Chapter 7 of https://brilliantlightpower.com/GUT/GUT_Volume_1/ covers two electron atoms.
> Helium's first ionization energy is −24.587387936(25) eV. This value was measured experimentally. The theoretic value of Helium atom's second ionization energy is −54.41776311(2) eV. The total ground state energy of the helium atom is −79.005154539(25) eV, or −2.90338583(13) Atomic units a.u., which equals −5.80677166(26) Ry.
Where are those calculations in chapter 7? I can't find them. I only find a few nice closed formulas. Hydrogen has closed formulas (almost), but Helium not.
If you compare it with the Wikipedia page, the energy of Helium has no closed formulas. There are four approximated method, and the list exclude the brute force method of using Gaussian[1] or Psi[2] that can solve any molecule with a high approximation using a decomposition into a big number of orbitals.
If they can give a nice closed formula for Helium (without magical ad hoc constants), it would be huge. Nobody has one, nobody believe there is one. It doesn't matter if the derivation of the formula makes no sense, the Bohr atom made no sense in 1913, but he had a nice formula that made sense like 20 years later.
[1] https://en.wikipedia.org/wiki/Gaussian_(software)
[2] https://en.wikipedia.org/wiki/PSI_(computational_chemistry)
I think it's an interesting method to use a classical model to aproximate atoms, and I surprised that it gives very close results, but I think at the end it's like the Bohr atom, that works in simple cases but fails in the complicated ones. I'll try to think an example where the quantum mechanics effects are big enough to cause a problem.