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\title{A no-go theorem for purely local \texorpdfstring{$(\rho,T)$}{(rho,T)} Dirac interpolation}

\author{%
  Dong Bai\thanks{College of Mechanics and Engineering Science,
    Hohai University, Nanjing 211100, China.
    E-mail: \href{mailto:bydonfancy@gmail.com}{\texttt{bydonfancy@gmail.com}}.}%
}

\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We prove a sharp impossibility result for local Lieb--Oxford-type lower
bounds on the indirect Coulomb energy of antisymmetric fermionic states.
For a normalized \(N\)-fermion wavefunction \(\Psi\) write \(\rho_\Psi\)
for its one-particle density, \(T_\Psi\) for its local kinetic energy
density, and \(E_{\rm xc}(\Psi)\) for its indirect (exchange-correlation)
energy.  Let \(\mathcal R_\Psi\) be the dimensionless local kinetic
ratio.  Two constants control the discussion: the Dirac exchange
constant \(C_D(q)=\tfrac{3}{4}(6/\pi)^{1/3}q^{-1/3}\) and the
\emph{one-cell self-interaction constant}
\(C_{\rm cell}=\tfrac{3}{5}(4\pi/3)^{1/3}\), which is strictly larger
than \(C_D(q)\) for every spin multiplicity \(q\ge 1\).  We show that any
bounded function \(F\) with
\(\limsup_{r\downarrow 0}F(r)<C_{\rm cell}\) cannot serve as the integrand
of a universal pointwise lower bound
\(E_{\rm xc}(\Psi)\ge -\!\int F(\mathcal R_\Psi)\rho_\Psi^{4/3}\) for
arbitrarily large \(N\).  The proof constructs antisymmetric many-cell
states whose local data
\((\rho_\Psi,T_\Psi)\) look like a flat Fermi sea but whose indirect
energy is dominated by one-particle self-interaction.  Consequently, no
universal pointwise bound depending only on \((\rho_\Psi,T_\Psi)\) can
have \(C_D(q)\) as its small-\(\mathcal R\) limit.
\end{abstract}

\section{Setting and statement}
\label{sec:statement}

A long-standing goal in density-functional analysis is to refine the
Lieb--Oxford inequality~\cite{LiebOxford1981} into a \emph{pointwise} lower
bound whose integrand depends only on local densities and whose
small-kinetic-ratio limit reproduces the Dirac exchange constant
\(C_D(q)\)~\cite{Dirac1930}.  Various ans\"atze in the physics literature
take the natural form
\[
  E_{\rm xc}(\Psi)\ge -\!\int F\bigl(\mathcal R_\Psi(x)\bigr)\,\rho_\Psi(x)^{4/3}\dd x ,
\]
with \(F\) bounded and \(F(r)\to C_D(q)\) as \(r\downarrow 0\).  The main
theorem of this paper rules out every such ansatz: there is a hard
\emph{cell self-interaction constant} \(C_{\rm cell}>C_D(q)\) such that no
\(F\) with \(\limsup_{r\downarrow 0}F(r)<C_{\rm cell}\) can give a universal
bound.

For a normalized \(N\)-fermion wavefunction \(\Psi\), write
\[
  \rho_\Psi(x)
  =
  N\int |\Psi(x,x_2,\ldots,x_N)|^2\dd x_2\cdots\dd x_N
\]
and
\[
  T_\Psi(x)
  =
  N\int |\nabla_x\Psi(x,x_2,\ldots,x_N)|^2\dd x_2\cdots\dd x_N .
\]
Let
\[
  \mathcal R_\Psi(x)
  =
  \frac{T_\Psi(x)}
  {\frac35(6\pi^2)^{2/3}q^{-2/3}\rho_\Psi(x)^{5/3}} .
\]
The indirect Coulomb energy is
\[
  E_{\rm xc}(\Psi)
  =
  \left\langle\Psi,\sum_{1\le i<j\le N}\frac1{|x_i-x_j|}\Psi\right\rangle
  -
  \frac12
  \iint \frac{\rho_\Psi(x)\rho_\Psi(y)}{|x-y|}\dd x\dd y .
\]

Let
\[
  C_D(q)=\frac34\left(\frac6\pi\right)^{1/3}q^{-1/3}
\]
be the Dirac constant and define the one-cell self-interaction constant
\[
  C_{\rm cell}
  =
  \frac35\left(\frac{4\pi}{3}\right)^{1/3}.
\]
Numerically,
\[
  C_{\rm cell}=0.967\ldots,
  \qquad
  C_D(1)=0.930\ldots,
\]
so \(C_{\rm cell}>C_D(q)\) for every \(q\ge1\).

\begin{theorem}[No purely local Dirac interpolation]
\label{thm:nogo}
Let \(F:[0,\infty]\to[0,\infty)\) be bounded above and suppose
\[
  \limsup_{r\downarrow0}F(r)<C_{\rm cell}.
\]
Then the bound
\[
  E_{\rm xc}(\Psi)
  \ge
  -\int_{\R^3}F(\mathcal R_\Psi(x))\rho_\Psi(x)^{4/3}\dd x
\]
cannot hold for all antisymmetric \(N\)-fermion wavefunctions, even if
\(N\) is allowed to be arbitrarily large.

In particular, no universal pointwise local bound depending only on
\(\rho_\Psi(x)\) and \(T_\Psi(x)\) can have the Dirac constant
\(C_D(q)\) as its small-\(\mathcal R\) local limit.
\end{theorem}

\paragraph{Plan of the paper.}
\Cref{sec:onecell} carries out the one-cell computation that identifies
\(C_{\rm cell}\) and constructs a smooth approximating sequence with
controlled boundary error.  \Cref{sec:manycells} translates that single
cell into a many-cell Slater determinant whose density and kinetic
profile place each cell deep in the regime \(F\le a<C_{\rm cell}\), and
deduces the contradiction that proves \cref{thm:nogo}.

\section{The one-cell computation}
\label{sec:onecell}

Let \(B\subset\R^3\) be the unit ball and
\[
  \rho_B(x)=\frac{\mathbf 1_B(x)}{|B|},
  \qquad
  |B|=\frac{4\pi}{3}.
\]
Then
\[
  \int\rho_B^{4/3}=|B|^{-1/3}.
\]
The Coulomb self-energy of the uniform unit ball of total charge one is
\[
  \frac12\iint\frac{\rho_B(x)\rho_B(y)}{|x-y|}\dd x\dd y
  =
  \frac35.
\]
Hence
\[
  \frac{
  \frac12\iint |x-y|^{-1}\rho_B(x)\rho_B(y)\dd x\dd y
  }{
  \int\rho_B^{4/3}
  }
  =
  \frac35\left(\frac{4\pi}{3}\right)^{1/3}
  =
  C_{\rm cell}.
\]

We shall use smooth approximations to \(\rho_B\).  Choose
\(\varphi_\varepsilon\in C_c^\infty(B)\), \(\|\varphi_\varepsilon\|_2=1\),
such that
\[
  |\varphi_\varepsilon|^2\to \rho_B
  \quad\text{in }L^{4/3}(\R^3)
\]
and in Coulomb energy.  Additionally, choose it so that
\(\varphi_\varepsilon\) is constant on \(|x|\le1-\varepsilon\).  Then
\[
  \frac12
  \iint\frac{|\varphi_\varepsilon(x)|^2|\varphi_\varepsilon(y)|^2}
  {|x-y|}\dd x\dd y
  =
  C_{\rm cell}
  \int|\varphi_\varepsilon|^{8/3}
  +o_\varepsilon(1),
\]
The contribution of the boundary layer
\[
  \{1-\varepsilon<|x|<1\}
\]
to \(\int|\varphi_\varepsilon|^{8/3}\) is \(o_\varepsilon(1)\).

\section{Many disjoint cells}
\label{sec:manycells}

\begin{proof}[Proof of \cref{thm:nogo}]
Let \(M=\sup F<\infty\).  By assumption, choose \(a<C_{\rm cell}\) and
\(\delta>0\) such that
\[
  F(r)\le a
  \qquad
  0< r\le\delta .
\]
If \(M=0\), the contradiction is immediate.  Otherwise choose
\(\varepsilon>0\) so small that the boundary-layer contribution to
\(\int|\varphi_\varepsilon|^{8/3}\) is less than
\[
  \frac{C_{\rm cell}-a}{4M}
  \int|\varphi_\varepsilon|^{8/3},
\]
and the Coulomb self-energy ratio of \(\varphi_\varepsilon\) is larger than
\[
  \frac{C_{\rm cell}+a}{2}.
\]

Choose a vector \(\kappa\in\R^3\) so small that, on the plateau where
\(\varphi_\varepsilon\) is constant,
\[
  \frac{|\kappa|^2}
  {\frac35(6\pi^2)^{2/3}q^{-2/3}
  |\varphi_\varepsilon|^{4/3}}
  \le \delta .
\]
Replace \(\varphi_\varepsilon\) by
\[
  e^{i\kappa\cdot x}\varphi_\varepsilon(x).
\]
This does not change its density or Coulomb self-energy.  On the plateau,
however, the ratio \(\mathcal R\) is now a positive number not exceeding
\(\delta\), so the estimate \(F(\mathcal R)\le a\) applies there.

Place \(N\) translated copies of \(\varphi_\varepsilon\) in pairwise disjoint
balls whose mutual distances tend to infinity.  Denote these orbitals by
\[
  u_1,\ldots,u_N .
\]
Because their supports are disjoint, they are orthonormal.  Put
\[
  \Psi_N=u_1\wedge\cdots\wedge u_N .
\]
This is an antisymmetric \(N\)-fermion wavefunction.

On each cell, the one-particle density is \(|u_j|^2\), and the cells are
disjoint.  Therefore
\[
  \rho_{\Psi_N}=\sum_{j=1}^N |u_j|^2,
  \qquad
  T_{\Psi_N}=\sum_{j=1}^N |\nabla u_j|^2 .
\]
Since the supports are disjoint, the local ratio
\(\mathcal R_{\Psi_N}\) is exactly the one-cell ratio on each cell.
Thus, on the plateau part of every cell, \(\mathcal R_{\Psi_N}\le\delta\).

The pair density between different cells is exactly the product of the
corresponding one-cell densities.  Hence all inter-cell Coulomb terms cancel
against the inter-cell Hartree terms in \(E_{\rm xc}\).  What remains is only
the self-interaction subtraction in each cell:
\[
  E_{\rm xc}(\Psi_N)
  =
  -N\,
  \frac12
  \iint\frac{|\varphi_\varepsilon(x)|^2|\varphi_\varepsilon(y)|^2}{|x-y|}
  \dd x\dd y
  .
\]
Consequently,
\[
  E_{\rm xc}(\Psi_N)
  \le
  -N\,\frac{C_{\rm cell}+a}{2}
  \int|\varphi_\varepsilon|^{8/3}
  .
\]

On the other hand,
\[
\begin{aligned}
  \int F(\mathcal R_{\Psi_N})\rho_{\Psi_N}^{4/3}
  &=
  N\int F(\mathcal R_{\varphi_\varepsilon})
       |\varphi_\varepsilon|^{8/3}                                      \\
  &\le
  N a\int |\varphi_\varepsilon|^{8/3}
  +
  N M
  \int_{\text{boundary layer}}|\varphi_\varepsilon|^{8/3}                 \\
  &\le
  N\frac{3a+C_{\rm cell}}{4}
  \int|\varphi_\varepsilon|^{8/3}.
\end{aligned}
\]
Since
\[
  \frac{C_{\rm cell}+a}{2}
  >
  \frac{3a+C_{\rm cell}}{4},
\]
we obtain
\[
  E_{\rm xc}(\Psi_N)
  <
  -
  \int F(\mathcal R_{\Psi_N})\rho_{\Psi_N}^{4/3}
\]
for large enough cell separation.  This contradicts the proposed universal
bound.
\end{proof}

\begin{remark}
The construction uses arbitrarily many fermions, not merely \(N=1\).  The
obstruction is local: every cell contains a single electron, so the
kinetic ratio is small on most of the cell, yet the indirect Coulomb
energy is dominated by one-particle self-interaction whose sharp local
constant is \(C_{\rm cell}\), strictly larger than the Dirac constant.
\end{remark}

\begin{remark}
A true local Dirac theorem must therefore include an additional datum
that excludes this construction.  Natural candidates are: a local
many-particle condition, a subtraction of one-particle self-interaction,
or a pair- or number-variance datum on a controlled spatial scale.
Without such an addition, the pair \((\rho_\Psi,T_\Psi)\) is insufficient.
\end{remark}

\bigskip
\noindent\textbf{Acknowledgements.}
LaTeX typesetting and bibliographic copy-editing were carried out with
assistance from Claude Opus 4.7 (Anthropic).  All mathematical content
and proofs are the author's own.

\begin{thebibliography}{99}

\bibitem{Dirac1930}
P.~A.~M. Dirac.
\newblock Note on exchange phenomena in the Thomas atom.
\newblock \emph{Math. Proc. Cambridge Philos. Soc.} \textbf{26}
  (1930), 376--385.

\bibitem{LiebOxford1981}
E.~H. Lieb and S.~Oxford.
\newblock Improved lower bound on the indirect Coulomb energy.
\newblock \emph{Int. J. Quantum Chem.} \textbf{19} (1981), 427--439.

\end{thebibliography}

\end{document}
