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For simple atoms like Hydrogen or Helium, the precision is very high now. But how about atoms like oxygen or even uranium? Could it be one in a million?

poisson
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    Certainly not 1 in a million. The issue is that the milti-electron wave equation cannot be solved (even with a numerical approach) and some type of mean field approximation must be made (Hartree-Fock, etc). These introduce errors (generally of a few %). As the size of the atom increases, relativistic efffects become important and more errors are introduced by their approximate treatment. – Lewis Miller Sep 15 '17 at 14:40

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