  A mapping is made between fermion exchange and excluded volume in the
quantum-classical isomorphism using polymer self-consistent field theory. Apart
from exchange, quantum particles are known to be exactly representable in
classical statistical mechanics as ring polymers, with contours that are
parametrized by the inverse thermal energy, often called the imaginary time.
Evidence in support of a previously used approximation for fermion exchange in
ring polymer self-consistent field theory is given, specifically, that the use
of all-contour interactions in the mean field picture instead of equal
imaginary time interactions is justified based on the symmetry of ring
polymers. It is also shown that the removal of forbidden thermal trajectories,
both those that violate excluded volume directly and those that represent
topologically inaccessible microstates, is equivalent to antisymmetric
exchange. The electron density of the beryllium atom is calculated with ring
polymer self-consistent field theory ignoring classical correlations, and very
good agreement is found with Hartree-Fock theory which also neglects Coulomb
correlations. The total binding energies agree to within less than 6%, which
while still far from chemical accuracy, is remarkable given that the field
theory equations are derived from first principles with zero free parameters.
The discrepancy between self-consistent field theory and Hartree-Fock theory is
attributed to classical Coulomb self-interactions which are included in
Hartree-Fock theory but not in self-consistent field theory. A potential method
to improve the agreement by more accurately representing electron-electron
self-interactions in self-consistent field theory is discussed, as are the
implications for quantum foundations of the quantum-classical mapping between
fermion exchange and thermal trajectory excluded volume.
