Excited States of Crystalline Point Defects with Multireference Density Matrix Embedding Theory

The Localized Active Space Method with Unitary Selective Coupled Cluster

We introduce a hybrid quantum-classical algorithm, the localized active space unitary selective coupled cluster singles and doubles (LAS-USCCSD) method. Derived from the localized active space unitary coupled cluster (LAS-UCCSD) method, LAS-USCCSD first performs a classical LASSCF calculation, then selectively identifies the most important parameters (cluster amplitudes used to build the multireference UCC ansatz) for restoring interfragment interaction energy using this reduced set of parameters with the variational quantum eigensolver method. We benchmark LAS-USCCSD against LAS-UCCSD by calculating the total energies of (H2)2, (H2)4, and trans-butadiene, and the magnetic coupling constant for a bimetallic compound [Cr2(OH)3(NH3)6]3+. For these systems, we find that LAS-USCCSD reduces the number of required parameters and thus the circuit depth by at least 1 order of magnitude, an aspect which is important for the practical implementation of multireference hybrid quantum-classical algorithms like LAS-UCCSD on near-term quantum computers.

Multiconfigurational Correlation at DFT + U Cost: On-Site Electron-Electron Interactions Yield a Block-Localized Configuration Interaction Hamiltonian

This work presents a first-principles wavefunction-in-DFT approach based on the Hubbard density functional theory (DFT) + U method. This approach begins with the standard DFT reference system of noninteracting electrons and introduces an electron-electron interaction projected onto DFT+U-type atomic states. The reference system's configuration interaction Hamiltonian is block-localized to these states and can be expressed in terms of state occupation numbers, state self-energies (which correspond to unscreened Hubbard U values), and the promotion energies of doubly excited Slater determinants. Simple approximations for the promotion energies provide multiconfigurational correlation energies without requiring explicit orbital localization/transform. Numerical results for fractionally occupied chromium atom, bonded chromium dimer, dissociating covalent bonds, and large active spaces show that the approach provides beyond-zero-sum accuracy at computational cost comparable to standard DFT+U.

Core Binding Energy Calculations: A Scalable Approach with the Quantum Embedding-Based Equation-of-Motion Coupled-Cluster Method

We investigated the use of density matrix embedding theory to facilitate the computation of core ionization energies (IPs) of large molecules at the equation-of-motion coupled-cluster singles doubles with perturbative triples (EOM-CCSD*) level in combination with the core-valence separation (CVS) approximation. The unembedded IP-CVS-EOM-CCSD* method with a triple-ζ basis set produced ionization energies within 1 eV of experiment with a standard deviation of ∼0.2 eV for the core65 data set. The embedded variant contributed very little systematic error relative to the unembedded method, with a mean unsigned error of 0.07 eV and a standard deviation of ∼0.1 eV, in exchange for accelerating the calculations by many orders of magnitude. By employing embedded EOM-CC methods, we computed the core ionization energies of the uracil hexamer, doped fullerene, and chlorophyll molecule, utilizing up to ∼4000 basis functions within 1 eV from experimental values. Such calculations are not currently possible with the unembedded EOM-CC method.

Accurate Excitation Energies of Point Defects from Fast Particle-Particle Random Phase Approximation Calculations

We present an efficient particle-particle random phase approximation (ppRPA) approach that predicts accurate excitation energies of point defects, including the nitrogen-vacancy (NV-) and silicon-vacancy (SiV0) centers in diamond and the divacancy center (VV0) in 4H silicon carbide, with errors of ±0.2 eV compared with experimental values. Starting from the (N + 2)-electron ground state calculated with density functional theory (DFT), the ppRPA excitation energies of the N-electron system are calculated as the differences between the two-electron removal energies of the (N + 2)-electron system. We demonstrate that the ppRPA excitation energies converge rapidly with a few hundred canonical active-space orbitals. We also show that active-space ppRPA has weak DFT starting-point dependence and is significantly cheaper than the corresponding ground-state DFT calculation. This work establishes ppRPA as an accurate and low-cost tool for investigating excited-state properties of point defects and opens up new opportunities for applications of ppRPA to periodic bulk materials.

On the Structure of Oxygen Deficient Amorphous Oxide Films

Understanding defects in amorphous oxide films and heterostructures is vital to improving performance of microelectronic devices, thin-film transistors, and electrocatalysis. However, to what extent the structure and properties of point defects in amorphous solids are similar to those in the crystalline phase are still debated. The validity of this analogy and the experimental and theoretical evidence of the effects of oxygen deficiency in amorphous oxide films are critically discussed. The authors start with the meaning and significance of defect models, such as "oxygen vacancy" in crystalline oxides, and then introduce experimental and computational methods used to study intrinsic defects in amorphous oxides and discuss their limitations and challenges. To test the validity of existing defect models, ab initio molecular dynamics is used with a non-local density functional to model the structure and electronic properties of oxygen-deficient amorphous alumina. Unlike some previous studies, the formation of deep defect states in the bandgap caused by the oxygen deficiency is found. Apart from atomistic structures analogous to crystal vacancies, the formation of more stable defect states characterized by the bond formation between under-coordinated Al ions is shown. The limitations of such defect models and how they may be overcome in simulations are discussed.

Excited State Properties of Point Defects in Semiconductors and Insulators Investigated with Time-Dependent Density Functional Theory

We present a formulation of spin-conserving and spin-flip hybrid time-dependent density functional theory (TDDFT), including the calculation of analytical forces, which allows for efficient calculations of excited state properties of solid-state systems with hundreds to thousands of atoms. We discuss an implementation on both GPU- and CPU-based architectures along with several acceleration techniques. We then apply our formulation to the study of several point defects in semiconductors and insulators, specifically the negatively charged nitrogen-vacancy and neutral silicon-vacancy centers in diamond, the neutral divacancy center in 4H silicon carbide, and the neutral oxygen-vacancy center in magnesium oxide. Our results highlight the importance of taking into account structural relaxations in excited states in order to interpret and predict optical absorption and emission mechanisms in spin defects.

Lessons Learned from Catalysis to Qubits: General Strategies to Build Accessible and Accurate First-Principles Models of Point Defects

Defects and dopants play critical roles in defining the properties of a material. Achieving a mechanistic understanding of how such properties arise is challenging with current experimental methods, and computational approaches suffer from significant modeling limitations that frequently require a posteriori fitting. Consequently, the pace of dopant discovery as a means of tuning material properties for a particular application has been slow. However, recent advances in computation have enabled researchers to move away from semiempirical schemes to reposition density functional theory as a predictive tool and improve the accessibility of highly accurate first-principles methods to all researchers. This Perspective discusses some of these recent achievements that provide more accurate first-principles geometric, thermodynamic, optical, and electronic properties simultaneously. Advancements related to supercells, basis sets, functionals, and optimization protocols, as well as suggestions for evaluating the quality of a computational model through comparison to experimental data, are discussed. Moreover, recent computational results in the fields of energy materials, heterogeneous catalysis, and quantum informatics are reviewed along with an evaluation of current frontiers and opportunities in the field of computational materials chemistry.

Optical Properties of Neutral F Centers in Bulk MgO with Density Matrix Embedding

The optical spectra of neutral oxygen vacancies (F0 centers) in the bulk MgO lattice are investigated using density matrix embedding theory. The impurity Hamiltonian is solved with the complete active space self-consistent field and second-order n-electron valence state perturbation theory (NEVPT2-DMET) multireference methods. To estimate defect-localized vertical excitation energies at the nonembedding and thermodynamic limits, a double extrapolation scheme is employed. The extrapolated NEVPT2-DMET vertical excitation energy value of 5.24 eV agrees well with the experimental absorption maxima at 5.03 eV, whereas the excitation energy value of 2.89 eV at the relaxed triplet defect-localized state geometry overestimates the experimental emission at 2.4 eV by only nearly 0.5 eV, indicating the involvement of the triplet-singlet decay pathway.

Density Matrix Embedding Using Multiconfiguration Pair-Density Functional Theory

We present a quantum embedding method for ground and excited states of extended systems that uses multiconfiguration pair-density functional theory (MC-PDFT) with densities provided by periodic density matrix embedding theory (pDMET). We compute local excitations in oxygen mono- and divacancies on a magnesium oxide (100) surface and find absolute deviations within 0.05 eV between pDMET using the MC-PDFT, denoted as pDME-PDFT, and the more expensive, nonembedded MC-PDFT approach. We further use pDME-PDFT to calculate local excitations in larger supercells for the monovacancy defect, for which the use of nonembedded MC-PDFT is prohibitively costly.

Periodic Bootstrap Embedding

Bootstrap embedding (BE) is a recently developed electronic structure method that has shown great success at treating electron correlation in molecules. Here, we extend BE to treat surfaces and solids where the wave function is represented in periodic boundary conditions using reciprocal space sums (i.e., k-point sampling). The major benefit of this approach is that the resulting fragment Hamiltonians carry no explicit dependence on the reciprocal space sums, allowing one to apply traditional nonperiodic electronic structure codes to the fragments even though the entire system requires careful consideration of periodic boundary conditions. Using coupled cluster singles and doubles (CCSD) as an example method to solve the fragment Hamiltonians, we present minimal basis set CCSD-in-HF results on 1D conducting polymers. We show that periodic BE-CCSD can typically recover ∼99.9% of the electron correlation energy. We further demonstrate that periodic BE-CCSD is feasible even for complex donor-acceptor polymers of interest to organic solar cells─despite the fact that the monomers are sufficiently large that even a Γ-point periodic CCSD calculation is prohibitive. We conclude that BE is a promising new tool for applying molecular electronic structure tools to solids and interfaces.

Local Excitations of a Charged Nitrogen Vacancy in Diamond with Multireference Density Matrix Embedding Theory

We investigate the negatively charged nitrogen-vacancy center in diamond using periodic density matrix embedding theory (pDMET). To describe the strongly correlated excited states of this system, the complete active space self-consistent field (CASSCF) followed by n-electron valence state second-order perturbation theory (NEVPT2) was used as the impurity solver. Since the NEVPT2-DMET energies show a linear dependence on the inverse of the size of the embedding subspace, we performed an extrapolation of the excitation energies to the nonembedding limit using a linear regression. The extrapolated NEVPT2-DMET first triplet-triplet excitation energy is 2.31 eV and that for the optically inactive singlet-singlet transition is 1.02 eV, both in agreement with the experimentally observed vertical excitation energies of ∼2.18 eV and ∼1.26 eV, respectively. This is the first application of pDMET to a charged periodic system and the first investigation of the NV^- defect using NEVPT2 for periodic supercell models.

Toward a Rigorous Theoretical Description of Photocatalysis Using Realistic Models

This Perspective aims at providing a road map to computational heterogeneous photocatalysis highlighting the knowledge needed to boost the design of efficient photocatalysts. A plausible computational framework is suggested focusing on static and dynamic properties of the relevant excited states as well of the involved chemistry for the reactions of interest. This road map calls for explicitly exploring the nature of the charge carriers, the excited-state potential energy surface, and its time evolution. Excited-state descriptors are introduced to locate and characterize the electrons and holes generated upon excitation. Nonadiabatic molecular dynamics simulations are proposed as a convenient tool to describe the time evolution of the photogenerated species and their propagation through the crystalline structure of photoactive material, ultimately providing information about the charge carrier lifetime. Finally, it is claimed that a detailed understanding of the mechanisms of heterogeneously photocatalyzed reactions demands the analysis of the excited-state potential energy surface.

A multi-fragment real-time extension of projected density matrix embedding theory: Non-equilibrium electron dynamics in extended systems

In this work, we derive a multi-fragment real-time extension of the projected density matrix embedding theory (pDMET) designed to treat non-equilibrium electron dynamics in strongly correlated systems. As in the previously developed static pDMET, the real time pDMET partitions the total system into many fragments; the coupling between each fragment and the rest of the system is treated through a compact representation of the environment in terms of a quantum bath. The real-time pDMET involves simultaneously propagating the wavefunctions for each separate fragment–bath embedding system along with an auxiliary mean-field wavefunction of the total system. The equations of motion are derived by (i) projecting the time-dependent Schrödinger equation in the fragment and bath space associated with each separate fragment and by (ii) enforcing the pDMET matching conditions between the global 1-particle reduced density matrix (1-RDM) obtained from the fragment calculations and the mean-field 1-RDM at all points in time. The accuracy of the method is benchmarked through comparisons to time-dependent density-matrix renormalization group and time-dependent Hartree–Fock (TDHF) theory; the methods were applied to a one- and two-dimensional single-impurity Anderson model and multi-impurity Anderson models with ordered and disordered distributions of the impurities. The results demonstrate a large improvement over TDHF and rapid convergence to the exact dynamics with an increase in fragment size. Our results demonstrate that the real-time pDMET is a promising and flexible method that balances accuracy and efficiency to simulate the non-equilibrium electron dynamics in heterogeneous systems of large size.

Bootstrap Embedding on a Quantum Computer

We extend molecular bootstrap embedding to make it appropriate for implementation on a quantum computer. This enables solution of the electronic structure problem of a large molecule as an optimization problem for a composite Lagrangian governing fragments of the total system, in such a way that fragment solutions can harness the capabilities of quantum computers. By employing state-of-art quantum subroutines including the quantum SWAP test and quantum amplitude amplification, we show how a quadratic speedup can be obtained over the classical algorithm, in principle. Utilization of quantum computation also allows the algorithm to match─at little additional computational cost─full density matrices at fragment boundaries, instead of being limited to 1-RDMs. Current quantum computers are small, but quantum bootstrap embedding provides a potentially generalizable strategy for harnessing such small machines through quantum fragment matching.

DFT-Based Multireference Diagnostics in the Solid State: Application to Metal-Organic Frameworks

When a many-body wave function of a system cannot be captured by a single determinant, high-level multireference (MR) methods are required to properly explain its electronic structure. MR diagnostics to estimate the magnitude of such static correlation have been primarily developed for molecular systems and range from low in computational cost to as costly as the full MR calculation itself. We report the first application of low-cost MR diagnostics based on the fractional occupation number calculated with finite-temperature DFT to solid-state systems. To compare the behavior of the diagnostics on solids and molecules, we select metal-organic frameworks (MOFs) as model materials because their reticular nature provides an intuitive way to identify molecular derivatives. On a series of closed-shell MOFs, we demonstrate that the DFT-based MR diagnostics are equally applicable to solids as to their molecular derivatives. The magnitude and spatial distribution of the MR character of a MOF are found to have a good correlation with those of its molecular derivatives, which can be calculated much more affordably in comparison to those of the full MOF. The additivity of MR character discussed here suggests the set of molecular derivatives to be a good representation of a MOF for both MR detection and ultimately for MR corrections, facilitating accurate and efficient high-throughput screening of MOFs and other porous solids.

Efficient Multiconfigurational Quantum Chemistry Approach to Single-Ion Magnets Based on Density Matrix Embedding Theory

Density matrix embedding theory (DMET) provides a systematic framework to combine low-level (e.g., Hartree-Fock approximation) and high-level correlated quantum chemistry methods to treat strongly correlated systems. In this work, we propose an efficient quantum embedding approach that combines DMET with the complete active space self-consistent field and subsequent state interaction treatment of spin-orbit coupling (CASSI-SO) and apply it to a theoretical description of single-ion magnets (SIMs). We have developed a novel regularized direct inversion of iterative subspace (R-DIIS) technique that ensures restricted open-shell Hartree-Fock converging to a physically correct ground state, which is found to be crucial for the efficacy of subsequent CASSI-SO calculation. The DMET+CASSI-SO approach can produce reliable zero-field splitting parameters in typical 3d-SIMs with dramatically reduced computational cost compared to its all-electron counterpart. This work therefore demonstrates the great potential of DMET-based multiconfigurational approaches for efficient ab initio study of magneto-structural correlations in complex molecular magnetic systems.

Periodic Density Matrix Embedding for CO Adsorption on the MgO(001) Surface

The adsorption of simple gas molecules to metal oxide surfaces is a primary step in many heterogeneous catalysis applications. Quantum chemical modeling of these reactions is a challenge in terms of both cost and accuracy, and quantum-embedding methods are promising, especially for localized chemical phenomena. In this work, we employ density matrix embedding theory (DMET) for periodic systems to calculate the adsorption energy of CO to the MgO(001) surface. Using coupled-cluster theory with single and double excitations and second-order Møller-Plesset perturbation theory as quantum chemical solvers, we perform calculations with embedding clusters up to 266 electrons in 306 orbitals, with the largest embedding models agreeing to within 1.2 kcal/mol of the non-embedding references. Moreover, we present a memory-efficient procedure of storing and manipulating electron repulsion integrals in the embedding space within the framework of periodic DMET.

Green's Function Formulation of Quantum Defect Embedding Theory

We present a Green's function formulation of the quantum defect embedding theory (QDET) where a double counting scheme is rigorously derived within the G₀W₀ approximation. We then show the robustness of our methodology by applying the theory with the newly derived scheme to several defects in diamond. Additionally, we discuss a strategy to obtain converged results as a function of the size and composition of the active space. Our results show that QDET is a promising approach to investigate strongly correlated states of defects in solids.

Defect engineering of oxide surfaces: dream or reality?

In this brief perspective we analyze the present status of the field of defect engineering of oxide surfaces. In particular we discuss the tools and techniques available to generate, identify, quantify, and characterize point defects at oxide surfaces and the main areas where these centers play a role in practical applications.