The coupling parameter, λ, was sampled at 24 equally spaced data points in the range λ ∈ (0,1]. At each point, the system was equilibrated for 0.25 ns followed by a 0.75 ns run to obtain u sl . The Hamiltonian derivatives, ∂u sl ∂λ , were computed using two distinct methods: numerical differentiation and polynomial fitting. The first method employed numerical differentiation of u sl through the central differences scheme. The second method involved fitting u sl with a second-order polynomial and subsequently obtaining its derivative. The resulting numerical and polynomial Hamiltonian derivatives were then integrated using the trapezoid rule and analytic methods, respectively, allowing W sl to be calculated using equations 16-18.