Only orbitals mainly centered on the border atoms were included in the projection operator, whereas molecular orbitals located on more distant atoms where treated in terms of an approximate, non-additive kinetic potential contribution. Bennie et al. later introduced a basis-set truncation scheme based on a Mulliken population analysis for the atoms of the active system . In Refs. 239, 240 these two strategies for top-down PbE were combined and tested for all three variants of PbE (Level-Shift, Hoffmann-type projection, and Huzinaga equation); the Huzinaga-operator strategy turned out to behave most favorably with respect to the basis-set truncation. In addition, an automatic basis-set extension was proposed for bottom-up PbE calculations in that work, in which the reference basis set is just the monomolecular basis set of the active system. Another strategy in terms of a dual-basis approach for embedding calculations was proposed by the Kállay group in 2018 .