In the Hartree-method for two distinguishable particles we approximate the ground state \( \vert\Phi_0\rangle \) of the full Hamiltonian \( \hat{H} \) as the product state
$$ \vert\Phi_0\rangle \approx \vert\Psi\rangle= \vert\phi^L_0 \phi^R_0\rangle, $$under the constraint that the Hartree orbitals are orthonormal.