where the angled brackets denote the ensemble average and the subscript "0" indicates that the solute X is absent during the simulation. In the QCT, the excess free energy is partitioned into three physical parts by manipulations involving insertion and withdrawal of a hard particle that carves out a cavity in the liquid: µ ex X = -kT ln⟨e -M λ /kT ⟩ 0 + kT ln⟨e -M λ /kT ⟩ ε X -kT ln⟨e -ε X /kT ⟩ M λ (3) where M λ is a repulsive potential (half-harmonic potential in the current work) that pushes solvent ions away to the distance λ . The first term (packing, PK) is the free energy change to grow a cavity of radius λ in the liquid. The second term (inner-shell, IS) is minus the free energy change to grow the same cavity in the liquid around the solute X and the subscript ε X indicates that the solute X is present during the simulation. The third term (longranged, LR) is the free energy change for inserting the solute X into the cavity center and the subscript M λ indicates that the solute X is absent and there is a cavity generated by the repulsive potential M λ during the simulation (referred to as "uncoupled" below).