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. All these formulas can be expressed as summations of products of binomial coefficients. 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 degener...

中文翻译：

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超级壳中包含的原子构型数的解析和数值表达式

对于原子中电子构型的数量，我们给出了三个明确的公式，即在各个简并g 1，g 2，…，g N的N个子壳中分布Q电子的方式的数量。使用生成函数形式主义获得新的表达式。第一个包含涉及多项式系数的和。第二个依赖于收集具有相同简并性的子壳的想法。第三个也收集具有相同简并性的子壳，并导致定义了一个二变量生成函数，从而允许推导递归关系。所有这些公式都可以表示为二项式系数的乘积之和。关于给定简并g的N个不同子壳上的种群分布，给出了该分布的最初时刻的解析表达式。