Extending the Projection-Based Embedding Technique to Open-Shell Systems Using the Huzinaga Equation

In this work, we present an approach for the embedding of wave function theory (WFT) and density functional theory (DFT) methods in a lower-level density functional approximation using the projection-based embedding (PbE) technique for open-shell systems. Our method is based on the Huzinaga equation, which is implemented in both spin-restricted and spin-unrestricted forms. While the unrestricted PbE approach has been previously reported in the literature and follows naturally from the theory for closed-shell systems, the restricted formulation required the development of a new theory, building on earlier work by Roothaan (Rev. Mod. Phys., 1960, 32, 179) as well as Shaik and Filatov (Chem. Phys. Lett., 1999, 304, 429). Our implementation allows for the use of arbitrary combinations of restricted and unrestricted wave functions for the high- and low-level methods, which can be advantageous for the full-system low-level calculations. The various spin-restricted and unrestricted wave function-based PbE schemes are thoroughly tested, examining how the error in reaction energies depends on the size of the subsystem treated at the high level. Additionally, we compared the performance of PbE to that of other focused multilevel approaches, such as vacuum embedding, "our-own n-layered integrated molecular orbital and molecular mechanics" (ONIOM), and multilevel local correlation (MLC). The results showed that MLC performed the best among the tested methods, while only those PbE and ONIOM variants were proved to be competitive whose low-level methods employed at most a generalized gradient approximation (GGA). It is not straightforward to determine whether PbE or ONIOM is generally more advantageous: the latter can sometimes be more accurate and computationally cheaper, while PbE offers greater robustness and the possibility of systematic improvement.