Q \cdot e^{-\beta H_0} \cdot e^{-\beta H'}}{\int e^{-\beta H_0} \cdot e^{-\beta H'}}$$ where $H_0$ is the Hamiltonian with the LJ potential and $H'$ the potential energy resulting from the Gaussian perturbation following form from Eq. \[eq:FD2\]. Equation \[eq:ensavg\] can be re-written as the ratio between two canonical averages over the unmodified Hamiltonian by multiplying by additional factors of $\int e^{-\beta H_0}$ and rearranging: $$\label{eq:perturbative}
\langle Q \rangle_H = \langle Q \cdot e^{-\beta H'} \rangle_{H_0} \cdot 
\frac{1}{\langle e^{-\beta H'} \rangle_{H_0}}$$ Since both canonical averages are over the unmodified potential only simulation with the low-fidelity potential is required. Therefore, computing the functional derivatives requires evaluating the Gaussian perturbation potentials on the trajectory obtained with the low-fidelity model. Such expressions are commonly used to compute free energies in thermodynamic integration and free energy perturbation approaches [@Straatsma1992; @Kollman1993].

Quantities for the canonical average in Eq. \[eq:perturbative\] are computed every 1 ps with a total of 48 ns of simulation time with the low-fidelity LJ potential, deemed well converged (see supplementary material). We evaluate these averages using a binned coordination number with a total of 2000 bins to compute $H'$ and $Q$ (as it contains contributions from the perturbation), described in more detail in the 