Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
J Chem Phys. 2013 Sep 21;139(11):114109. doi: 10.1063/1.4821217.
Asymptotic equivalence of the shell-model and local-density (LDA) descriptions of Coulombic systems confined by radially symmetric potentials in two and three dimensions is demonstrated. Tight upper bounds to the numerical constants that enter the LDA expressions for the Madelung energy are derived and found to differ by less than 0.5% from the previously known approximate values. Thanks to the variational nature of the shell-model approximate energies, asymptotic expressions for other properties, such as mean radial positions of the particles and number densities, are also obtained. A conjecture that generalizes the present results to confining potentials with arbitrary symmetries is formulated.
本文证明了在二维和三维空间中，由径向对称势约束的库仑系统的壳模型和局域密度（LDA）描述的渐近等价性。推导出了 LDA 中马德隆能表达式中出现的数值常数的紧上界，并发现它们与之前已知的近似值相差不到 0.5%。由于壳模型近似能的变分性质，还得到了其他性质的渐近表达式，例如粒子的平均径向位置和数密度。本文还提出了一个猜想，将目前的结果推广到具有任意对称性的约束势。
