where W γδ is an element of the density matrix of a fragment in the environment weighted by the occupied molecular orbital (MO) energies (ε), i.e., W γδ = N f occ i=1 ε i C γi C δi with C being MO coefficients. This operator models the effect of exchange repulsion and thus effectively prevents the electrons in the core region from penetrating into the environment by modeling the effect of exchange repulsion. It is approximate because we do not solve the Huzinaga-Cantu equations, which would require either a calculation on the full system or an iterative procedure. Instead, the fragment densities are determined in isolation, and the repulsion is dampened by a transferable prefactor. Exchange repulsion is a crucial component of the PDE model (and generally in quantum embedding) needed to avoid variational collapse, whereas it can sometimes be neglected when using conventional classical embedding, though with the risk of erroneous results.