  We derive two path integral Monte Carlo estimators for the derivative of the
potential of mean force (PMF), which may be numerically integrated to yield the
PMF. For the first estimator, we perform the differentiation on the exact path
integral, and for the second, we perform the differentiation on the path
integral after discretization. These estimators are successfully validated
against reference results for the harmonic oscillator and Lennard-Jones dimer
systems using constrained path integral Monte Carlo (PIMC) simulations.
Specifically, the estimators reproduce both the derivative of the PMF, as well
as the PMF itself, for the model systems at multiple temperatures. In Paper II,
these estimators are implemented alongside path integral molecular dynamics
(PIMD) with a constrained path integral Langevin equation thermostat for use
with more general systems and potentials.
