In a previous work we introduced a localdensity-fitting NEO-Hartree-Fock (LDF-NEO-HF) program, which exhibits a small computational footprint, allowing for the routine calculation of several quantum protons in extended systems on a regular workstation. Furthermore, we introduced a rather simple protocol to on-the-fly determine the positions of quantum nuclei. The latter works by updating the nuclei positions to the respective nuclear orbital charge centroids during the self-consistent field (SCF) cycles. The issue of nuclear orbitals being overly polarised can also be addressed by the constrained-NEO ansatz. In these earlier works, however, we failed to include dynamical correlation e↵ects. In particular, the electronproton correlation is deemed significant for obtaining nuclear densities and energetics, albeit their relative impact in NEO calculations is still disputed. In general, one can argue that the question of correlation in multicomponent approaches is still today a very current topic. In the following section we introduce the basics of second-order Møller-Plesset perturbation-theory within the NEO framework, as well as a short overview of density fitting and local correlation treatments.