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The system in question is H$_{2}$. Why does the Hartree Fock orbital give true ground state energy for this system?

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The premisse is not correct. Also for H2 electron correlation is important. As the ground state is a spin singlet, there is no Pauli corration but there still is Coulomb correlation. The ground state therefore contains contributions from multiple Slater determinants even in this relatively simple case.

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