Density functional theory based embedding approaches for transition-metal complexes

Calculating Molecular Polarizabilities Using Exact Frozen Density Embedding with External Orthogonality

Frozen density embedding (FDE) with freeze-thaw cycles is a formally exact embedding scheme. In practice, this method is limited to systems with small density overlaps when approximate nonadditive kinetic energy functionals are used. It has been shown that the use of approximate nonadditive kinetic energy functionals can be avoided when external orthogonality (EO) is enforced, and FDE can then generate exact results even for strongly overlapping subsystems. In this work, we present an implementation of exact FDEc-EO (coupled FDE TDDFT with EO) for the calculation of polarizabilities in the Amsterdam density functional program package. EO is enforced using the level-shift projection operator method, which ensures that orbitals between fragments are orthogonal. For pure functionals, we show that only the symmetric EO contributions to the induced density matrix are needed. This leads to a simplified implementation for the calculation of polarizability that can exactly reproduce the supermolecular TDDFT results. We further discuss the limitation of exact FDEc-EO in interpreting subsystem polarizabilities due to the nonunique partitioning of the total density. We show that this limitation is due to the fact that subsystem polarizability partitioning is dependent on how the subsystems are initially polarized. As supermolecular virtual orbitals are exactly reproduced, this dependence is attributed to the description of the occupied orbitals. In contrast, for excitations of subsystems that are localized within one subsystem, we show that the excitation energies are stable with respect to the order of polarization. This observation shows that impacts from the nonunique nature of exact FDE on subsystem properties can be minimized by better fragmentation of the supermolecular systems if the property is localized. For global properties like polarizability, this is not the case, and nonuniqueness remains independent of the fragmentation used.

Automated, Consistent, and Even-Handed Selection of Active Orbital Spaces for Quantum Embedding

A widely used strategy to reduce the computational cost of quantum-chemical calculations is to partition the system into an active subsystem, which is the focus of the computational efforts, and an environment that is treated at a lower computational level. The system partitioning is mostly based on localized molecular orbitals. When reaction paths or energy differences are to be calculated, it is crucial to keep the orbital space consistent for all structures. Inconsistencies in orbital space can lead to unpredictable errors on the potential energy surface. While successful strategies to ensure this consistency have been established for organic and even metal-organic systems, these methods often fail for metal clusters or nanoparticles with a high density of near-degenerate and delocalized molecular orbitals. However, such systems are highly relevant for catalysis. Accurate yet feasible quantum-mechanical ab initio calculations are therefore highly desired. In this work, we present an approach based on the subsystem projected atomic orbital decomposition algorithm that allows us to ensure automated and consistent partitioning even for systems with delocalized and near-degenerate molecular orbitals and demonstrate the validity of this method for the binding energies of small molecules on transition-metal clusters.

Corresponding Active Orbital Spaces along Chemical Reaction Paths

The accuracy of reaction energy profiles calculated with multiconfigurational electronic structure methods and corrected by multireference perturbation theory depends crucially on consistent active orbital spaces selected along the reaction path. However, it has been challenging to choose molecular orbitals that can be considered corresponding in different molecular structures. Here, we demonstrate how active orbital spaces can be selected consistently along reaction coordinates in a fully automatized way. The approach requires no structure interpolation between reactants and products. Instead, it emerges from a synergy of the Direct Orbital Selection orbital mapping ansatz combined with our fully automated active space selection algorithm autoCAS. We demonstrate our algorithm for the potential energy profile of the homolytic carbon-carbon bond dissociation and rotation around the double bond of 1-pentene in the electronic ground state. However, our algorithm also applies to electronically excited Born-Oppenheimer surfaces.

On the accuracy of orbital based multi-level approaches for closed-shell transition metal chemistry

In this work, we investigate the accuracy of the local molecular orbital molecular orbital (LMOMO) scheme and projection-based wave function-in-density functional theory (WF-in-DFT) embedding for the prediction of reaction energies and barriers of typical reactions involving transition metals. To analyze the dependence of the accuracy on the system partitioning, we apply a manual orbital selection for LMOMO as well as the so-called direct orbital selection (DOS) for both approaches. We benchmark these methods on 30 closed shell reactions involving 16 different transition metals. This allows us to devise guidelines for the manual selection as well as settings for the DOS that provide accurate results within an error of 2 kcal mol^-1 compared to local coupled cluster. To reach this accuracy, on average 55% of the occupied orbitals have to be correlated with coupled cluster for the current test set. Furthermore, we find that LMOMO gives more reliable relative energies for small embedded regions than WF-in-DFT embedding.

Projection-Based Density Matrix Renormalization Group in Density Functional Theory Embedding

The density matrix renormalization group (DMRG) method has already proved itself as a very efficient and accurate computational method, which can treat large active spaces and capture the major part of strong correlation. Its application on larger molecules is, however, limited by its own computational scaling as well as demands of methods for treatment of the missing dynamical electron correlation. In this work, we present the first step in the direction of combining DMRG with density functional theory (DFT), one of the most employed quantum chemical methods with favorable scaling, by means of the projection-based wave function (WF)-in-DFT embedding. On two proof-of-concept but important molecular examples, we demonstrate that the developed DMRG-in-DFT approach provides a very accurate description of molecules with a strongly correlated fragment.

Orbital Pair Selection for Relative Energies in the Domain-Based Local Pair Natural Orbital Coupled-Cluster Method

For the accurate computation of relative energies, domain-based local pair natural orbital coupled-cluster [DLPNO-CCSD(T0)] has become increasingly popular. Even though DLPNO-CCSD(T0) shows a formally linear scaling of the computational effort with the system size, accurate predictions of relative energies remain costly. Therefore, multi-level approaches are attractive that focus the available computational resources on a minor part of the molecular system, e.g., a reaction center, where changes in the correlation energy are expected to be the largest. We present a pair-selected multi-level DLPNO-CCSD(T0) ansatz that automatically partitions the orbital pairs according to their contribution to the overall correlation energy change in a chemical reaction. To this end, the localized orbitals are mapped between structures in the reaction; all pair energies are approximated through computationally efficient semi-canonical second-order Møller--Plesser perturbation theory, and the orbital pairs for which the pair energies change significantly are identified. This multi-level approach is significantly more robust than our previously suggested, orbital selection-based multi-level DLPNO-CCSD(T0) ansatz [ J. Chem. Phys. 2021, 155, 224102] for reactions showing only small changes in the occupied orbitals. At the same time, it is even more efficient without added input complexity or accuracy loss compared to the full DLPNO-CCSD(T0) calculation. We demonstrate the accuracy of the multi-level approach for a total of 128 chemical reactions and potential energy curves of weakly interacting complexes from the S66x8 benchmark set.

Direct orbital selection within the domain-based local pair natural orbital coupled-cluster method

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.

"Dividing and Conquering" and "Caching" in Molecular Modeling

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, most important methodological advancements in more than half century of molecular modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes "dividing and conquering" and/or "caching" in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but results are not transferable. Deep learning has been utilized to realize more efficient and accurate ways of "dividing and conquering" and "caching" along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science. This framework is based on a third class of algorithm that facilitates molecular modeling through partially transferable in resolution "caching" of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for "dividing and conquering" and "caching" in complex molecular systems.