information about the physics of the systems. The overall shapes correspond to the atomic shells as in the radial distribution function – notably more pointed for the solid case. At very low separation distances the FD goes to zero as atoms hit the soft-wall repulsion of the potentials; for the extreme liquid system the atoms occupy smaller separations.

To assess the accuracy of the calculation the functional derivative was also computed at equally spaced values of $r_i$ using the brute force approach described above. For each value of $r_i$ we perform four MD simulations with varying perturbations (identical to those in the perturbative approach) added to the LJ potential as shown in Fig. \[fig:calcFD\](a) and obtain the functional derivative with the same numerical derivatives as the perturbative approach. For each separate perturbation the simulation was run for 16 ns (split between 16 independent systems). The results of the brute force approach are shown in red in Fig. \[fig:FD\]; the two methods of calculating the functional derivative are nearly identical with only small numerical discrepancies.

Error correction using functional derivatives {#sec:results}
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To demonstrate our approach we now use the functional derivative calculated perturbatively in the previous section to correct the 