whereby the indices A stand for N c classical nuclei, j for the N e electrons and the dashed indices for the N p quantum protons. The potentials and densities are marked with e and p accordingly. In the case of the e↵ective potential ⌫ p e↵ felt by protons, one has the interaction with classical nuclei, electrons, other quantum protons and the electron-proton (⌫ ep c ) correlation potential. The electrons and quantum protons couple through the Coulomb interaction and electron-proton correlation. The electronelectron correlation bears the same form as in Born-Oppenheimer calculations.