Interplay between incompleteness of the basis sets and approximations for the bi-functional for the non-additive kinetic potential in Frozen-Density Embedding Theory

Restricting the atom-centered basis functions of the embedded subsystem to those centered on the subsystem itself (the monomer expansion approximation) is commonly used in Frozen-Density Embedding Theory (FDET) based simulations as well as in conventional QM/MM type of simulations. Despite its widespread use, its influence on the FDET results has rarely been investigated independently from another approximation the one used for the non-additive kinetic-energy component of the embedding potential.In the present work, we disentangle these two sources of error by comparing monomer and supermolecular basis expansions for a series of alkali-metal cation-radical complexes exhibiting widely different tendencies toward charge transfer.Additionally, we show that the effective environment LUMO introduced in our previous work, Englert et al. [J. Chem. Phys. 164, 084122 (2026)], remains a good predictive tool even if the monomer expansion is employed. It provides a unified physical interpretation of the influence of both the basis-set restriction and the approximation to the non-additive kinetic potential. These results clarify the conditions under which the monomer expansion constitutes a reliable approximation and provide practical guidelines for setting up approximations used in the FDET eigenvalue equation.