Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.
J Chem Phys. 2011 Mar 28;134(12):124305. doi: 10.1063/1.3569128.
A shell model of an assembly of N equicharged particles subject to an arbitrary radial confining potential N W(r), where W(r) is parameterized in terms of an auxiliary function Λ(t), is presented. The validity of the model requires that Λ(t) is strictly increasing and concave for any t ∈ (0, 1), Λ'(0) is infinite, and Λ(t) = -t(-1) Λ'(t)/Λ''(t) is finite at t = 0. At the bulk limit of N → ∞, the model is found to correctly reproduce the energy per particle pair and the mean crystal radius R(N), which are given by simple functionals of Λ(t) and Λ'(t), respectively. Explicit expressions for an upper bound to the cohesive energy and the large-N asymptotics of R(N) are obtained for the first time. In addition, variational formulation of the cohesive energy functional leads to a closed-form asymptotic expression for the shell occupancies. All these formulae involve the constant ξ that enters the expression -(ξ/2) n(3/2) for the leading angular-correlation correction to the minimum energy of n electrons on the surface of a sphere with a unit radius (the solution of the Thomson problem). The approximate energies, which constitute rigorous upper bounds to their exact counterparts for any value of N, include the cohesive term that is not accounted for by the mean-field (fluidlike) theory and its simple extensions but completely neglect the surface-energy correction proportional to N.
提出了一种 N 个等电荷粒子组装体的壳模型，该组装体受任意径向约束势 N W(r)的作用，其中 W(r)用辅助函数 Λ(t)来参数化。该模型的有效性要求 Λ(t)在任何 t ∈ (0, 1)上严格递增和凹，Λ'(0)是无穷大，并且在 t = 0 时 Λ(t) = -t(-1) Λ'(t)/Λ''(t)是有限的。在 N → ∞的体相极限下，发现该模型正确地再现了每对粒子的能量和平均晶体半径 R(N)，它们分别是 Λ(t)和 Λ'(t)的简单泛函。首次获得了内聚能的上限和 R(N)的大 N 渐近表达式的显式表达式。此外，内聚能泛函的变分公式导致了壳占据的封闭形式的渐近表达式。所有这些公式都涉及常数 ξ，它出现在具有单位半径的球体表面上 n 个电子的最低能量的主要角相关修正表达式中-(ξ/2) n(3/2)（Thomson 问题的解）。对于任何 N 值，这些近似能量都构成了其精确对应物的严格上限，包括均场（类流体）理论及其简单扩展未考虑的内聚项，但完全忽略了与 N 成正比的表面能修正。
