Thomas-Fermi theory

Fermi–Thomas theory

Sometimes called the "statistical theory" , it was invented by L.H. Thomas [a13] and E. Fermi [a2], shortly after E. Schrödinger invented his quantum-mechanical wave equation, in order to approximately describe the electron density, $\rho ( x )$, $x \in \mathbf{R} ^ { 3 }$, 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

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[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.