We derive an explicit expression for the functional derivative of the subleading term in the strong interaction limit expansion of the generalized Levy-Lieb functional for the special case of two electrons in one dimension. The expression is derived from the zero-point-energy (ZPE) functional, which is valid if the quantum state reduces to strongly correlated electrons in the strong coupling limit. The explicit expression is confirmed numerically and respects the relevant sum rule. We also show that the ZPE potential is able to generate a bond midpoint peak for homonuclear dissociation and is properly of purely kinetic origin. Unfortunately, the ZPE diverges for Coulomb systems, whereas the exact peaks should be finite.