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In other words, how does theory ensure that the multi-electron wavefunction is seperable into variables r1 and r2?

Even in Hartree form (where things like exhange interaction and exclusion principle is not readily captured) -- the way we write down Psi(r1,r2) is:

[tex]\Psi(r1,r2) = \phi (r1)\phi(r2)[/tex]

I learned from Wikipedia that this is really an ansatz in the HF theory. But is there any convincing reason that the well-versed quantum camp could deliver here?