compute the functional derivative in a MD simulations performed using the Lennard-Jones potential and shows that this information can be used to infer thermodynamic properties corresponding to various other potentials without re-running the simulation. The functional uncertainty quantification approach is quite generally applicable and we believe it will be useful to quantify uncertainties in a variety of materials models.

Thermodynamic quantities using coordination number
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The calculation of the functional derivative using the perturbative approach (Eq. \[eq:FD2\] and \[eq:perturbative\]) is expanded upon here, beginning with the contribution to the Hamiltonian from the perturbation. This quantity can be directly calculated with a sum over all pairs of atoms as the potential energy in any MD simulation: $$\label{eq:PEall}
H' = \sum_{i<j} \phi'(r_{ij})$$ This would require modification of the MD code to calculate this quantity within the force loop or summing over a saved atomic trajectory. This expression can be replaced: $$\label{eq:PEcoord}
H' = \frac{N}{2} \sum_{k} \phi'(r_k) \cdot c(r_k)$$ where $c(r)$ is the average coordination number at a given separation distance $r$, discretized into $k$ bins and $N$ the number of atoms. This is neither invasive to the code, nor requires significant storage or computation past the low-fidelity simulation.

With the potential energy 