Note that the implementation of pDME-PDFT differs from the one of NEVPT2-DMET in the following way: while pDME-PDFT evaluates the total energy using the density and on-top pair-density of the whole system (see eqs. 4 and 5), NEVPT2-DMET applies the NEVPT2 method only to the embedding space. Since pDME-PDFT is agnostic to the way in which the embedding calculation has been performed, it is designed to recover in part the effects of dynamic electron correlation even for inactive electrons, which are not correlated in the underlying trial wave function. In contrast, NEVPT2-DMET can not describe electron correlation beyond the embedding space. Moreover, pDME-PDFT has a lower cost scaling with respect to embedding space size compared to NEVPT2-DMET, making it potentially more advantageous both in terms of accuracy and cost reduction.