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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.
Bibliography:
Video 11. 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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