as both potentials tend to zero at the same cutoff.

The functional error was numerically integrated using the trapezoid rule to obtain the total correction for the QoI. In order to verify these correction predictions we performed explicit MD simulations with the high-fidelity modified LJ potentials. Figure \[fig:results\] compares the potential energy (a) and pressure (b) differences explicitly simulated with low and high-fidelity potentials (gray) to the corrections obtained with FUQ (with colors matching those in Fig. \[fig:potentials\], as well as the hybrid LJ-Morse potential discussed in sub-section \[sec:resultsMorse\] in white). These results are shown in detail in Tables \[tbl:PE\] and \[tbl:P\] in the appendix.

For almost all cases, the FUQ predictions are in excellent agreement with the direct simulations. Excluding only the Sine 6 and 7 potentials for the ambient liquid (discussed below) average error is 0.600% and 1.70% for potential energy and pressure, respectively. We stress that these very accurate corrections are obtained only making use of the the simulation with the unmodified LJ potential, no additional MD simulations are required.

The ability of FUQ to correct the QoIs is impacted greatly by the degree of phase space overlap between the high and low-fidelity simulations. This can be 