论文标题
Supershell中包含的原子构型数的分析和数值表达式
Analytical and numerical expressions for the number of atomic configurations contained in a supershell
论文作者
论文摘要
我们为原子中的电子配置数量提出了三个明确的公式，即在$ n $ degneracies $ g_1 $，$ g_2 $，$ g_2 $，...，$ g_n $中分发$ q $电子的方法。新表达式是使用生成功能形式主义获得的。第一个包含涉及多项式系数的总和。第二个依赖于收集具有相同堕落的子壳的想法。第三个还收集具有相同退化的子壳，并导致两变量生成函数的定义，从而允许递归关系的推导。关于给定退化$ g $的$ n $不同子壳的人口分布，给出了该分布第一瞬间的分析表达式。通过计算累积物来分析具有任何变性的子壳的一般情况。提供任何顺序的累积物以及累积生成函数的相当简单的表达式。使用革兰氏特征膨胀，给出了分析分布的简单近似值，以正态分布乘以赫尔米特多项式的总和。 Edgeworth的扩展也已进行了测试。它的准确性等于革兰氏阴性的准确性时，几个术语的准确性都很少，但是当截断顺序增加时，它的差异要快得多。尽管原子超壳中的例子说明了这种分析，但它也适用于更通用的组合问题，例如费米亚分布。
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.
