As in Hartree Fock Approximation Slater has used slater determinant to maintain the antisymmetry!
Why? Because electrons are spin-1/2 particles; and therefore -- according to the spin-statistics theorem -- must obey Fermi statistics. Which is a GOOD THING for the stability of matter in and around us.
The electronic Hamiltonian commutes with the operator permutation of identical particles. The eigenfunctions of this second operator are either symmetric or antisymmetric, that means that there are two kind of particles, those described by symmetric functions under permutation (bosons) and those described by antisymmetric functions under permutation (fermions). Electrons belong to the second class (fermions). Then, the eigenfunctions of a set of indistinguishable electrons must be antisymmetric under permutation of electron positions.
The key point is that the permutation of identical particles commute with the Hamiltonian operator of a system of identical interacting particles and there is no perturbing operator to be enclosed in the Hamiltonian that can destroy this commutation.
Then, the eigenfunctions of any system of identical particles, interacting or not, must be either symmetric or antisymmetric. If a system is described by a symmetric/antisymmetric function there is not possible perturbation that will change this symmetry.
Systems of electrons (fermions, in general) must be described by antisymmetric functions, while systems of photons (bosons, in general ) must be described by symmetric functions.
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