The extent of ion
pairing in solution is an important phenomenon
to rationalize 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 wave function based
machine learning potentials with the random phase approximation (RPA)
and second order Møller–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.