Accurate embedding through potential reconstruction: A comparison of different strategies

Potential reconstruction is a powerful strategy for deriving accurate (sometimes called "exact") embedding potentials in the context of density-dependent embedding methods. It is particularly useful for partitioning covalent bonds in such fragment-based electronic-structure methods. While the general approach is well defined and easily explained, there are a number of choices to be made in practice, concerning, e.g., the specific reconstruction algorithm, the assignment of electrons to subsystems, or the initial guess potential. A general choice to be made is whether "exact" embedding potentials shall be derived for pre-defined target densities (top-down) or for approximate fragment densities that can be iteratively defined (bottom-up). Here, we compare the pros and cons of a variety of different variants of potential reconstruction, both in terms of conceptual issues and concerning their accuracy and efficiency. We also present several algorithmic improvements that can be crucial in critical cases of potential reconstruction, namely, we show (i) that a combination of basis-set and grid-based potential reconstruction schemes can lead to improved resulting densities, (ii) that similarly the combination of real-space and matrix-representation based potential reconstruction gives great advantages, and (iii) that the potential-matrix reconstruction by Zhang and Carter [J. Chem. Phys. 148, 034105 (2018)] can be made much more efficient by avoiding an explicit Hessian calculation. Additionally, we demonstrated (iv) that a double reconstruction, meaning a reconstruction of both the supersystem potential and the subsystem potential, may lead to beneficial error cancellation. We also address the question of consistent energetics derived from such reconstructed potentials.