Predicting properties for the Morse potential from LJ further highlights the challenge of attempting to correct a prediction made with a low-fidelity model. In this case the discrepancy between the two models is very large for short interatomic distances, leading to near zero phase space overlap and an inaccurate FUQ prediction for all physical cases. The red line in Figure \[fig:hybrid\] shows this discrepancy as a function of interatomic separation with the functional derivatives from both liquid conditions in blue to demonstrate the sensitivity to changes in the potential at these distances. The short range repulsion is described with an exponential for Morse while LJ uses an inverse power of 12. Thus the discrepancy between the two potentials diverges for short distances and the functional derivative correction, being first order, produces extremely large errors. These results are therefore omitted from Figure \[fig:results\].

Our approach can, nevertheless, be useful in such circumstances, but the high-fidelity model cannot be fully replaced by the low-fidelity one. Instead, we create a hybrid potential in Eq. \[eq:hybrid\] that smoothly switches from the low-fidelity LJ to the high-fidelity Morse between 2.8 and 2.4Å  using error functions, leading to a discrepancy of reasonable magnitude shown in green 