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Because we are studying a fermionic quantum system (system of electrons). The independent electron solution can’t just be a sum of all single electron solutions -as it would be in the classical case- because the wave function has to be anti-symmetric for a system of fermionic particles. This anti-symmetry needs to be specifically taken into account for the quantum system, which is done by the Slater determinant (obtained by the Hartree-Fock method).
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