# Upper and lower bounds to atomic radial position moments

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## Abstract

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.

## Keywords

Upper bounds Lower bounds Position moments Radial moments Helium atom## Mathematics Subject Classification

35P15 49J40## Notes

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