A Pair-Weighted Reciprocal-Cost Representation of the von Weizsäcker Functional
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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BibTeX
@misc{simons2026pairweighted,
title = {{A Pair-Weighted Reciprocal-Cost Representation of the von Weizsäcker Functional}},
author = {Megan Simons and Jonathan Washburn},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03677},
eprint = {2611.03677},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03677},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Pair-Weighted Reciprocal-Cost Representation of the von Weizsäcker Functional
AU - Megan Simons
AU - Jonathan Washburn
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03677
UR - https://arxiv.org/abs/2611.03677
AB - 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 −h^2/24∑_k∫(∂_k^2u)^2,dr+O(h^4) under the stated regularity conditions. Identifying u=√ρ and restoring the factor ℏ^2/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 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 1.5×10^−4 relative over R=1--6,a_0. 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.
ER -