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No closed form solution, but good approximation that can be calculated by hand with the Hartree-Fock method, see hartree-Fock method for the helium atom.
Video 12.
Quantum Chemistry 9.2 - Helium Atom Energy Approximations by TMP Chem (2016)
. Source. Video gives the actual numerical value of various methods, second order perturbation theory being very close. But it the says that in the following videos will only do Hartree-Fock method.


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