超壳层中原子组态数的解析和数值表达式
Analytical and numerical expressions for the number of atomic configurations contained in a supershell
摘要：
本文给出了原子中电子组态数的三个显式公式，即$Q$电子在各简并态$g_1$，$g_2$，.的$N$亚壳层中分布的方式数，$g_N$。新的表达式得到使用生成函数的形式主义。第一个包含涉及多项式系数的总和。第二个依赖于收集具有相同简并度的子壳的想法。第三个也收集子壳具有相同的退化，并导致一个双变量生成函数的定义，允许推导递归关系。对于给定简并度$g$的$N$不同子壳上的布居分布，给出了该分布的一阶矩的解析表达式.通过计算累积量，分析了具有任意退化度的子壳的一般情况。给出了任意阶累积量的一个相当简单的表达式，以及累积量的母函数。使用Gram-Charlier展开，分析的分布在一个正态分布乘以一个厄米多项式的总和的简单近似。埃奇沃思的扩张也得到了测试。当保留较少项时，它的精度与Gram-Charlier精度相当，但当截断阶数增加时，它的发散速度要快得多。虽然这种分析是通过原子超壳层中的例子来说明的，但它也适用于更一般的组合问题，例如费米子分布。
英文摘要：
We present three explicit formulas for the number of electronic configurations in an atom, i.e. the number of ways to distribute $Q$ electrons in $N$ subshells of respective degeneracies $g_1$, $g_2$, ..., $g_N$. The new expressions are obtained using the generating-function formalism. The first one contains sums involving multinomial coefficients. The second one relies on the idea of gathering subshells having the same degeneracy. A third one also collects subshells with the same degeneracy and leads to the definition of a two-variable generating function, allowing the derivation of recursion relations. Concerning the distribution of population on $N$ distinct subshells of a given degeneracy $g$, analytical expressions for the first moments of this distribution are given. The general case of subshells with any degeneracy is analyzed through the computation of cumulants. A fairly simple expression for the cumulants at any order is provided, as well as the cumulant generating function. Using Gram-Charlier expansion, simple approximations of the analyzed distribution in terms of a normal distribution multiplied by a sum of Hermite polynomials are given. The Edgeworth expansion has also been tested. Its accuracy is equivalent to the Gram-Charlier accuracy when few terms are kept, but it is much more rapidly divergent when the truncation order increases. While this analysis is illustrated by examples in atomic supershells it also applies to more general combinatorial problems such as fermion distributions.
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