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The Hartree product is a system (many-particle) wavefunction, given as a combination of wavefunctions of the individual particle wavefunctions. It is inherently mean-field (assumes the particles are independent) and is the unsymmetrized version of the Slater determinant ansatz in the Hartree–Fock method.
For two particles,
This is not satisfactory for fermions, such as electrons, because the resulting wave function is not antisymmetric. An antisymmetric wave function can be mathematically described using the Slater determinant.