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 [Hégely 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.

In projection-based embedding (PbE) the subsystem partitioning of a chemical system is based on localized orbitals. We demonstrate how the localization step can lead to inconsistent orbital spaces along reaction paths, with severe consequences for reaction barriers and energies. We propose an orbital alignment procedure that resolves this problem without manual input. The usefulness of this alignment is demonstrated for a reaction benchmark set in combination with a direct orbital selection approach to automatize PbE calculations of double hybrid-in-nonhybrid density functional theory (DFT) and wave-function-in-DFT type for reaction energies and barriers. We show how the embedded calculations are accelerated in comparison to the corresponding supersystem calculations for realistic example reactions, using a new implementation of domain-based local pair natural orbital coupled cluster with single, double, and perturbative triple excitations [DLPNO-CCSD­(T0)]. The embedded calculations yield results within an error margin below 4 kJ mol–1 for the reaction barrier and energy when compared to the supersystem calculation. The calculations can be executed in a user-friendly, black-box-like fashion with minimal manual input.

Projection-based embedding (PbE) has become increasingly popular in recent years due to its simplicity and robustness. It is a very promising method for highly accurate calculations of reaction barriers and reaction energies via embedding of a correlated wavefunction or sophisticated density functional theory (DFT) method for the reaction center into a more cost effective DFT description of the environment. PbE enables an arbitrary partitioning of the supersystem orbitals into subsystems. In most applications so far, the selection of orbitals for the active system was directly linked to the selection of “active atoms.” We propose an inexpensive approach that automatically selects orbitals as active that change during the reaction and that assigns all remaining orbitals to the environment. This approach is directly coupled to the reaction under investigation and does not rely on any specification of active atoms. We compare different variants of this approach for the selection of orbitals along the reaction path for embedding of Adamo and Barone’s hybrid functional (known as PBE0) into Perdew, Burke, and Ernzerhof’s exchange-correlation functional (PBE), a method dubbed as PBE0–in–PBE embedding, based on orbitalwise partial charges and the kinetic energy. The most successful comparison scheme is based on shellwise intrinsic atomic orbital charges. We show for a set of six reactions of different types that the corresponding errors in reaction energies and barriers converge quickly to zero with the extension of the active-orbital space.

Domain-based local pair natural orbital coupled cluster (DLPNO-CC) has become increasingly popular to calculate relative energies (e.g., reaction energies and reaction barriers). It can be applied within a multi-level DLPNO-CC-in-DLPNO-CC ansatz to reduce the computational cost and focus the available computational resources on a specific subset of the occupied orbitals. We demonstrate how this multi-level DLPNO-CC ansatz can be combined with our direct orbital selection (DOS) approach [M. Bensberg and J. Neugebauer, J. Chem. Phys. 150, 214106 (2019)] to automatically select orbital sets for any multi-level calculation. We find that the parameters for the DOS procedure can be chosen conservatively such that they are transferable between reactions. The resulting automatic multi-level DLPNO-CC method requires no user input and is extremely robust and accurate. The computational cost is easily reduced by a factor of 3 without sacrificing accuracy. We demonstrate the accuracy of the method for a total of 61 reactions containing up to 174 atoms and use it to predict the relative stability of conformers of a Ru-based catalyst.

Density functional theory based embedding approaches for the description of chemical reactions are reviewed for their applicability to transition metal species.