Chemical Physics · arXiv:2611.03677 v1 · submitted 20 Oct 2026
A Pair-Weighted Reciprocal-Cost Representation of the von Weizsäcker Functional
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“We investigate pair-weighted lattice energies constructed from the reciprocal ratio cost J(x) = 1/2(x + x−1)−1 and their connection to the von Weizsäcker kinetic-energy functional.”
more of the abstract
When the pair weight depends only on the common multiplicative amplitude scale and is quadratic under uniform amplitude rescaling, it takes the form B(u,v) = Cuv. For the normalization C = 1, the weighted edge cost reduces exactly to 1/2(u−v)2 for arbitrary positive neighboring amplitudes, without requiring a small-deviation expansion. On a Cartesian lattice, the resulting discrete energy converges to 1/2∫|∇u|2,dr, with leading correction −(h2)/24∑k∫(∂k2u)2,dr + O(h4) under the stated regularity conditions. Identifying u = √ρ and restoring the factor ℏ2/m yields the von Weizsäcker functional. By contrast, the unweighted ratio cost approaches a scale-invariant log-Dirichlet action and does not have kinetic-energy dimensions. Numerical benchmarks confirm the expected lattice convergence and assess the discretization error in smooth, nodal, cusp, and molecular cases. For H2+ densities obtained from aug-cc-pV5Z orbitals, a four-grid least-squares h2 + h4 extrapolation agrees with orbital kinetic energies within 1.5 × 10−4 relative over R = 1–6,a0. Alternative extrapolations indicate a conservative sensitivity of up to approximately 5 × 10−4. The construction therefore provides a conditional exact discrete representation of the von Weizsäcker sector rather than an approximation to the complete noninteracting kinetic functional.
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The paper
Abstract
We investigate pair-weighted lattice energies constructed from the reciprocal ratio cost J(x) = 1/2(x + x−1)−1 and their connection to the von Weizsäcker kinetic-energy functional. When the pair weight depends only on the common multiplicative amplitude scale and is quadratic under uniform amplitude rescaling, it takes the form B(u,v) = Cuv. For the normalization C = 1, the weighted edge cost reduces exactly to 1/2(u−v)2 for arbitrary positive neighboring amplitudes, without requiring a small-deviation expansion. On a Cartesian lattice, the resulting discrete energy converges to 1/2∫|∇u|2,dr, with leading correction −(h2)/24∑k∫(∂k2u)2,dr + O(h4) under the stated regularity conditions. Identifying u = √ρ and restoring the factor ℏ2/m yields the von Weizsäcker functional. By contrast, the unweighted ratio cost approaches a scale-invariant log-Dirichlet action and does not have kinetic-energy dimensions. Numerical benchmarks confirm the expected lattice convergence and assess the discretization error in smooth, nodal, cusp, and molecular cases. For H2+ densities obtained from aug-cc-pV5Z orbitals, a four-grid least-squares h2 + h4 extrapolation agrees with orbital kinetic energies within 1.5 × 10−4 relative over R = 1–6,a0. Alternative extrapolations indicate a conservative sensitivity of up to approximately 5 × 10−4. The construction therefore provides a conditional exact discrete representation of the von Weizsäcker sector rather than an approximation to the complete noninteracting kinetic functional.
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abstract compiled · 2026-11-04
The authors' main claim, as the classifier picked it out 1.00
“The construction therefore provides a conditional exact discrete representation of the von Weizs\"acker sector rather than an approximation to the complete noninteracting kinetic functional.”
How the main claim is stated: asserted 0.99
What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.96 · human participants 0.01 · controlled comparison 0.21 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.84 · mentions code or data availability 0.03
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v1). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| We investigate pair-weighted lattice energies constructed from the reciprocal ratio cost $J(x)=\frac{1}{2}(x+x^{-1})-1$ and their connection to the von Weizs\"acker kinetic-energy functional. | contribution 0.99 | 0.88 | 0.09 | asserted 0.79 |
| When the pair weight depends only on the common multiplicative amplitude scale and is quadratic under uniform amplitude rescaling, it takes the form $B(u,v)=Cuv$. | method 0.60 | 0.68 | 0.05 | asserted 0.99 |
| For the normalization $C=1$, the weighted edge cost reduces exactly to $\frac{1}{2}(u-v)^2$ for arbitrary positive neighboring amplitudes, without requiring a small-deviation expansion. | result 0.97 | 0.85 | 0.15 | asserted 1.00 |
| On a Cartesian lattice, the resulting discrete energy converges to $\frac{1}{2}\int |\nabla u|^2,d\mathbf r$, with leading correction $-\frac{h^2}{24}\sum_k\int(\partial_k^2u)^2,d\mathbf r+O(h^4)$ under the stated regularity conditions. | result 0.98 | 0.86 | 0.14 | asserted 1.00 |
| Identifying $u=\sqrt{\rho}$ and restoring the factor $\hbar^2/m$ yields the von Weizs\"acker functional. | interpretation 0.58 | 0.78 | 0.20 | asserted 1.00 |
| By contrast, the unweighted ratio cost approaches a scale-invariant log-Dirichlet action and does not have kinetic-energy dimensions. | interpretation 0.53 | 0.52 | 0.06 | asserted 1.00 |
| Numerical benchmarks confirm the expected lattice convergence and assess the discretization error in smooth, nodal, cusp, and molecular cases. | method 0.56 | 0.93 | 0.05 | asserted 1.00 |
| For $\mathrm{H}_2^+$ densities obtained from aug-cc-pV5Z orbitals, a four-grid least-squares $h^2+h^4$ extrapolation agrees with orbital kinetic energies within $1.5\times10^{-4}$ relative over $R=1$--$6,a_0$. | result 0.98 | 0.94 | 0.07 | asserted 1.00 |
| Alternative extrapolations indicate a conservative sensitivity of up to approximately $5\times10^{-4}$. | result 0.65 | 0.90 | 0.04 | hedged possibility 0.87 |
| The construction therefore provides a conditional exact discrete representation of the von Weizs\"acker sector rather than an approximation to the complete noninteracting kinetic functional. | interpretation 0.73 | 0.83 | 0.82 | asserted 0.99 |
What kind of paper the classifier reports this is: theoretical result 0.83 · evidence from proof or derivation 0.87
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