Self-consistent embedding theory for locally correlated configuration interaction wave functions in condensed matter

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.

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.

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.

Adsorption energies on transition metal surfaces: towards an accurate and balanced description

Density functional theory predictions of binding energies and reaction barriers provide invaluable data for analyzing chemical transformations in heterogeneous catalysis. For high accuracy, effects of band structure and coverage, as well as the local bond strength in both covalent and non-covalent interactions, must be reliably described and much focus has been put on improving functionals to this end. Here, we show that a correction from higher-level calculations on small metal clusters can be applied to improve periodic band structure adsorption energies and barriers. We benchmark against 38 reliable experimental covalent and non-covalent adsorption energies and five activation barriers with mean absolute errors of 2.2 kcal mol^-1, 2.7 kcal mol^-1, and 1.1 kcal mol^-1, respectively, which are lower than for functionals widely used and tested for surface science evaluations, such as BEEF-vdW and RPBE.

Frozen-Density Embedding for Including Environmental Effects in the Dirac-Kohn-Sham Theory: An Implementation Based on Density Fitting and Prototyping Techniques

Frozen density embedding (FDE) represents an embedding scheme in which environmental effects are included from first-principles calculations by considering the surrounding system explicitly by means of its electron density. In the present paper, we extend the full four-component relativistic Dirac-Kohn-Sham (DKS) method, as implemented in the BERTHA code, to include environmental and confinement effects with the FDE scheme (DKS-in-DFT FDE). The implementation, based on the auxiliary density fitting techniques, has been enormously facilitated by BERTHA's python API (PyBERTHA), which facilitates the interoperability with other FDE implementations available through the PyADF framework. The accuracy and numerical stability of this new implementation, also using different auxiliary fitting basis sets, has been demonstrated on the simple NH₃-H₂O system, in comparison with a reference nonrelativistic implementation. The computational performance has been evaluated on a series of gold clusters (Au_n, with n = 2, 4, 8) embedded into an increasing number of water molecules (5, 10, 20, 40, and 80 water molecules). We found that the procedure scales approximately linearly both with the size of the frozen surrounding environment (consistent with the underpinnings of the FDE approach) and with the size of the active system (in line with the use of density fitting). Finally, we applied the code to a series of heavy (Rn) and super-heavy elements (Cn, Fl, Og) embedded in a C₆₀ cage to explore the confinement effect induced by C₆₀ on their electronic structure. We compare the results from our simulations, with respect to more-approximate models employed in the atomic physics literature. Our results indicate that the specific interactions described by FDE are able to improve upon the cruder approximations currently employed, and, thus, they provide a basis from which to generate more-realistic radial potentials for confined atoms.

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.

First-principles calculations of hybrid inorganic-organic interfaces: from state-of-the-art to best practice

The computational characterization of inorganic-organic hybrid interfaces is arguably one of the technically most challenging applications of density functional theory. Due to the fundamentally different electronic properties of the inorganic and the organic components of a hybrid interface, the proper choice of the electronic structure method, of the algorithms to solve these methods, and of the parameters that enter these algorithms is highly non-trivial. In fact, computational choices that work well for one of the components often perform poorly for the other. As a consequence, default settings for one materials class are typically inadequate for the hybrid system, which makes calculations employing such settings inefficient and sometimes even prone to erroneous results. To address this issue, we discuss how to choose appropriate atomistic representations for the system under investigation, we highlight the role of the exchange-correlation functional and the van der Waals correction employed in the calculation and we provide tips and tricks how to efficiently converge the self-consistent field cycle and to obtain accurate geometries. We particularly focus on potentially unexpected pitfalls and the errors they incur. As a summary, we provide a list of best practice rules for interface simulations that should especially serve as a useful starting point for less experienced users and newcomers to the field.

Combining wavefunction frozen-density embedding with one-dimensional periodicity

We present the combination of wavefunction frozen-density embedding (FDE) with a periodic repetition in one dimension (1D) for molecular systems in the KOALA program. In this periodic orbital-uncoupled FDE ansatz, no wavefunction overlap is taken into account, and only the electron density of the active subsystem is computed explicitly. This density is relaxed in the presence of the environment potential, which is obtained by translating the updated active subsystem density, yielding a fully self-consistent solution at convergence. Treating only one subsystem explicitly, the method allows for the calculation of local properties in condensed molecular systems, while no orbital band structure is obtained preventing the application, e.g., to systems with metallic bonding. In order to illustrate possible applications of the new implementation, selected case studies are presented, ranging from ground-state dipole moments using configuration interaction methods via excitation energies using time-dependent density-functional theory to ionization potentials obtained from equation-of-motion correlation methods. Different levels of approximations are assessed, revealing that an active subsystem consisting of two or three molecules leads to results that are converged with respect to the environment contributions.

A Fock-operator complete active space self-consistent field (CAS-SCF) method combined with frozen-density embedding

We report the implementation of a Fock-operator complete-active space self-consistent field (CAS-SCF) method combined with frozen-density embedding (FDE) into the KOALA quantum-chemistry program. The implementation is based on configuration interaction from an unrestricted reference determinant and is able to treat electronic configurations such as singlet, triplet, or quintet states embedded in a molecular environment. In order to account for possible spin polarization effects, the FDE contribution is extended to the unrestricted case. We assess the convergence obtained with the implementation at the example of a stretched lithium dimer with significant multi-reference character. The efficiency of the implementation enables the orbital optimization for 25 states in a state-average SA[S₀-S₁₀,T₁-T₁₂,Q₁-Q₂]-CAS(10,10)-SCF calculation for the retinal molecule using a def2-TZVP basis. The FDE ansatz leads to orbitals localized by definition on the target system, thus facilitating the orbital selection required for CAS methods in complex environments.

Embedding Methods for Quantum Chemistry: Applications from Materials to Life Sciences

Quantum mechanical embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and improve scaling for macro-molecular systems containing hundreds if not thousands of atoms. The field of embedding has grown increasingly broad with many approaches of different intersecting flavors. In this perspective, we lay out the methods into two streams: QM:MM and QM:QM, showcasing the advantages and disadvantages of both. We provide a review of the literature, the underpinning theories including our contributions, and we highlight current applications with select examples spanning both materials and life sciences. We conclude with prospects and future outlook on embedding, and our view on the use of universal test case scenarios for cross-comparisons of the many available (and future) embedding theories.

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.

Electronic Communication as a Transferable Property of Molecular Bridges?

Electronic communication through molecular bridges is important for different types of experiments, such as single-molecule conductance, electron transfer, superexchange spin coupling, and intramolecular singlet fission. In many instances, the chemical structure of the bridge determines how the two parts it is connecting communicate, and does so in ways that are transferable between these different manifestations (for example, high conductance often correlates with strong antiferromagnetic spin coupling, and low conductance due to destructive quantum interference correlates with ferromagnetic coupling). Defining electronic communication as a transferable property of the bridge can help transfer knowledge between these different areas of research. Examples and limits of such transferability are discussed here, along with some possible directions for future research, such as employing spin-coupled and mixed-valence systems as structurally well-controlled proxies for understanding molecular conductance and for validating first-principles theoretical methodologies, building conceptual understanding for the growing experimental work on intramolecular singlet fission, and developing measures for the transferability of electronic communication as a bridge property.

Computational Approach to Molecular Catalysis by 3d Transition Metals: Challenges and Opportunities

Computational chemistry provides a versatile toolbox for studying mechanistic details of catalytic reactions and holds promise to deliver practical strategies to enable the rational in silico catalyst design. The versatile reactivity and nontrivial electronic structure effects, common for systems based on 3d transition metals, introduce additional complexity that may represent a particular challenge to the standard computational strategies. In this review, we discuss the challenges and capabilities of modern electronic structure methods for studying the reaction mechanisms promoted by 3d transition metal molecular catalysts. Particular focus will be placed on the ways of addressing the multiconfigurational problem in electronic structure calculations and the role of expert bias in the practical utilization of the available methods. The development of density functionals designed to address transition metals is also discussed. Special emphasis is placed on the methods that account for solvation effects and the multicomponent nature of practical catalytic systems. This is followed by an overview of recent computational studies addressing the mechanistic complexity of catalytic processes by molecular catalysts based on 3d metals. Cases that involve noninnocent ligands, multicomponent reaction systems, metal-ligand and metal-metal cooperativity, as well as modeling complex catalytic systems such as metal-organic frameworks are presented. Conventionally, computational studies on catalytic mechanisms are heavily dependent on the chemical intuition and expert input of the researcher. Recent developments in advanced automated methods for reaction path analysis hold promise for eliminating such human-bias from computational catalysis studies. A brief overview of these approaches is presented in the final section of the review. The paper is closed with general concluding remarks.

Photophysical properties and the NO photorelease mechanism of a ruthenium nitrosyl model complex investigated using the CASSCF-in-DFT embedding approach

A complete state-averaged active space self-consistent field (SA-CASSCF) calculation by means of the SA-CASSCF(18,14)-in-BP86 Miller-Manby embedding approach was performed to explore the ground and excited electronic states of the trans-[RuCl(NO)(NH₃)₄]²⁺ complex. Insights into the NO photodissociation mechanism and Ru-NO bonding properties are provided. In addition, spin-orbit (SO) interactions were taken into account to describe and characterize the spin-forbidden transitions observed at the low-energy regions of the trans-[RuCl(NO)(NH₃)₄]²⁺ UV-Vis spectrum. The SA-CASSCF(18,14)-in-BP86 electronic spectrum is in great agreement with the experimental data of Schreiner [Schreiner et al., Inorg. Chem., 1972, 11, 880].

Explicit vs. implicit electronic polarisation of environment of an embedded chromophore in frozen-density embedding theory

A comparison of strategies to account for environment polarisation in Frozen Density Embedding Theory (FDET).

Analytical nuclear excited-state gradients for the Tamm-Dancoff approximation using uncoupled frozen-density embedding

We report the derivation and implementation of analytical nuclear gradients for excited states using time-dependent density functional theory using the Tamm-Dancoff approximation combined with uncoupled frozen-density embedding using density fitting. Explicit equations are presented and discussed. The implementation is able to treat singlet as well as triplet states and functionals using the local density approximation, the generalized gradient approximation, combinations with Hartree-Fock exchange (hybrids), and range-separated functionals such as CAM-B3LYP. The new method is benchmarked against supermolecule calculations in two case studies: The solvatochromic shift of the (vertical) fluorescence energy of 4-aminophthalimide on solvation, and the first local excitation of the benzonitrile dimer. Whereas for the 4-aminophthalimide-water complex deviations of about 0.2 eV are obtained to supermolecular calculations, for the benzonitrile dimer the maximum error for adiabatic excitation energies is below 0.01 eV due to a weak coupling of the subsystems. © 2017 Wiley Periodicals, Inc.

Reduced density matrix embedding. General formalism and inter-domain correlation functional

An embedding method for a one-electron reduced density matrix (1-RDM) is proposed.

Frozen density embedding with non-integer subsystems' particle numbers

We extend the frozen density embedding theory to non-integer subsystems' particles numbers. Different features of this formulation are discussed, with special concern for approximate embedding calculations. In particular, we highlight the relation between the non-integer particle-number partition scheme and the resulting embedding errors. Finally, we provide a discussion of the implications of the present theory for the derivative discontinuity issue and the calculation of chemical reactivity descriptors.

Embedded correlated wavefunction schemes: theory and applications

Conspectus Ab initio modeling of matter has become a pillar of chemical research: with ever-increasing computational power, simulations can be used to accurately predict, for example, chemical reaction rates, electronic and mechanical properties of materials, and dynamical properties of liquids. Many competing quantum mechanical methods have been developed over the years that vary in computational cost, accuracy, and scalability: density functional theory (DFT), the workhorse of solid-state electronic structure calculations, features a good compromise between accuracy and speed. However, approximate exchange-correlation functionals limit DFT's ability to treat certain phenomena or states of matter, such as charge-transfer processes or strongly correlated materials. Furthermore, conventional DFT is purely a ground-state theory: electronic excitations are beyond its scope. Excitations in molecules are routinely calculated using time-dependent DFT linear response; however applications to condensed matter are still limited. By contrast, many-electron wavefunction methods aim for a very accurate treatment of electronic exchange and correlation. Unfortunately, the associated computational cost renders treatment of more than a handful of heavy atoms challenging. On the other side of the accuracy spectrum, parametrized approaches like tight-binding can treat millions of atoms. In view of the different (dis-)advantages of each method, the simulation of complex systems seems to force a compromise: one is limited to the most accurate method that can still handle the problem size. For many interesting problems, however, compromise proves insufficient. A possible solution is to break up the system into manageable subsystems that may be treated by different computational methods. The interaction between subsystems may be handled by an embedding formalism. In this Account, we review embedded correlated wavefunction (CW) approaches and some applications. We first discuss our density functional embedding theory, which is formally exact. We show how to determine the embedding potential, which replaces the interaction between subsystems, at the DFT level. CW calculations are performed using a fixed embedding potential, that is, a non-self-consistent embedding scheme. We demonstrate this embedding theory for two challenging electron transfer phenomena: (1) initial oxidation of an aluminum surface and (2) hot-electron-mediated dissociation of hydrogen molecules on a gold surface. In both cases, the interaction between gas molecules and metal surfaces were treated by sophisticated CW techniques, with the remainder of the extended metal surface being treated by DFT. Our embedding approach overcomes the limitations of conventional Kohn-Sham DFT in describing charge transfer, multiconfigurational character, and excited states. From these embedding simulations, we gained important insights into fundamental processes that are crucial aspects of fuel cell catalysis (i.e., O2 reduction at metal surfaces) and plasmon-mediated photocatalysis by metal nanoparticles. Moreover, our findings agree very well with experimental observations, while offering new views into the chemistry. We finally discuss our recently formulated potential-functional embedding theory that provides a seamless, first-principles way to include back-action onto the environment from the embedded region.

Accurate basis set truncation for wavefunction embedding

Density functional theory (DFT) provides a formally exact framework for performing embedded subsystem electronic structure calculations, including DFT-in-DFT and wavefunction theory-in-DFT descriptions. In the interest of efficiency, it is desirable to truncate the atomic orbital basis set in which the subsystem calculation is performed, thus avoiding high-order scaling with respect to the size of the MO virtual space. In this study, we extend a recently introduced projection-based embedding method [F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput. 8, 2564 (2012)] to allow for the systematic and accurate truncation of the embedded subsystem basis set. The approach is applied to both covalently and non-covalently bound test cases, including water clusters and polypeptide chains, and it is demonstrated that errors associated with basis set truncation are controllable to well within chemical accuracy. Furthermore, we show that this approach allows for switching between accurate projection-based embedding and DFT embedding with approximate kinetic energy (KE) functionals; in this sense, the approach provides a means of systematically improving upon the use of approximate KE functionals in DFT embedding.