Accurate and systematically improvable density functional theory embedding for correlated wavefunctions

State-of-the-art local correlation methods enable affordable gold standard quantum chemistry for up to hundreds of atoms

In this feature, we review the current capabilities of local electron correlation methods up to the coupled cluster model with single, double, and perturbative triple excitations [CCSD(T)], which is a gold standard in quantum chemistry. The main computational aspects of the local method types are assessed from the perspective of applications, but the focus is kept on how to achieve chemical accuracy (i.e., <1 kcal mol-1 uncertainty), as well as on the broad scope of chemical problems made accessible. The performance of state-of-the-art methods is also compared, including the most employed DLPNO and, in particular, our local natural orbital (LNO) CCSD(T) approach. The high accuracy and efficiency of the LNO method makes chemically accurate CCSD(T) computations accessible for molecules of hundreds of atoms with resources affordable to a broad computational community (days on a single CPU and 10-100 GB of memory). Recent developments in LNO-CCSD(T) enable systematic convergence and robust error estimates even for systems of complicated electronic structure or larger size (up to 1000 atoms). The predictive power of current local CCSD(T) methods, usually at about 1-2 order of magnitude higher cost than hybrid density functional theory (DFT), has become outstanding on the palette of computational chemistry applicable for molecules of practical interest. We also review more than 50 LNO-based and other advanced local-CCSD(T) applications for realistic, large systems across molecular interactions as well as main group, transition metal, bio-, and surface chemistry. The examples show that properly executed local-CCSD(T) can contribute to binding, reaction equilibrium, rate constants, etc. which are able to match measurements within the error estimates. These applications demonstrate that modern, open-access, and broadly affordable local methods, such as LNO-CCSD(T), already enable predictive computations and atomistic insight for complicated, real-life molecular processes in realistic environments.

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.

Linear Scaling Incremental Scheme for Correlation Energies with Embedding Generated Virtuals

A novel incremental scheme is presented including an incremental expansion of the virtual space for the calculation of electron correlation energies, which is compatible with any size-extensive correlation method and scales asymptotically linear for large molecules. The performance is studied for organic molecules, water clusters, and a La(III)-water complex, where the compatibility with pseudopotentials is also examined. The computational requirements are already reduced tremendously for medium-sized water clusters and hydrocarbons with respect to the canonical CCSD as well as the ordinary incremental scheme references. Correlation energies within chemical accuracy have been observed for all studied systems. The novelty of the method is that relatively small virtual spaces are used in combination with tuples of localized occupied spaces. The corresponding orthonormal occupied and virtual orbitals are obtained from QM/QM embedding calculations and can thus be used with standard quantum chemistry codes for correlation calculations. It is presented how relevant virtual spaces are selected and the correlation energies are linked in the new virtual space expansion.

Development of analytic gradients for the Huzinaga quantum embedding method and its applications to large-scale hybrid and double hybrid DFT forces

The theory of analytic gradients is presented for the projector-based density functional theory (DFT) embedding approach utilizing the Huzinaga-equation. The advantages of the Huzinaga-equation-based formulation are demonstrated. In particular, it is shown that the projector employed does not appear in the Lagrangian, and the potential risk of numerical problems is avoided at the evaluation of the gradients. The efficient implementation of the analytic gradient theory is presented for approaches where hybrid DFT, second-order Møller-Plesset perturbation theory, or double hybrid DFT are embedded in lower-level DFT environments. To demonstrate the applicability of the method and to gain insight into its accuracy, it is applied to equilibrium geometry optimizations, transition state searches, and potential energy surface scans. Our results show that bond lengths and angles converge rapidly with the size of the embedded system. While providing structural parameters close to high-level quality for the embedded atoms, the embedding approach has the potential to relax the coordinates of the environment as well. Our demonstrations on a 171-atom zeolite and a 570-atom protein system show that the Huzinaga-equation-based embedding can accelerate (double) hybrid gradient computations by an order of magnitude with sufficient active regions and enables affordable force evaluations or geometry optimizations for molecules of hundreds of atoms.

Multiscale Frozen Density Embedding/Molecular Mechanics Approach for Simulating Magnetic Response Properties of Solvated Systems

We present a three-layer hybrid quantum mechanical/quantum embedding/molecular mechanics approach for calculating nuclear magnetic resonance (NMR) shieldings and J-couplings of molecular systems in solution. The model is based on the frozen density embedding (FDE) and polarizable fluctuating charges (FQ) and fluctuating dipoles (FQFμ) force fields and permits the accurate ab initio description of short-range nonelectrostatic interactions by means of the FDE shell and cost-effective treatment of long-range electrostatic interactions through the polarizable force field FQ(Fμ). Our approach's accuracy and potential are demonstrated by studying NMR spectra of Brooker's merocyanine in aqueous and nonaqueous solutions.

Introducing the embedded random phase approximation: H2 dissociative adsorption on Cu(111) as an exemplar

The random phase approximation (RPA) as a means of treating electron correlation recently has been shown to outperform standard density functional theory (DFT) approximations in a variety of cases. However, the computational cost of the RPA is substantially more than DFT, especially when aiming to study extended surfaces. Properly accounting for sufficient surface ensemble size, Brillouin zone sampling, and vacuum separation of periodic images in standard periodic-planewave-based DFT code raises the cost to achieve converged results. Here, we show that sub-system embedding schemes enable use of the RPA for modeling heterogeneous reactions at reduced computational cost. We explore two different embedded RPA (emb-RPA) approaches, periodic emb-RPA and cluster emb-RPA. We use the (experimentally and theoretically) well-studied H2 dissociative adsorption on Cu(111) as our exemplar, and first perform full periodic RPA calculations as a benchmark. The full RPA results match well the semi-empirical barrier fit to experimental observables and others derived from high-level computations, e.g., from recent embedded n-electron valence second order perturbation theory [Zhao et al., J. Chem. Theory Comput. 16(11), 7078-7088 (2020)] and quantum Monte Carlo [Doblhoff-Dier et al., J. Chem. Theory Comput. 13(7), 3208-3219 (2017)] simulations. Among the two emb-RPA approaches tested, the cluster emb-RPA accurately reproduces the energy profile (maximum error of 50 meV along the reaction pathway) while reducing the computational cost by approximately two orders of magnitude. We therefore expect that the embedded cluster approach will enable wider RPA implementation in heterogeneous catalysis.

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.

Static embedding with pair coupled cluster doubles based methods

Quantum embedding methods have recently been significantly developed to model large molecular structures. This work proposes a novel wave function theory in a density functional theory (WTF-in-DFT) embedding scheme based on pair-coupled cluster doubles (pCCD)-type methods. While pCCD can reliably describe strongly-correlated systems with mean-field-like computational cost, the large extent of the dynamic correlation can be accounted for by (linearized) coupled-cluster corrections on top of the pCCD wave function. Here we focus on the linearized coupled-cluster singles and doubles (LCCSD) ansatz for electronic ground states and its extension to excited states within the equation of motion (EOM) formalism. We test our EOM-pCCD-LCCSD-in-DFT approach for the vertical excitation energies of the hydrogen-bonded water-ammonia complex, micro-solvated thymine, and uranyl tetrahalides (UO2X42-, X = F, Cl, Br). Furthermore, we assess the quality of the embedding potential using an orbital entanglement and correlation analysis. The approximate embedding models successfully capture changes in the excitation energies going from bare fragments to supramolecular structures and represent a promising computational method for excited states in large molecular systems.

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.

Performance of Multilevel Methods for Excited States

The performance of multilevel quantum chemical approaches, which utilize an atom-based system partitioning scheme to model various electronic excited states, is studied. The considered techniques include the mechanical-embedding (ME) of "our own N-layered integrated molecular orbital and molecular mechanics" (ONIOM) method, the point charge embedding (PCE), the electronic-embedding (EE) of ONIOM, the frozen density-embedding (FDE), the projector-based embedding (PbE), and our local domain-based correlation method. For the investigated multilevel approaches, the second-order algebraic-diagrammatic construction [ADC(2)] approach was utilized as the high-level method, which was embedded in either Hartree-Fock or a density functional environment. The XH-27 test set of Zech et al. [ J. Chem. Theory Comput., 2018, 14, 4028] was used for the assessment, where organic dyes interact with several solvent molecules. With the selection of the chromophores as active subsystems, we conclude that the most reliable approach is local domain-based ADC(2) [L-ADC(2)], and the least robust schemes are ONIOM-ME and ONIOM-EE. The PbE, FDE, and PCE techniques often approach the accuracy of the L-ADC(2) scheme, but their precision is far behind. The results suggest that a more conservative subsystem selection algorithm or the inclusion of subsystem charge-transfers is required for the atom-based cost-efficient methods to produce high-accuracy excitation energies.

Challenges of modeling nanostructured materials for photocatalytic water splitting

Understanding the water splitting mechanism in photocatalysis is a rewarding goal as it will allow producing clean fuel for a sustainable life in the future. However, identifying the photocatalytic mechanisms by modeling photoactive nanoparticles requires sophisticated computational techniques based on multiscale modeling. In this review, we will survey the strengths and drawbacks of currently available theoretical methods at different length and accuracy scales. Understanding the surface-active site through Density Functional Theory (DFT) using new, more accurate exchange-correlation functionals plays a key role for surface engineering. Larger scale dynamics of the catalyst/electrolyte interface can be treated with Molecular Dynamics albeit there is a need for more generalizations of force fields. Monte Carlo and Continuum Modeling techniques are so far not the prominent path for modeling water splitting but interest is growing due to the lower computational cost and the feasibility to compare the modeling outcome directly to experimental data. The future challenges in modeling complex nano-photocatalysts involve combining different methods in a hierarchical way so that resources are spent wisely at each length scale, as well as accounting for excited states chemistry that is important for photocatalysis, a path that will bring devices closer to the theoretical limit of photocatalytic efficiency.

Full Configuration Interaction Quantum Monte Carlo treatment of fragments embedded in a periodic mean field

We present an embedded fragment approach for high-level quantum chemical calculations on local features in periodic systems. The fragment is defined as a set of localized orbitals (occupied and virtual) corresponding to a converged periodic Hartree-Fock solution. These orbitals serve as the basis for the in-fragment post-Hartree Fock treatment. The embedding field for the fragment, consisting of the Coulomb and exchange potential from the rest of the crystal, is included in the fragment's one-electron Hamiltonian. As an application of the embedded fragment approach we investigate the performanceof full configuration interaction quantum Monte Carlo (FCIQMC) with the adaptive shift. As the orbital choice we use the natural orbitals from the distinguishable cluster method with singles and doubles. FCIQMC is a stochastic approximation to the full CI method and can be routinely applied to much larger active spaces than the latter. This makes this method especially attractive in the context of open shell defects in crystals, where fragments of adequate size can be ratherlarge. As a test case we consider dissociation of a fluorine atom from a fluorographane surface. This process poses a challenge for high-level electronic structure models as both the static and dynamic correlations are essential here. Furthermore the active space for an adequate fragment (32 electrons in 173 orbitals) is already quite large even for FCIQMC. Despite this, FCIQMC delivers accurate dissociation and total energies.

Second-Order Active-Space Embedding Theory

Quantum embedding schemes are a promising way to extend multireference computations to large molecules with strong correlation effects localized on a small number of atoms. This work introduces a second-order active-space embedding theory [ASET(2)] which improves upon mean-field frozen embedding by treating fragment-environment interactions via an approximate canonical transformation. The canonical transformation employed in ASET(2) is formulated using the driven similarity renormalization group. The ASET(2) scheme is benchmarked on the N═N bond dissociation in pentyldiazene, the S₀ to S₁ excitation in 1-octene, and the interaction energy of the O₂-benzene complex. The ASET(2) explicit treatment of fragment-environment interactions beyond the mean-field level generally improves the accuracy of embedded computations, and it becomes necessary to achieve an accurate description of excitation energies of 1-octene and the singlet-triplet gap of the O₂-benzene complex.

Employing Pseudopotentials to Tackle Excited-State Electron Spill-Out in Frozen Density Embedding Calculations

In frozen density embedding (FDE), the properties of a target molecule are computed in the presence of an effective embedding potential, which accounts for the attractive and repulsive contributions of the environment. The formally exact embedding potential, however, is in practice calculated using explicit kinetic-energy functionals for which the resulting potentials are in many cases not repulsive enough to account fully for Pauli repulsion by the electrons of the environment and to compensate thereby the strong electron-nuclear attraction. For the excited states on the target molecule, this leads to charge spill-out when diffuse basis functions are included, which allow that valence electrons are excited to those regions of the environment where the strong nuclear attraction is not sufficiently compensated by repulsive contributions. To reduce this insufficiency, we propose in the present work the inclusion of atomic all-electron pseudopotentials for all environment atoms on top of the conventional embedding potential. In the current work, the pseudopotentials are applied for computing vertical excitation energies of local excited states in complex systems employing the second-order algebraic diagrammatic construction (ADC(2)) scheme. The proposed approach leads to significantly reduced charge spill-out and an improved agreement of FDE and supermolecular calculations in the frozen solvent approximation. In particular, when diffuse functions are employed, the mean absolute deviation (MAD) is reduced from 0.27 to 0.05 eV for the investigated cases.

Surface science using coupled cluster theory via local Wannier functions and in-RPA-embedding: The case of water on graphitic carbon nitride

A first-principles study of the adsorption of a single water molecule on a layer of graphitic carbon nitride is reported employing an embedding approach for many-electron correlation methods. To this end, a plane-wave based implementation to obtain intrinsic atomic orbitals and Wannier functions for arbitrary localization potentials is presented. In our embedding scheme, the localized occupied orbitals allow for a separate treatment of short-range and long-range correlation contributions to the adsorption energy by a fragmentation of the simulation cell. In combination with unoccupied natural orbitals, the coupled cluster ansatz with single, double, and perturbative triple particle-hole excitation operators is used to capture the correlation in local fragments centered around the adsorption process. For the long-range correlation, a seamless embedding into the random phase approximation yields rapidly convergent adsorption energies with respect to the local fragment size. Convergence of computed binding energies with respect to the virtual orbital basis set is achieved employing a number of recently developed techniques. Moreover, we discuss fragment size convergence for a range of approximate many-electron perturbation theories. The obtained benchmark results are compared to a number of density functional calculations.

Local-Field Effects in Linear Response Properties within a Polarizable Frozen Density Embedding Method

In this work, we present a polarizable frozen density embedding (FDE) method for calculating polarizabilities of coupled subsystems. The method (FDE-pol) combines a FDE method with an explicit polarization model such that the expensive freeze/thaw cycles can be bypassed, and approximate nonadditive kinetic potentials are avoided by enforcing external orthogonality between the subsystems. To describe the polarization of the frozen environment, we introduce a Hirshfeld partition-based density-dependent method for calculating the atomic polarizabilities of atoms in molecules, which alleviates the need to fit the atomic parameters to a specific system of interest or to a larger general set of molecules. We show that the Hirshfeld partition-based method predicts molecular polarizabilities close to the basis set limit, and thus, a single basis set-dependent scaling parameter can be introduced to improve the agreement against the reference polarizability data. To test the model, we characterized the uncoupled and coupled response of small interacting molecular complexes. Here, the coupled response properties include the perturbation of the frozen system due to the external perturbation which is ignored in the uncoupled response. We show that FDE-pol can accurately reproduce both the exact uncoupled polarizability and the coupled polarizabilities of the supermolecular systems. Using damped response theory, we also demonstrate that the coupled frequency-dependent polarizability can be described by including local field effects. The results emphasize the necessity of including local-field effects for describing the response properties of coupled subsystems, as well as the importance of accurate atomic polarizability models.

Projector-Based Quantum Embedding for Molecular Systems: An Investigation of Three Partitioning Approaches

Projector-based embedding is a relatively recent addition to the collection of methods that seek to utilize chemical locality to provide improved computational efficiency. This work considers the interactions between the different proposed procedures for this method and their effects on the accuracy of the results. The interplay between the embedded background, projector type, partitioning scheme, and level of atomic orbital (AO) truncation are investigated on a selection of reactions from the literature. The Huzinaga projection approach proves to be more reliable than the level-shift projection when paired with other procedural options. Active subsystem partitioning from the subsystem projected AO decomposition (SPADE) procedure proves slightly better than the combination of Pipek-Mezey localization and Mulliken population screening (PMM). Along with these two options, a new partitioning criteria is proposed based on subsystem von Neumann entropy and the related subsystem orbital occupancy. This new method overlaps with the previous PMM method, but the screening process is computationally simpler. Finally, AO truncation proves to be a robust option for the tested systems when paired with the Huzinaga projection, with satisfactory results being acquired at even the most severe truncation level.

Three-Layer Multiscale Approach Based on Extremely Localized Molecular Orbitals to Investigate Enzyme Reactions

Quantum mechanics/molecular mechanics (QM/MM) calculations are widely used embedding techniques to computationally investigate enzyme reactions. In most QM/MM computations, the quantum mechanical region is treated through density functional theory (DFT), which offers the best compromise between chemical accuracy and computational cost. Nevertheless, to obtain more accurate results, one should resort to wave function-based methods, which however lead to a much larger computational cost already for relatively small QM subsystems. To overcome this drawback, we propose the coupling of our QM/ELMO (quantum mechanics/extremely localized molecular orbital) approach with molecular mechanics, thus introducing the three-layer QM/ELMO/MM technique. The QM/ELMO strategy is an embedding method in which the chemically relevant part of the system is treated at the quantum mechanical level, while the rest is described through frozen ELMOs. Since the QM/ELMO method reproduces results of fully QM computations within chemical accuracy and with a much lower computational effort, it can be considered a suitable strategy to extend the range of applicability and accuracy of the QM/MM scheme. In this paper, other than briefly presenting the theoretical bases of the QM/ELMO/MM technique, we will also discuss its validation on the well-tested deprotonation of acetyl coenzyme A by aspartate in citrate synthase.

Natural Charge-Transfer Analysis: Eliminating Spurious Charge-Transfer States in Time-Dependent Density Functional Theory via Diabatization, with Application to Projection-Based Embedding

For many types of vertical excitation energies, linear-response time-dependent density functional theory (LR-TDDFT) offers a useful degree of accuracy combined with unrivaled computational efficiency, although charge-transfer excitation energies are often systematically and dramatically underestimated, especially for large systems and those that contain explicit solvent. As a result, low-energy electronic spectra of solution-phase chromophores often contain tens to hundreds of spurious charge-transfer states, making LR-TDDFT needlessly expensive in bulk solution. Intensity borrowing by these spurious states can affect intensities of the valence excitations, altering electronic bandshapes. At higher excitation energies, it is difficult to distinguish spurious charge-transfer states from genuine charge-transfer-to-solvent (CTTS) excitations. In this work, we introduce an automated diabatization that enables fast and effective screening of the CTTS acceptor space in bulk solution. Our procedure introduces "natural charge-transfer orbitals" that provide a means to isolate orbitals that are most likely to participate in a CTTS excitation. Projection of these orbitals onto solvent-centered virtual orbitals provides a criterion for defining the most important solvent molecules in a given excitation and be used as an automated subspace selection algorithm for projection-based embedding of a high-level description of the CTTS state in a lower-level description of its environment. We apply this method to an ab initio molecular dynamics trajectory of I^-(aq) and report the lowest-energy CTTS band in the absorption spectrum. Our results are in excellent agreement with the experiment, and only one-third of the water molecules in the I^-(H₂O)₉₆ simulation cell need to be described with LR-TDDFT to obtain excitation energies that are converged to <0.1 eV. The tools introduced herein will improve the accuracy, efficiency, and usability of LR-TDDFT in solution-phase environments.

Accurate Electronic Excitation Energies in Full-Valence Active Space via Bootstrap Embedding

Fragment embedding has been widely used to circumvent the high computational scaling of using accurate electron correlation methods to describe the electronic ground states of molecules and materials. However, similar applications that utilize fragment embedding to treat electronic excited states are comparably less reported in the literature. The challenge here is twofold. First, most fragment embedding methods are most effective when the property of interest is local, but the change of the wave function upon excitation is nonlocal in general. Second, even for local excitations, an accurate estimate of, for example, the excitation energy can still be challenging owing to the need for a balanced treatment of both the ground and the excited states. In this work, we show that bootstrap embedding (BE), a fragment embedding method developed recently by our group, is promising toward describing general electronic excitations. Numerical simulations show that the excitation energies in full-valence active space (FVAS) can be well-estimated by BE to an error of ∼0.05 eV using relatively small fragments, for both local excitations and the excitations of some large dye molecules that exhibit strong charge-transfer characters. We hence anticipate BE to be a promising solution to accurately describing the excited states of large chemical systems.

A polarizable three-layer frozen density embedding/molecular mechanics approach

We present a novel multilayer polarizable embedding approach in which the system is divided into three portions, two of which are treated using density functional theory and their interaction is based on frozen density embedding (FDE) theory, and both also mutually interact with a polarizable classical layer described using an atomistic model based on fluctuating charges (FQ). The efficacy of the model is demonstrated by extending the formalism to linear response properties and applying it to the simulation of the excitation energies of organic molecules in aqueous solution, where the solute and the first solvation shell are treated using FDE, while the rest of the solvent is modeled using FQ charges.

Multilevel Density Functional Theory

Following recent developments in multilevel embedding methods, we introduce a novel density matrix-based multilevel approach within the framework of density functional theory (DFT). In this multilevel DFT, the system is partitioned in an active and an inactive fragment, and all interactions are retained between the two parts. The decomposition of the total system is performed upon the density matrix. The orthogonality between the two parts is maintained by solving the Kohn–Sham equations in the MO basis for the active part only, while keeping the inactive density matrix frozen. This results in the reduction of computational cost. We outline the theory and implementation and discuss the differences and similarities with state-of-the-art DFT embedding methods. We present applications to aqueous solutions of methyloxirane and glycidol.

Energy-Based Molecular Orbital Localization in a Specific Spatial Region

We present a novel energy-based localization procedure able to localize molecular orbitals into predefined spatial regions. The method is defined in a multiscale framework based on the multilevel Hartree-Fock approach. In particular, the system is partitioned into active and inactive fragments. The localized molecular orbitals are obtained maximizing the repulsion between the two fragments. The method is applied to several cases including both conjugated and non-conjugated systems. Our multiscale approach is compared with reference values for both ground-state properties, such as dipole moments, and local excitation energies. The proposed approach is useful to extend the application range of high-level electron correlation methods. In fact, the reduced number of molecular orbitals can lead to a large reduction in the computational cost of correlated calculations.

Local embedding of coupled cluster theory into the random phase approximation using plane waves

We present an embedding approach to treat local electron correlation effects in periodic environments. In a single consistent framework, our plane wave based scheme embeds a local high-level correlation calculation [here, Coupled Cluster (CC) theory], employing localized orbitals, into a low-level correlation calculation [here, the direct Random Phase Approximation (RPA)]. This choice allows for an accurate and efficient treatment of long-range dispersion effects. Accelerated convergence with respect to the local fragment size can be observed if the low-level and high-level long-range dispersions are quantitatively similar, as is the case for CC in RPA. To demonstrate the capabilities of the introduced embedding approach, we calculate adsorption energies of molecules on a surface and in a chabazite crystal cage, as well as the formation energy of a lattice impurity in a solid at the level of highly accurate many-electron perturbation theories. The absorption energy of a methane molecule in a zeolite chabazite is converged with an error well below 20 meV at the CC level. As our largest periodic benchmark system, we apply our scheme to the adsorption of a water molecule on titania in a supercell containing more than 1000 electrons.

Combining multilevel Hartree–Fock and multilevel coupled cluster approaches with molecular mechanics: a study of electronic excitations in solutions

We present the coupling of different quantum-embedding approaches with a third molecular-mechanics layer, which can be either polarizable or non-polarizable.

Bootstrap Embedding For Large Molecular Systems

Recent developments in quantum embedding theories have provided attractive approaches to correlated calculations for large systems. In this work, we extend our previous work [J. Chem. Theory Comput. 2019, 15, 4497-4506; J. Phys. Chem. Lett. 2019, 10, 6368-6374] on bootstrap embedding (BE) to enable correlated ab initio calculations at the coupled cluster with singles and doubles (CCSD) level for large molecules. We introduce several new algorithmic developments that significantly reduce the computational cost of BE, while maintaining its accuracy. The resulting implementation scales as O(N³) for the integral transform and O(N) for the CCSD calculation. Numerical results on a series of conjugated molecules suggest that BE with reasonably sized fragments can recover more than 99.5% of the total correlation energy of a full CCSD calculation, while the required computational resources (time and storage) compare favorably to one popular local correlation scheme: domain localized pair natural orbital (DLPNO). The largest BE calculation in this work involves ∼2900 basis functions and can be performed on a single node with 16 CPU cores and 64 GB of memory in a few days. We anticipate that these developments represent an important step toward the application of BE to solve practical problems.

Localized Molecular Orbital-Based Embedding Scheme for Correlated Methods

Embedding strategies currently provide the best compromise between accuracy and computational cost in modeling chemical properties and processes of large and complex systems. In this framework, different methods have been proposed all over the years, from the very popular QM/MM approaches to the more recent and very promising density matrix and density functional embedding techniques. Here, we present a further development of the quantum mechanics/extremely localized molecular orbital technique (QM/ELMO) method, a recently proposed multiscale embedding strategy in which the chemically active region of the investigated system is treated at a fully quantum mechanical level, while the rest is described by frozen extremely localized molecular orbitals previously transferred from proper libraries or tailor-made model molecules. In particular, in this work we discuss and assess in detail the extension of the QM/ELMO approach to density functional theory and post-Hartree-Fock techniques by evaluating its performances when it is used to describe chemical reactions, bond dissociations, and intermolecular interactions. The preliminary test calculations have shown that, in the investigated cases, the new embedding strategy enables the results of the corresponding fully quantum mechanical computations to be reproduced within chemical accuracy in almost all the cases but with a significantly reduced computational cost, especially when correlated post-Hartree-Fock strategies are used to describe the quantum mechanical subsystem. In light of the obtained results, we already envisage the future application of the new correlated QM/ELMO techniques to the investigation of more challenging problems, such as the modeling of enzyme catalysis, the study of excited states of biomolecules, and the refinement of macromolecular X-ray crystal structures.

A zeroth-order active-space frozen-orbital embedding scheme for multireference calculations

Multireference computations of large-scale chemical systems are typically limited by the computational cost of quantum chemistry methods. In this work, we develop a zeroth-order active space embedding theory [ASET(0)], a simple and automatic approach for embedding any multireference dynamical correlation method based on a frozen-orbital treatment of the environment. ASET(0) is combined with the second-order multireference driven similarity renormalization group and tested on several benchmark problems, including the excitation energy of 1-octene and bond-breaking in ethane and pentyldiazene. Finally, we apply ASET(0) to study the singlet-triplet gap of p-benzyne and 9,10-anthracyne diradicals adsorbed on a NaCl surface. Our results show that despite its simplicity, ASET(0) is a powerful and sufficiently accurate embedding scheme applicable when the coupling between the fragment and the environment is in the weak to medium regime.

Density functional theory based embedding approaches for transition-metal complexes

Transition metal species are commonly discussed by considering the metal atom embedded in a ligand environment. This apparently makes them interesting targets for modern embedding strategies based on Kohn-Sham density functional theory (DFT), which aim at modelling accurate predictions for large systems by combining different quantum chemical methods. In this perspective, we will focus on subsystem density functional theory and projection-based embedding. We review the developments in the field for transition metal species, demonstrate benefits, drawbacks and analyse error sources of the different strategies using the example of chromium hexacarbonyle, before giving a perspective where the field is currently heading.

First-principles studies of strongly correlated states in defect spin qubits in diamond

Using a recently developed quantum embedding theory, we present first-principles calculations of strongly correlated states of spin defects in diamond. Using this theory, effective Hamiltonians are constructed, which can be solved by classical and quantum computers; the latter promise a much more favorable scaling as a function of system size than the former. In particular, we report a study on the neutral group-IV vacancy complexes in diamond, and we discuss their strongly correlated spin-singlet and spin-triplet excited states. Our results provide valuable predictions for experiments aimed at optical manipulation of these defects for quantum information technology applications.

Atom-Based Bootstrap Embedding For Molecules

Recent developments in quantum embedding have offered an attractive approach to describing electron correlation in molecules. However, previous methods such as density matrix embedding theory (DMET) require rigid partitioning of the system into fragments, which creates significant ambiguity for molecules. Bootstrap embedding (BE) is more flexible because it allows overlapping fragments, but when done on an orbital-by-orbital basis, BE introduces ambiguity in defining the connectivity of the orbitals. In this Letter, we present an atom-based fragment definition that significantly augments BE's performance in molecules. The resulting method, which we term atom-based BE, is very effective at recovering valence electron correlation in moderate-sized bases and delivers near-chemical-accuracy results using extrapolation. We anticipate atom-based BE may lead to a low-scaling and highly accurate approach to electron correlation in large molecules.

Prediction of many-electron wavefunctions using atomic potentials: extended basis sets and molecular dissociation

A one-electron Schrödinger equation based on special one-electron potentials for atoms is shown to exist that produces orbitals for an arbitrary molecule that are sufficiently accurate to be used without modification to construct single- and multi-determinant wavefunctions. The exact Hamiltonian is used to calculate the energy variationally and to generate configuration interaction expansions. Earlier work on equilibrium geometries is extended to larger basis sets and molecular dissociation. For a test set of molecules representing different bonding environments, a single set of invariant atomic potentials gives wavefunctions with energies that deviate from configuration interaction energies based on SCF orbitals by less than 0.04 eV per bond or valence electron pair. On a single diagonalization of the Fock matrix, the corresponding errors are reduced 0.01 eV. Atomization energies are also in good agreement with CI values based on canonical SCF orbitals. Configuration interaction applications to single bond dissociations of water and glycine, and multiple bond dissociations of ethylene and oxygen produce dissociation energy curves in close agreement with CI calculations based on canonical SCF orbitals for the entire range of internuclear distances.