Automatic basis-set adaptation in projection-based embedding

Projection-based embedding (PbE) is an exact embedding method within density-functional theory (DFT) that has received increasing attention in recent years. Several different variants have been described in the literature, but no systematic comparison has been presented so far. The truncation of the basis is critical for the efficiency of this class of approaches. Here, we employ a basis-set truncation scheme previously used for level-shift embedding in a top-down fashion, and we present an own basis-set extension scheme for bottom-up type PbE. We compare its accuracy for the level-shift technique [Manby et al., J. Chem. Theory Comput. 8, 2564-2568 (2012)] and an empirically corrected variant, the external-orthogonality approach by Khait and Hoffmann [Annu. Rep. Comput. Chem. 8, 53-70 (2012)] and the approach based on the Huzinaga equation transferred to the DFT context [Hegely et al., J. Chem. Phys. 145, 064107 (2016)]. Concerning the reproduction in total energies, we show that the Huzinaga method yields the most stable results concerning a basis-set truncation in top-down embedding. For the practically more relevant calculation of energy differences, the efficient level-shift technique yields very promising results due to error cancellation. In bottom-up embedding, we observe convergence issues in cases where constraints in the Lagrange formalism cannot be fulfilled due to basis-set incompleteness.