Before we demonstrate the capabilities of the method, there is one last point to discuss for the general implementation scheme of local approximations within multicomponent methods. As previously discussed in our LDF-NEO-RHF implementation, the major fraction of the computational time arises from the largely delocalized canonical orbitals of the electrons, whereas the nuclei are rather compact and local. Therefore, we analysed the impact of the individual contributions of the multicomponent correlation method on the total computational timing. For this analysis we compare the NEO-PNO-LMP2 timings of the DNA test systems. Thereby, the most demanding electron-electron interaction is already significantly reduced as shown above. However, the impact of the electron-proton and proton-proton to the total computational times has not been separately discussed. In Fig. we explicitly show their contribution. By increasing the number of quantum protons from 3 9 the absolute computational time for the proton-proton correlation rises only from 0.03 s to 0.20 s and therefore any further local approximation is irrelevant. The electron-proton correlation is computationally much more demanding, since it also scales with the number of electrons. As previously mentioned, our implementation makes e↵ective use of the nuclear fitting basis in order to reduce the timings. The fraction of the computational timing for the electron-proton correlation rises from 1.87 8.83 %, again going from 3 9 protons. So even for a large system with 9 quantum protons and 980 electrons the computational time of the electronproton correlation part is small compared to the dominating computational time of the electronelectron correlation. By dissecting the computational time of the electron-proton correlation further it can be observed that the computationally most demanding part is indeed due to the canonical electronic three-index integrals E(A 0 |ia) within the nuclear fitting basis A 0 . The integral assembly of K ii 0 aa 0 , which has nominally the higher computational scaling with asymptotically fifth order, only increases from 0.17 1.78 %. So in general there is also little gain in introducing local approximations to the nuclei when computing the electron-proton correlation. In order to reduce the computa-tional time in future versions di↵erent strategies can be employed. The first strategy and probably most complex would be to compute the electron-proton correlation with PNO approximations. However, the simpler and more ecient implementation scheme would be to introduce further and tighter screening procedures which only compute electronic three-index integrals whereby the electronic orbitals i and j are close to the proton center. The third strategy would be to develop a tailored nuclear fitting basis set A 0 with less functions than the 10s10p10d10f fitting basis set. This would significantly reduce the computational time for the computation of the three-index integrals E(A 0 |ia). We leave this point for future work and demonstrate in the following the actual capabilities of a composite NEO approach as mentioned earlier, where the level of the electronicelectronic correlation is further increased by explicit correlation and more accurate correlation methods.