predictions for Morse from the hybrid potential (white symbols) lie near the unity line in Fig. \[fig:histpoints\] (describing good agreement between the phase space of each trajectory) in contrast to Morse from LJ (gray symbols), furthest from the line. We note that the high-fidelity model needs to be used during the actual simulation; however, only approximately 12% of the atomic force calculations fall in the range requiring the high-fidelity model. Thus, if the high-fidelity model was significantly more intensive than the low-fidelity the methodology would still greatly reduce computational cost.

Discussion
==========

The perturbative method to obtain the functional derivatives described here is similar to techniques in free energy methods: free energy perturbation and thermodynamic integration [@Bash1987; @Karplus1990]. One subset of these methods most comparable to FUQ and often used in biology simulations, referred to as computer alchemy, is utilized to calculate free energy changes along non-physical paths as potentials are slowly turned on or off for the various molecules or solvents of interest [@Gumbart2012; @He2014; @Wescott2002; @Lawrenz2012]. These methods share with the work described here the need to sample from the phase space trajectory of an initial state (often a simple reference state for free energy calculations, e.g. the 