Challenging the Lieb–Oxford bound in a systematic way

Figures & data

Table 1. Values Λ[ρ] for some simple spherical two-electron trial densities ρ(r) in three dimensions (N = 2, D = 3), obtained numerically from Equations (A9(A9) $$N_e\left(s\right)\equiv {\int }_{\left|\mathbf{r}\right|\le s}{\mathrm{d}}^{D}r\rho \left(r\right),$$(A9) )–(A11(A11) $$V_{\mathrm{ee}}^{\mathrm{SCE}}\left[\rho \right]=\frac{1}{2}\int {\mathrm{d}}^{D}r\frac{\rho \left(r\right)}{r+f\left(r\right)}.$$(A11) ) of Appendix A. In the last two rows, we consider densities with compact support: ‘droplet’ corresponds to the case of a sphere of uniform density [24], and the density proportional to r⁻³ [11] has been evaluated for R₁ = 10³ and R₂ = 10⁵. [Atomic units are used, where r is a dimensionless radial coordinate.]

ρ(r)∝	Λ[ρ]	ρ(r)∝	Λ[ρ]
$e^{-10{(r-1)}^2}$	1.499	$e^{-50{(r-1)}^2}$	1.262
(1 + r)^−4	1.562	$e^{-r^2}$	1.689
(1 + r)^−5	1.637	e^−r	1.69905
(1 + r)^−6	1.662	r e^−r	1.69866
(1 + r)^−7	1.674	r^1/2e^−r	1.70097
(1 + r)^−10	1.687	r^1/3e^−r	1.70095
1 − r,  r ≤ 1	1.638	r^−3,  r ∈ [R1, R2]	1.145
Droplet	1.498	$cos\left(r\right),r\le \frac{\pi }{2}$	1.627

Table 2. Exact values $\Lambda [\rho +ϵ{\sigma }_{a_2}]$ for various values of ε < 0, compared with the first-order expansion (see Section 3.3).

ε	$\Lambda [\rho +ϵ{\sigma }_{a_2}]$	Λ[ρ] + εI(a2)
0	1.699052	1.699052
−0.01	1.700487	1.701354
−0.015	1.700833	1.702505
−0.02	1.700868	1.703656
−0.0207	1.700843	1.703817

Table 3. For different values of N, we compare the Λ[ρ] obtained from atomic densities (values from [21]) with the ones obtained from spherical droplets of uniform density (values from [24]).

N	Λ[ρ] atomic	Λ[ρ] droplet
3	1.713	1.550
4	1.731	1.603
5	1.747	1.627
6	1.767	1.657
10	1.816	1.708

Figure 1. Values of Λ[ρ] for the fixed density profile ρ(r)∝r^1/2e^−r compared with those for spheres of uniform density (droplets) as a function of the particle number N. The values for ρ(r)∝r^1/2e^−r are significantly higher than those for uniform densities. The size extrapolation for uniform densities is also shown, where the fitting parameters are a = 1.918, b = −0.3253, c = −0.2791.

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