Improved Lieb-Oxford bound on the indirect and exchange energies

Abstract

The Lieb-Oxford inequality provides a lower bound on the Coulomb energy of a classical system of $N$ identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree-Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.

Figures (3)

• Figure 1: Left: Critical points of the function $b\mapsto \Theta(a,b)-\Theta(b,b)/2$ in terms of $a\in[0,1]$. The maximum in \ref{['eq:def_g_neg_corr']} is attained at $b=b(a)$, which is the minimum of the two positive roots (thicker curve on the picture). It is equal to $a$ for $a\leqslant3/8$ and is then decreasing and reaches 0 at $a=1$. Right: The corresponding function $g(a)=\Theta(a,b(a))-\Theta(b(a),b(a))/2$, which equals $\Theta(a,a)/2=a^6(1-2a)$ for $a\leqslant3/8$.
• Figure 2: Plot of the numerical approximations of the function $r\mapsto r^{-3}f(r)$, for $\mu=\nu=\sigma$ (delta measure of the sphere) on the left and $\mu=\nu=\mathcal{B}=(3/4\pi){ {\mathds 1} }_{B_1}$ (uniform measure of the unit ball) on the right. In these two cases, the solution $f$ seems to coincide with the function $g$ from Lemma \ref{['lem:G']}. We used here the exact formula for $\Psi_{\mu\nu}$, which we discretized into the matrix $\psi$ by \ref{['eq:matrix_Psi_exact']} on a grid with $M=1000$ points per unit length and $R=20$ to compute the approximation $F$ to the function $f$. The shapes in the two cases are very similar but the ball gives a much lower function, hence a much better approximation $c_{\rm LO}\leqslant 1.6044$.
• Figure 3: Plot of the two radial measures $r\mapsto r^2\mu(r)$ (left) and $r\mapsto r^2\nu(r)$ (right) found by the BFGS algorithm for $K=50$, $M=100$, $R=10$. We obtain the upper bound $c_{\rm LO}\leqslant 1.5765$ claimed in \ref{['eq:LO_final']}.

Theorems & Definitions (19)

• Theorem 1: Main estimate
• Remark 2
• Remark 3: Link with the Lieb-Oxford proof
• Theorem 4: Lieb-Oxford bound for negatively--correlated states
• Corollary 5: Lieb-Oxford for negatively-correlated homogeneous processes
• proof : Proof of Theorem \ref{['thm:negative_correlations']}
• proof : Proof of Corollary \ref{['cor:Jellium']}
• Remark 6: Optimality of $g$
• Lemma 7: Case of the sphere
• Lemma 8