The exchange potential in path integral studies: Analytical justification

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Abstract
We present analytical justification for our previously described exchange pseudopotential. We show how the fermi quantum partition function can be constructed from the Boltzmann (distinguishable particle) wave functions if the states that correspond to like-spin electrons occupying the same quantum state are excluded. A class of weighting functions that satisfy this constraint approximately is discussed. Our previous pseudopotential falls under this class. Essentially, our pseudopotential forces the unwanted states to have high energy and, hence, to make negligible contribution to the partition function. Exchange potentials of the form discussed in this article should be useful for studying systems where the (allowed) correlated Boltzmann wave functions have negligible amplitude for like-spin fermion–fermion distances less than the diameter of the individual particle wave packets. For example, in the case of two spin-up (or spin-down) fermions, if one fermion is located at r, then ‖Ψ(r,q)‖2 is negligible if q≂r. This should be the case for systems where a tight binding model is appropriate or for systems with strong interparticle repulsions.