  The extent of ion pairing in solution is an important phenomenon to
rationalise transport and thermodynamic properties of electrolytes. A
fundamental measure of this pairing is the potential of mean force (PMF)
between solvated ions. The relative stabilities of the paired and solvent
shared states in the PMF and the barrier between them are highly sensitive to
the underlying potential energy surface. However direct application of accurate
electronic structure methods is challenging, since long simulations are
required. We develop wavefunction based machine learning potentials with the
Random Phase Approximation (RPA) and second order Moller-Plesset (MP2)
perturbation theory for the prototypical system of Na and Cl ions in water. We
show both methods in agreement, predicting the paired and solvent shared states
to have similar energies (within 0.2 kcal/mol). We also provide the same
benchmarks for different DFT functionals as well as insight into the PMF based
on simple analyses of the interactions in the system.
