  Feynman and Hibbs were the first to variationally determine an effective
potential whose associated classical canonical ensemble approximates the exact
quantum partition function. We examine the existence of a map between the local
potential and an effective classical potential which matches the exact quantum
equilibrium density and partition function. The usefulness of such a mapping
rests in its ability to readily improve Born-Oppenheimer potentials for use
with classical sampling. We show that such a map is unique and must exist. To
explore the feasibility of using this result to improve classical molecular
mechanics, we numerically produce a map from a library of randomly generated
one-dimensional potential/effective potential pairs then evaluate its
performance on independent test problems. We also apply the map to simulate
liquid para-hydrogen, finding that the resulting radial pair distribution
functions agree well with path integral Monte Carlo simulations. The surprising
accessibility and transferability of the technique suggest a quantitative route
to adapting Born-Oppenheimer potentials, with a motivation similar in spirit to
the powerful ideas and approximations of density functional theory.
