QM:QM embedding using electronic densities within an ONIOM framework: energies and analytic gradients

Accurate calculations of large systems remain a challenge in electronic structure theory. Hybrid energy techniques are a promising family of methods for treating such systems. Expanding on previous developments, we present a QM:QM electronic embedding model whereby the high-level region is polarized by the electron density of the low-level region within an ONIOM framework. A direct Coulomb embedding model as well a more computationally efficient model involving a density fitting expansion are considered. We also develop a generalized theory for the first derivatives of these classes of QM:QM electronic embedding schemes, which requires solution of a single set of self-consistent field response equations. Two initial test cases are presented and discussed.