A procedure is developed to generate rigorous rough upper bounds to radial moments [ r 2n ] with integer n ≥ − 2 for the ground and excited states of atoms and molecules. These rough upper bounds to [ r 2n ] enable the calculation of accurate upper and lower bounds to the lesser moment [ r n ] through existing and new formulas that utilize a lower bound to the square overlap of a trial function and true wave function (in this case through an energy lower bound via the Eckart formula). Error bars to [ r −1 ], [ r ], [ r 2 ], and [ r 4 ] are calculated for the ground state of the non-relativistic infinite nuclear mass helium atom to yield expectation values within 0.037%, 0.018%, 0.039%, and 0.24% respectively, of the true values. As a byproduct of this investigation, a new formula for the error bar to observables is derived which is a slight improvement upon similar error bars. Also Lieb’s limit on the number of electrons that an atom can bind is reproduced.

中文翻译：

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原子径向位置矩的上限和下限

开发了一种程序来为原子和分子的基态和激发态生成径向矩 [ r 2n ] 的严格粗略上限，其中整数 n ≥ − 2。[ r 2n ] 的这些粗略上界可以通过现有的和新的公式计算较小矩 [ rn ] 的准确上下界，这些公式利用试函数和真实波函数的平方重叠的下界（在此通过 Eckart 公式的能量下界的情况）。[ r -1 ]、[ r ]、[ r 2 ] 和[ r 4 ] 的误差条是针对非相对论无限核质量氦原子的基态计算的，以产生0.037%、0.018% 以内的期望值，分别为真值的 0.039% 和 0.24%。作为这项调查的副产品，推导出了一个新的可观测误差线公式，它对类似的误差线略有改进。也再现了 Lieb 对原子可以结合的电子数量的限制。