr><tr><td valign="top">[a7]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 950: Line 958:
 
(1977)
 
(1977)
  
pp. 22–116</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">
+
pp. 22–116</td></tr><tr><td valign="top">[a8]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 966: Line 974:
 
(1994)
 
(1994)
  
pp. 513–591</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">
+
pp. 513–591</td></tr><tr><td valign="top">[a9]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 982: Line 990:
 
(1994)
 
(1994)
  
pp. 77–124</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">
+
pp. 77–124</td></tr><tr><td valign="top">[a10]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 998: Line 1,006:
 
(1995)
 
(1995)
  
pp. 10646–10665</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">
+
pp. 10646–10665</td></tr><tr><td valign="top">[a11]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 1,018: Line 1,026:
 
pp. 145–167
 
pp. 145–167
  
(Edition: Second)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">
+
(Edition: Second)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">
  
 
E.H. Lieb,  
 
E.H. Lieb,  
Line 1,040: Line 1,048:
 
pp. 269–303
 
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">
+
((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,  
 
L.H. Thomas,  
Line 1,052: Line 1,060:
 
(1927)
 
(1927)
  
pp. 542–548</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">
+
pp. 542–548</td></tr><tr><td valign="top">[a14]</td> <td valign="top">
  
 
W. Thirring,  
 
W. Thirring,  
Line 1,064: Line 1,072:
 
(1983)
 
(1983)
  
pp. 209–277</TD></TR></table>
+
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.

Revision as of 17:01, 1 July 2020

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 ) : = \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 < 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=50420