Zhang and Zhang first introduced the MFCC method in the context of calculating protein-ligand interaction energies and later it was also used for nucleic-acid-ligand interaction energies. In short, the MFCC method proposes the following: In order to calculate protein-ligand interaction energies, the protein is partitioned into a set of amino-acid-based fragments by cutting the peptide bonds (see figure ). To saturate the open valencies caused by cutting covalent bonds, the fragments are capped by groups that consist of a number of atoms from the neighboring fragments. The procedure thus leads to a set of (partly) overlapping fragments. The capping groups of adjacent capped fragments are combined to create new fragments called conjugate caps (concap). The total protein-ligand interaction energy is then approximated as the sum of the capped-fragment-ligand interaction energies, minus the sum of the concap-ligand interaction energies, thus removing double-counting due to the overlapping groups. The accuracy of the method can be systematically improved by increasing the size of the capping groups. Shortly after the original method was presented, it was extended to (real-space) electronic densities (and thereby total energies), electrostatic potentials, and total dipole moments, and later the method was further extended in many different directions. Order Capped fragments Concap Applying the MFCC procedure for the individual quantities of the embedding potential in the PDE model is not straightforward for the construction of the electrostatic and exchangerepulsion operators (eqs 1 and 3 respectively). Whereas MFCC can be directly applied to derive the localized polarizabilities needed for the induction operator (eq 2), it is more involved for fragment density and overlap matrices, and it is entirely unclear how to handle MO energies and coefficients. However, the operators only contain contributions that depend on one fragment, f , of the environment at a time. Therefore, the fragmentation procedure can be simply realized at the level of Fock matrix contributions, which is the strategy we present here. The electrostatic and exchange-repulsion contributions to the Fock matrix are first calculated fragment-wise, after which all of the contributions are combined according to the MFCC approach. The Fock matrix contributions are thus obtained according to