  We derive two path integral estimators for the derivative of the quantum
mechanical 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.
