A trend already briefly touched in our original article is to entirely avoid the need for a NAKE functional and the corresponding NAKP potential by enforcing the mutual orthogonality of the subsystems. One flavor of such an approach was introduced in a work by Khait and Hoffmann , where explicit Lagrangian multipliers were introduced in the variational optimization procedure to ensure mutual orthogonality of the subsystem orbitals. While it was erroneously combined with a NAKE approximation in this original article, the use of additional NAKE components was avoided in later work . Subsequent work demonstrated that this method makes it possible to describe bond-dissociation processes . In parallel, Manby, Miller and co-workers had proposed an embedding scheme in which external orthogonality was introduced in terms of a level-shift operator, which shifted the occupied orbitals of the environment systems to very high energies and hence made them effectively orthogonal to the occupied orbitals of the active subsystem . While the external orthogonality represents a fundamental difference to the original formulation of sDFT and FDE, all other ingredients in energy and potential expressions (apart from a perturbative correction for finite level-shift parameters ) are identical in this level-shift embedding. Hence, the latter can, from the sDFT perspective, be regarded as an FDE variant in which the NAKP has pragmatically been replaced by a projection operator, so that the non-additive kinetic energy simply vanishes. As might seem obvious, the two schemes are closely related to each other. An independent implementation of the level-shift approach was presented by Chulhai and Jensen in 2015 and applied to subsystems connected by aromatic bonds. A third variant of PbE was introduced by Kállay and co-workers in 2016 , which explicitly related to the partitioning of Slater determinant wavefunctions in the context of Hartree-Fock calculations developed by Huzinaga and Cantu . The projection operator derived in that work is essentially identical to that by Khait and Hoffmann, and only differs in the way that Hermiticity of the resulting Fock operator is ensured.