I am using a Split-Operator Fourier Transform (SOFT) technique to solve the time-dependent electronic Schrödinger Equation (TDSE) for a Hydrogen molecule under the Born-Oppenheimer approximation. So I have two electrons (1 and 2) which are moving and two protons ($a$ and $b$) which are stationary. The wavefunction $\psi$ is thus a function of 6 variables, the three position coordinates of each electron.
Because I am using SOFT, there are no orbitals or Slater determinants, everything is a scalar grid of either real or complex values in position or momentum space. That includes the wavefunction and the operators.
In this setup I am $not$ assuming that a many-electron wavefunction can be broken down into a Slater determinant of one-electron spin-orbitals, in fact I am not assuming anything about the wave function. $\psi$ is initialized as a random grid and then ground state is reached by iterativelly solving the TDSE, propagating it in imaginary time.
In this regime, I was thinking that the mechanism by which the electronic wavefunction is guaranteed to be antisymmetric is gone. After all, the SE does not take into account the antisymmetry, automatically, it has to be hand-pushed into it by using an appropriate ansatz along with abstract spin "eigenstates".
At this point I realized that using SOFT to solve the many-body TDSE may be fundamentally flawed due to the lack of anti-symmetry. Not wanting to give it up just yet I wondered if maybe I could move the anti-symmetrization from the ansatz of $\psi$ to the Hamiltonian.
So I have to ask: Is there a way to construct a many-body Hamiltonian for the SE that will guarantee a ground-state anti-symmetric wavefunction?