A Pair-Weighted Reciprocal-Cost Representation of the von Weizsäcker Functional

Abstract

We investigate pair-weighted lattice energies constructed from the reciprocal ratio cost $J(x)=\frac{1}{2}(x+x^{-1})-1$ 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 $\frac{1}{2}(u-v)^2$ for arbitrary positive neighboring amplitudes, without requiring a small-deviation expansion. On a Cartesian lattice, the resulting discrete energy converges to $\frac{1}{2}\int |\nabla u|^2,d\mathbf r$, 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. Identifying $u=\sqrtρ$ and restoring the factor $\hbar^2/m$ 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 $\mathrm{H}_2^+$ 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 $R=1$--$6,a_0$. Alternative extrapolations indicate a conservative sensitivity of up to approximately $5\times10^{-4}$. The construction therefore provides a conditional exact discrete representation of the von Weizsäcker sector rather than an approximation to the complete noninteracting kinetic functional.