Ionization potentials of solids: the importance of vertex corrections

Band engineering of layered oxyhalide photocatalysts for visible-light water splitting

The band structure offers fundamental information on electronic properties of solid state materials, and hence it is crucial for solid state chemists to understand and predict the relationship between the band structure and electronic structure to design chemical and physical properties. Here, we review layered oxyhalide photocatalysts for water splitting with a particular emphasis on band structure control. The unique feature of these materials including Sillén and Sillén-Aurivillius oxyhalides lies in their band structure including a remarkably high oxygen band, allowing them to exhibit both visible light responsiveness and photocatalytic stability unlike conventional mixed anion compounds, which show good light absorption, but frequently encounter stability issues. For band structure control, simple strategies effective in mixed-anion compounds, such as anion substitution forming high energy p orbitals in accordance with its electronegativity, is not effective for oxyhalides with high oxygen bands. We overview key concepts for band structure control of oxyhalide photocatalysts such as lone-pair interactions and electrostatic interactions. The control of the band structure of inorganic solid materials is a crucial challenge across a wide range of materials chemistry fields, and the insights obtained by the development of oxyhalide photocatalysts are expected to provide knowledge for diverse materials chemistry.

Reference Energies for Valence Ionizations and Satellite Transitions

Upon ionization of an atom or a molecule, another electron (or more) can be simultaneously excited. These concurrently generated states are called "satellites" (or shakeup transitions) as they appear in ionization spectra as higher-energy peaks with weaker intensity and larger width than the main peaks associated with single-particle ionizations. Satellites, which correspond to electronically excited states of the cationic species, are notoriously challenging to model using conventional single-reference methods due to their high excitation degree compared to the neutral reference state. This work reports 42 satellite transition energies and 58 valence ionization potentials (IPs) of full configuration interaction quality computed in small molecular systems. Following the protocol developed for the quest database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; and Loos, P.-F. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1517], these reference energies are computed using the configuration interaction using a perturbative selection made iteratively (CIPSI) method. In addition, the accuracy of the well-known coupled-cluster (CC) hierarchy (CC2, CCSD, CC3, CCSDT, CC4, and CCSDTQ) is gauged against these new accurate references. The performances of various approximations based on many-body Green's functions (GW, GF2, and T-matrix) for IPs are also analyzed. Their limitations in correctly modeling satellite transitions are discussed.

Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex

The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.

Ab Initio Modeling of Mixed-Dimensional Heterostructures: A Path Forward

Understanding the electronic structure of mixed-dimensional heterostructures is essential for maximizing their application potential. However, accurately modeling such interfaces is challenging due to the complex interplay between the subsystems. We employ a computational framework integrating first-principles methods, including GW, density functional theory (DFT), and the polarizable continuum model, to elucidate the electronic structure of mixed-dimensional heterojunctions formed by free-base phthalocyanines and monolayer molybdenum disulfide. We assess the impact of dielectric screening across various scenarios, from isolated molecules to organic films on a substrate-supported monolayer. Our findings show that while polarization effects cause significant renormalization of molecular energy levels, band energies and alignments in the most relevant setup can be accurately predicted through DFT simulations of the individual subsystems. Additionally, we analyze orbital hybridization, revealing potential pathways for interfacial charge transfer. This study offers new insights into hybrid inorganic/organic interfaces and provides a practical computational protocol suitable for scaled-up studies.

Computational Discovery of Intermolecular Singlet Fission Materials Using Many-Body Perturbation Theory

Intermolecular singlet fission (SF) is the conversion of a photogenerated singlet exciton into two triplet excitons residing on different molecules. SF has the potential to enhance the conversion efficiency of solar cells by harvesting two charge carriers from one high-energy photon, whose surplus energy would otherwise be lost to heat. The development of commercial SF-augmented modules is hindered by the limited selection of molecular crystals that exhibit intermolecular SF in the solid state. Computational exploration may accelerate the discovery of new SF materials. The GW approximation and Bethe-Salpeter equation (GW+BSE) within the framework of many-body perturbation theory is the current state-of-the-art method for calculating the excited-state properties of molecular crystals with periodic boundary conditions. In this Review, we discuss the usage of GW+BSE to assess candidate SF materials as well as its combination with low-cost physical or machine learned models in materials discovery workflows. We demonstrate three successful strategies for the discovery of new SF materials: (i) functionalization of known materials to tune their properties, (ii) finding potential polymorphs with improved crystal packing, and (iii) exploring new classes of materials. In addition, three new candidate SF materials are proposed here, which have not been published previously.

Low-Scaling Algorithms for GW and Constrained Random Phase Approximation Using Symmetry-Adapted Interpolative Separable Density Fitting

We present low-scaling algorithms for GW and constrained random phase approximation based on a symmetry-adapted interpolative separable density fitting (ISDF) procedure that incorporates the space-group symmetries of crystalline systems. The resulting formulations scale cubically, with respect to system size, and linearly with the number of k-points, regardless of the choice of single-particle basis and whether a quasiparticle approximation is employed. We validate these methods through comparisons with published literature and demonstrate their efficiency in treating large-scale systems through the construction of downfolded many-body Hamiltonians for carbon dimer defects embedded in hexagonal boron nitride supercells. Our work highlights the efficiency and general applicability of ISDF in the context of large-scale many-body calculations with k-point sampling beyond density functional theory.

Fully Dynamic G3W2 Self-Energy for Finite Systems: Formulas and Benchmark

Over the years, Hedin's GW self-energy has been proven to be a rather accurate and simple approximation to evaluate electronic quasiparticle energies in solids and in molecules. Attempts to improve over the simple GW approximation, the so-called vertex corrections, have been constantly proposed in the literature. Here, we derive, analyze, and benchmark the complete second-order term in the screened Coulomb interaction W for finite systems. This self-energy named G3W2 contains all the possible time orderings that combine 3 Green's functions G and 2 dynamic W. We present the analytic formula and its imaginary frequency counterpart, with the latter allowing us to treat larger molecules. The accuracy of the G3W2 self-energy is evaluated on well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. Its link with the simpler static approximation, named SOSEX for static screened second-order exchange, is analyzed, which leads us to propose a more consistent approximation named 2SOSEX. In the end, we find that neither the G3W2 self-energy nor any of the investigated approximations to it improve over one-shot G0W0 with a good starting point. Only quasi-particle self-consistent GW HOMO energies are slightly improved by addition of the G3W2 self-energy correction. We show that this is due to the self-consistent update of the screened Coulomb interaction, leading to an overall sign change of the vertex correction to the frontier quasiparticle energies.

Comparing Self-Consistent GW and Vertex-Corrected G0W0 (G0W0Γ) Accuracy for Molecular Ionization Potentials

We test the performance of self-consistent GW and several representative implementations of vertex-corrected G0W0 (G0W0Γ). These approaches are tested on benchmark data sets covering full valence spectra (first ionization potentials and some inner valence shell excitations). For small molecules, when comparing against state-of-the-art wave function techniques, our results show that full self-consistency in the GW scheme either systematically outperforms vertex-corrected G0W0 or gives results of at least comparative quality. Moreover, G0W0Γ results in additional computational cost when compared to G0W0 or self-consistent GW. The dependency of G0W0Γ on the starting mean-field solution is frequently more dominant than the magnitude of the vertex correction itself. Consequently, for molecular systems, self-consistent GW performed on the imaginary axis (and then followed by modern analytical continuation techniques) offers a more reliable approach to make predictions of ionization potentials.

Fermi-Level Pinning in ErAs Nanoparticles Embedded in III-V Semiconductors

Embedding rare-earth monopnictide nanoparticles into III-V semiconductors enables unique optical, electrical, and thermal properties for THz photoconductive switches, tunnel junctions, and thermoelectric devices. Despite the high structural quality and control over growth, particle size (<3 nm), and density, the underlying electronic structure of these nanocomposite materials has only been hypothesized. Structural and electronic properties of ErAs nanoparticles with different shapes and sizes (cubic to spherical, 1.14, 1.71, and 2.28 nm) in AlAs, GaAs, InAs, and their alloys are investigated using first-principles calculations, revealing that spherical nanoparticles have lower formation energies. For the lowest-energy nanoparticles, the Fermi level is pinned near midgap in GaAs and AlAs but resonant in the conduction band in InAs. The Fermi level is shifted down as the particle size increases and is pinned on an absolute energy scale considering the band alignment at AlAs/GaAs/InAs interfaces, offering insights into the rational design of these nanomaterials.

Band Alignment of Oxides by Learnable Structural-Descriptor-Aided Neural Network and Transfer Learning

The band alignment of semiconductors, insulators, and dielectrics is relevant to diverse material properties and device structures utilizing their surfaces and interfaces. In particular, the ionization potential and electron affinity are fundamental quantities that describe surface-dependent band-edge positions with respect to the vacuum level. Their accurate and systematic determination, however, demands elaborate experiments or simulations for well-characterized surfaces. Here, we report machine learning for the band alignment of nonmetallic oxides using a high-throughput first-principles calculation data set containing about 3000 oxide surfaces. Our neural network accurately predicts the band positions for relaxed surfaces of binary oxides simply by using the information on bulk structures and surface termination planes. Moreover, we extend the model to naturally include multiple-cation effects and transfer it to ternary oxides. The present approach enables the band alignment of a vast number of solid surfaces, thereby opening the way to a systematic understanding and materials screening.

Total Energy beyond GW: Exact Results and Guidelines for Approximations

The total energy and electron addition and removal spectra can, in principle, be obtained exactly from the one-body Green's function (GF). In practice, the GF is obtained from an approximate self-energy. In the framework of many-body perturbation theory, we derive different expressions that are based on an approximate self-energy, but that yield nevertheless, in principle, the exact exchange-correlation contribution to the total energy for any interaction strength. Response functions play a crucial role, which explains why, for example, ingredients from time-dependent density functional theory can be used to build these approximate self-energies. We show that the key requirement for obtaining exact results is the consistent combination of ingredients. Also when further approximations are made, as it is necessary in practice, this consistency remains the key to obtain good results. All findings are illustrated using the exactly solvable symmetric Hubbard dimer.

Absolute energy levels of liquid water from many-body perturbation theory with effective vertex corrections

We demonstrate the importance of addressing the Γ vertex and thus going beyond the GW approximation for achieving the energy levels of liquid water in many-body perturbation theory. In particular, we consider an effective vertex function in both the polarizability and the self-energy, which does not produce any computational overhead compared with the GW approximation. We yield the band gap, the ionization potential, and the electron affinity in good agreement with experiment and with a hybrid functional description. The achieved electronic structure and dielectric screening further lead to a good description of the optical absorption spectrum, as obtained through the solution of the Bethe-Salpeter equation. In particular, the experimental peak position of the exciton is accurately reproduced.

The three channels of many-body perturbation theory: GW, particle-particle, and electron-hole T-matrix self-energies

We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known GW self-energy, as well as the particle-particle and electron-hole T-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory. The performance of these schemes on strongly correlated systems (B2 and C2) is also discussed.

Two-Component GW Calculations: Cubic Scaling Implementation and Comparison of Vertex-Corrected and Partially Self-Consistent GW Variants

We report an all-electron, atomic orbital (AO)-based, two-component (2C) implementation of the GW approximation (GWA) for closed-shell molecules. Our algorithm is based on the space-time formulation of the GWA and uses analytical continuation (AC) of the self-energy, and pair-atomic density fitting (PADF) to switch between AO and auxiliary basis. By calculating the dynamical contribution to the GW self-energy at a quasi-one-component level, our 2C-GW algorithm is only about a factor of 2-3 slower than in the scalar relativistic case. Additionally, we present a 2C implementation of the simplest vertex correction to the self-energy, the statically screened G3W2 correction. Comparison of first ionization potentials (IPs) of a set of 67 molecules with heavy elements (a subset of the SOC81 set) calculated with our implementation against results from the WEST code reveals mean absolute deviations (MAD) of around 70 meV for G0W0@PBE and G0W0@PBE0. We check the accuracy of our AC treatment by comparison to full-frequency GW calculations, which shows that in the absence of multisolution cases, the errors due to AC are only minor. This implies that the main sources of the observed deviations between both implementations are the different single-particle bases and the pseudopotential approximation in the WEST code. Finally, we assess the performance of some (partially self-consistent) variants of the GWA for the calculation of first IPs by comparison to vertical experimental reference values. G0W0@PBE0 (25% exact exchange) and G0W0@BHLYP (50% exact exchange) perform best with mean absolute deviations (MAD) of about 200 meV. Explicit treatment of spin-orbit effects at the 2C level is crucial for systematic agreement with experiment. On the other hand, eigenvalue-only self-consistent GW (evGW) and quasi-particle self-consistent GW (qsGW) significantly overestimate the IPs. Perturbative G3W2 corrections increase the IPs and therefore improve the agreement with experiment in cases where G0W0 alone underestimates the IPs. With a MAD of only 140 meV, 2C-G0W0@PBE0 + G3W2 is in best agreement with the experimental reference values.

Prediction of Stable Surfaces of Metal Oxides through the Unsaturated Coordination Index

This study proposes the unsaturated coordination index, σ, as a potential descriptor of the stability of metal-oxide surfaces cleaved from bulk. The value of σ, the number of missing bonds per unit area, can be obtained very quickly using only crystallographic data, namely, the bulk geometry. The surface energies of various binary oxides, with and without atom relaxation, were calculated. Their correlations with σ had good coefficients of determination (R2) values, particularly in high-symmetry crystals. The proposed descriptor is very useful for an initial evaluation of stable metal-oxide surfaces without conducting any surface model calculations.

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

Many-body screening effects in liquid water

The screening arising from many-body excitations is a crucial quantity for describing absorption and inelastic X-ray scattering (IXS) of materials. Similarly, the electron screening plays a critical role in state-of-the-art approaches for determining the fundamental band gap. However, ab initio studies of the screening in liquid water have remained limited. Here, we use a combined analysis based on the Bethe-Salpeter equation and time-dependent density functional theory. We first show that absorption spectra at near-edge energies are insufficient to assess the accuracy by which the screening is described. Next, when the energy range under scrutiny is extended, we instead find that the IXS spectra are highly sensitive and allow for the selection of the optimal theoretical scheme. This leads to good agreement with experiment over a large range of transferred energies and momenta, and enables establishing the elusive fundamental band gap of liquid water at 9.3 eV.

Embedding vertex corrections in GW self-energy: Theory, implementation, and outlook

The vertex function (Γ) within the Green's function formalism encapsulates information about all higher-order electron-electron interaction beyond those mediated by density fluctuations. Herein, we present an efficient approach that embeds vertex corrections in the one-shot GW correlation self-energy for isolated and periodic systems. The vertex-corrected self-energy is constructed through the proposed separation-propagation-recombination procedure: the electronic Hilbert space is separated into an active space and its orthogonal complement denoted as the "rest;" the active component is propagated by a space-specific effective Hamiltonian different from the rest. The vertex corrections are introduced by a rescaled time-dependent nonlocal exchange interaction. The direct Γ correction to the self-energy is further updated by adjusting the rescaling factor in a self-consistent post-processing cycle. Our embedding method is tested mainly on donor-acceptor charge-transfer systems. The embedded vertex effects consistently and significantly correct the quasiparticle energies of the gap-edge states. The fundamental gap is generally improved by 1-3 eV upon the one-shot GW approximation. Furthermore, we provide an outlook for applications of (embedded) vertex corrections in calculations of extended solids.

Natural virtual orbitals for the GW method in the random-phase approximation and beyond

The increasingly popular GW method is becoming a convenient tool to determine vertical ionization energies in molecular systems. However, depending on the formalism used and the range of orbitals investigated, it may be hampered by a steep computational scaling. To alleviate this issue, correlated natural virtual orbitals (NVOs) based on second-order Møller-Plesset (MP2) and direct MP2 correlation energies are implemented, and the resulting correlated NVOs are tested on GW quasiparticle energies. Test cases include the popular GW variants G₀W₀ and evGW₀ as well as more elaborate vertex corrections. We find that for increasingly larger molecular systems and basis sets, NVOs considerably improve efficiency. Furthermore, we test the performance of the truncated (frozen) NVO ansatz on the GW100 test set. For the latter, it is demonstrated that, using a carefully chosen truncation threshold, NVOs lead to a negligible loss in accuracy while providing speedups of one order of magnitude. Furthermore, we compare the resulting quasiparticle energies to very accurate vertical ionization energies obtained from coupled-cluster theory with singles, doubles, and noniterative triples [CCSD(T)], confirming that the loss in accuracy introduced by truncating the NVOs is negligible compared to the methodical errors in the GW approximation. It is also demonstrated that the choice of basis set impacts results far more than using a suitably truncated NVO space. Therefore, at the same computational expense, more accurate results can be obtained using NVOs. Finally, we provide improved reference CCSD(T) values for the GW100 test set, which have been obtained using the def2-QZVPP basis set.

Doping Limits of Phosphorus, Arsenic, and Antimony in CdTe

Low p-type doping is a limiting factor to increase CdTe thin-film solar-cell efficiency toward the theoretical Shockley-Queisser limit of 33%. Previous calculations predict relatively high ionization energies for group-V acceptors (P, As, and Sb), and they are plagued by self-compensation, forming AX centers, severely limiting hole concentration. However, recent experiments on CdTe single crystals indicate a much more favorable scenario, where P, As, and Sb behave as shallow acceptors. Using hybrid functional calculations, we solve this puzzle by showing that the ionization energies significantly decrease with the supercell size. When including the effects of spin-orbit coupling and extrapolating the results to the dilute limit, we find these impurities behave as hydrogenic-like shallow acceptors, and AX centers are unstable and do not limit p-type doping. We address the differences between our results and previous theoretical predictions and show that our ionization energies predict hole concentrations that agree with recent temperature-dependent Hall measurements.

In2Se3, In2Te3, and In2(Se,Te)3 Alloys as Photovoltaic Materials

In its lowest-energy three-dimensional (3D) hexagonal crystal structure (γ phase), In₂Se₃ has a direct band gap of ∼1.8 eV and displays high absorption coefficient, making it a promising semiconductor material for optoelectronics. Incorporation of Te allows for tuning the band gap, adding flexibility to device design and extending the application range. Here we report results of hybrid density functional theory calculations to assess the electronic and optical properties of γ-In₂Se₃, γ-In₂Te₃, and γ-In₂(Se_1-xTe_x)₃ alloys, and initial experiments on the growth and characterization of γ-In₂Se₃ thin films. The predicted band gap of 1.84 eV for γ-In₂Se₃ is in good agreement with the absorption onset derived from transmission and reflection spectra of thin films. We show that incorporation of Te gives γ-In₂(Se_1-xTe_x)₃ alloys with a band gap ranging from 1.84 eV down to 1.23 eV, thus covering the optimal band gap range for single-junction solar cells. In addition, the γ-In₂Se₃/γ-In₂(Se_1-xTe_x)₃ bilayer could be employed in tandem solar-cell architectures absorbing at E_g ≈ 1.8 eV and at E_g ≤ 1.4 eV, toward overcoming the ∼33% efficiency set by the Shockley-Queisser limit for single junction solar cells. We also discuss band gap bowing and mixing enthalpies, aiming at adding γ-In₂Se₃, γ-In₂Te₃, and γ-In₂(Se_1-xTe_x)₃ alloys to the available toolbox of materials for solar cells and other optoelectronic applications.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Efficient and improved prediction of the band offsets at semiconductorheterojunctions from meta-GGA density functionals: a benchmark study

Accurate theoretical prediction of the band offsets at interfaces of semiconductor heterostructures can of-ten be quite challenging. Although density functional theory has been reasonably successful to carry outsuch calculations and efficient and accurate semilocal functionals are desirable to reduce the computational cost. In general, the semilocal functionals based on the generalized gradient approximation (GGA) significantly underestimate the bulk band gaps. This, in turn, results in inaccurate estimates of the band offsets at the heterointerfaces. In this paper, we investigate the performance of several advanced meta-GGA functionals in the computational prediction of band offsets at semiconductor heterojunctions. In particular, we investigate the performance of r 2 SCAN (revised strongly-constrained and appropriately-normed functional), rMGGAC (revised semilocal functional based on cuspless hydrogen model and Pauli kinetic energy density functional), mTASK (modified Aschebrock and Kümmel meta-GGA functional), and LMBJ (local modified Becke-Johnson) exchange-correlation functionals. Our results strongly suggest that these meta-GGA functionals for supercell calculations perform quite well, especially, when compared to computationally more demanding GW calculations. We also present band offsets calculated using ionization potentials and electron affinities, as well as band alignment via the branch point energies. Overall, our study shows that the aforementioned meta-GGA functionals can be used within the DFT framework to estimate the band offsets in semiconductor heterostructures with predictive accuracy.

Numerically Precise Benchmark of Many-Body Self-Energies on Spherical Atoms

We investigate the performance of beyond-GW approaches in many-body perturbation theory by addressing atoms described within the spherical approximation via a dedicated numerical treatment based on B-splines and spherical harmonics. We consider the GW, second Born (2B), and GW + second order screened exchange (GW+SOSEX) self-energies and use them to obtain ionization potentials from the quasi-particle equation (QPE) solved perturbatively on top of independent-particle calculations. We also solve the linearized Sham–Schlüter equation (LSSE) and compare the resulting xc potentials against exact data. We find that the LSSE provides consistent starting points for the QPE but does not present any practical advantage in the present context. Still, the features of the xc potentials obtained with it shed light on possible strategies for the inclusion of beyond-GW diagrams in the many-body self-energy. Our findings show that solving the QPE with the GW+SOSEX self-energy on top of a PBE or PBE0 solution is a viable scheme to go beyond GW in finite systems, even in the atomic limit. However, GW shows a comparable performance if one agrees to use a hybrid starting point. We also obtain promising results with the 2B self-energy on top of Hartree–Fock, suggesting that the full time-dependent Hartree–Fock vertex may be another viable beyond-GW scheme for finite systems.

S66x8 noncovalent interactions revisited: new benchmark and performance of composite localized coupled-cluster methods

The S66x8 noncovalent interactions benchmark has been re-evaluated at the “sterling silver” level. Against this, a selection of computationally more economical alternatives has been assayed, ranging from localized CC to double hybrids and SAPT(DFT).

Electronic Structure of Molecules, Surfaces, and Molecules on Surfaces with the Local Modified Becke-Johnson Exchange-Correlation Potential

The knowledge of electronic properties of matter is the key to the understanding of its properties and to propose useful applications. To model hybrid organic/inorganic systems with the plane-wave approach, large supercells with many atoms are usually necessary to minimize artificial interactions between periodic images. For such systems, accurate approximations to the exchange-correlation functional of density functional theory, such as hybrid functionals, become computationally expensive, and cheaper approaches need to be considered. Here, we apply the local modified Becke-Johnson exchange-correlation potential to free molecules and surfaces and study its accuracy for calculated ionization potentials. This quantity being important to understand the band alignment of composite heterogeneous systems, we demonstrate the application of the potential to the electronic structure calculation of an exemplary composite semiconductor/molecule system, namely, a F6-TCNNQ molecule adsorbed on a hydrogenated Si(111) surface.

Pros and cons of the time-dependent hybrid density functional approach for calculating the optical spectra of solids: a case study of CeO2

The prediction of optical spectra of complex solids remains a great challenge for first-principles calculations due to the huge computational cost of the state-of-the-art many-body perturbation theory based GW-Bethe Salpeter equation (BSE) approach. An alternative method is the time-dependent density-functional theory (TDDFT) based on hybrid exchange-correlation functionals, which involves the essential ingredients of electron-hole interactions in its formalism in contrast to its local/semi-local functional counterparts. In this work, we investigate the optical absorption spectra of ceria (CeO2), a prototypical lanthanide oxide with a 4f0 configuration, utilizing TDDFT based on four well-established hybrid functionals for ground state DFT calculations. All four functionals reproduce well the excitonic features of the experimental optical spectra, in spite of the significant differences in their band structures arising from different hybridization parameters (i.e. the fraction of the Hartree-Fock exchange and the screening parameter). It is demonstrated that the apparently weak dependence of the resulting optical spectra on the employed functionals is quite universal and applies to simple semiconductors such as Si and GaAs and insulator LiF as well. This study highlights the feasibility of TDDFT based on existing hybrids to describe optical spectra of solids, and also, points out the difficulty of obtaining accurate exciton binding energies using these hybrid functionals due to the strong functional dependence of quasi-particle band structures.

Assessing the G0W0Γ0(1) Approach: Beyond G0W0 with Hedin's Full Second-Order Self-Energy Contribution

We present and benchmark a self-energy approach for quasiparticle energy calculations that goes beyond Hedin's GW approximation by adding the full second-order self-energy (FSOS-W) contribution. The FSOS-W diagram involves two screened Coulomb interaction (W) lines, and adding the FSOS-W to the GW self-energy can be interpreted as first-order vertex correction to GW (GWΓ⁽¹⁾). Our FSOS-W implementation is based on the resolution-of-identity technique and exhibits better than O(N⁵) scaling with system size for small- to medium-sized molecules. We then present one-shot GWΓ⁽¹⁾ (G₀W₀Γ₀⁽¹⁾) benchmarks for the GW100 test set and a set of 24 acceptor molecules. For semilocal or hybrid density functional theory starting points, G₀W₀Γ₀⁽¹⁾ systematically outperforms G₀W₀ for the first vertical ionization potentials and electron affinities of both test sets. Finally, we demonstrate that a static FSOS-W self-energy significantly underestimates the quasiparticle energies.

Optimized effective potentials from the random-phase approximation: Accuracy of the quasiparticle approximation

The optimized effective potential (OEP) method presents an unambiguous way to construct the Kohn-Sham potential corresponding to a given diagrammatic approximation for the exchange-correlation functional. The OEP from the random-phase approximation (RPA) has played an important role ever since the conception of the OEP formalism. However, the solution of the OEP equation is computationally fairly expensive and has to be done in a self-consistent way. So far, large scale solid state applications have, therefore, been performed only using the quasiparticle approximation (QPA), neglecting certain dynamical screening effects. We obtain the exact RPA-OEP for 15 semiconductors and insulators by direct solution of the linearized Sham-Schlüter equation. We investigate the accuracy of the QPA on Kohn-Sham bandgaps and dielectric constants, and comment on the issue of self-consistency.

Predicting Wettability and the Electrochemical Window of Lithium-Metal/Solid Electrolyte Interfaces

The development of solid electrolytes (SEs) is expected to enhance the safety of lithium-ion batteries. Additionally, a viable SE could allow the use of a Li-metal negative electrode, which would increase energy density. Recently, several antiperovskites have been reported to exhibit high ionic conductivities, prompting investigations of their use as an SE. In addition to having a suitable conductivity, phenomena at the interface between an SE and an electrode are also of great importance in determining the viability of an SE. For example, interfacial interactions can change the positions of the band edges of the SE, altering its stability against undesirable oxidation or reduction. Furthermore, the wettability of the SE by the metallic anode is desired to enable low interfacial resistance and uniform metal plating and stripping during cycling. The present study probes several properties of the SE/electrode interface at the atomic scale. Adopting the antiperovskite SE Li₃OCl (LOC)/Li-metal anode interface as a model system, the interfacial energy, work of adhesion, wettability, band edge shifts, and the electrochemical window are predicted computationally. The oxygen-terminated interface was determined to be the most thermodynamically stable. Moreover, the large calculated work of adhesion for this system implies that Li will wet LOC, suggesting the possibility for low interfacial resistance. Nevertheless, these strong interfacial interactions come at a cost to electrochemical stability: strong interfacial bonding lowers the energy of the conduction band minimum (CBM) significantly and narrows the local band gap by 30% in the vicinity of the interface. Despite this interface-induced reduction in electrochemical window, the CBM in LOC remains more negative than the Li/Li⁺ redox potential, implying stability against reduction by the anode. In sum, this study illustrates a comprehensive computational approach to assessing electrode/electrolyte interfacial properties in solid-state batteries.

The GW Compendium: A Practical Guide to Theoretical Photoemission Spectroscopy

The GW approximation in electronic structure theory has become a widespread tool for predicting electronic excitations in chemical compounds and materials. In the realm of theoretical spectroscopy, the GW method provides access to charged excitations as measured in direct or inverse photoemission spectroscopy. The number of GW calculations in the past two decades has exploded with increased computing power and modern codes. The success of GW can be attributed to many factors: favorable scaling with respect to system size, a formal interpretation for charged excitation energies, the importance of dynamical screening in real systems, and its practical combination with other theories. In this review, we provide an overview of these formal and practical considerations. We expand, in detail, on the choices presented to the scientist performing GW calculations for the first time. We also give an introduction to the many-body theory behind GW, a review of modern applications like molecules and surfaces, and a perspective on methods which go beyond conventional GW calculations. This review addresses chemists, physicists and material scientists with an interest in theoretical spectroscopy. It is intended for newcomers to GW calculations but can also serve as an alternative perspective for experts and an up-to-date source of computational techniques.

Ab Initio Lifetime and Concomitant Double-Excitation Character of Plasmons at Metallic Densities

The accurate calculation of excited state properties of interacting electrons in the condensed phase is an immense challenge in computational physics. Here, we use state-of-the-art equation-of-motion coupled-cluster theory with single and double excitations (EOM-CCSD) to calculate the dynamic structure factor, which can be experimentally measured by inelastic x-ray and electron scattering. Our calculations are performed on the uniform electron gas at densities corresponding to Wigner-Seitz radii of r_{s}=5, 4, and 3 corresponding to the valence electron densities of common metals. We compare our results to those obtained using the random-phase approximation (RPA), which is known to provide a reasonable description of the collective plasmon excitation and which resums only a small subset of the polarizability diagrams included in EOM-CCSD. We find that EOM-CCSD, instead of providing a perturbative improvement on the RPA plasmon, predicts a many-state plasmon resonance, where each contributing state has a double-excitation character of 80% or more. This finding amounts to an ab initio treatment of the plasmon linewidth, which is in good quantitative agreement with previous diagrammatic calculations, and highlights the strongly correlated nature of lifetime effects in condensed-phase electronic structure theory.

Simple eigenvalue-self-consistent Δ ¯ G W 0

We show that a rigid scissors-like GW self-consistency approach, labeled here Δ ¯ G W 0 , can be trivially implemented at zero additional cost for large scale one-shot G ₀ W ₀ calculations. The method significantly improves one-shot G ₀ W ₀ and for large systems is very accurate. Δ ¯ G W 0 is similar in spirit to evGW ₀ where the self-consistency is only applied on the eigenvalues entering Green's function, while both W and the eigenvectors of Green's function are held fixed. Δ ¯ G W 0 further assumes that the shift of the eigenvalues is rigid scissors-like so that all occupied states are shifted by the same amount and analogously for all the unoccupied states. We show that this results in a trivial modification of the time-dependent G ₀ W ₀ self-energy, enabling an a posteriori self-consistency cycle. The method is applicable for our recent stochastic-GW approach, thereby enabling self-consistent calculations for giant systems with thousands of electrons. The accuracy of Δ ¯ G W 0 increases with the system size. For molecules, it is up to 0.4-0.5 eV away from coupled-cluster single double triple (CCSD(T)), but for tetracene and hexacene, it matches the ionization energies from both CCSD(T) and evGW ₀ to better than 0.05 eV. For solids, as exemplified here by periodic supercells of semiconductors and insulators with 6192 valence electrons, the method matches evGW ₀ quite well and both methods are in good agreement with the experiment.

Combining Wave Function Methods with Density Functional Theory for Excited States

We review state-of-the-art electronic structure methods based both on wave function theory (WFT) and density functional theory (DFT). Strengths and limitations of both the wave function and density functional based approaches are discussed, and modern attempts to combine these two methods are presented. The challenges in modeling excited-state chemistry using both single-reference and multireference methods are described. Topics covered include background, combining density functional theory with single-configuration wave function theory, generalized Kohn-Sham (KS) theory, global hybrids, range-separated hybrids, local hybrids, using KS orbitals in many-body theory (including calculations of the self-energy and the GW approximation), Bethe-Salpeter equation, algorithms to accelerate GW calculations, combining DFT with multiconfigurational WFT, orbital-dependent correlation functionals based on multiconfigurational WFT, building multiconfigurational wave functions from KS configurations, adding correlation functionals to multiconfiguration self-consistent-field (MCSCF) energies, combining DFT with configuration-interaction singles by means of time-dependent DFT, using range separation to combine DFT with MCSCF, embedding multiconfigurational WFT in DFT, and multiconfiguration pair-density functional theory.

Accuracy of dielectric-dependent hybrid functionals in the prediction of optoelectronic properties of metal oxide semiconductors: a comprehensive comparison with many-body GW and experiments

Understanding the electronic structure of metal oxide semiconductors is crucial to their numerous technological applications, such as photoelectrochemical water splitting and solar cells. The needed experimental and theoretical knowledge goes beyond that of pristine bulk crystals, and must include the effects of surfaces and interfaces, as well as those due to the presence of intrinsic defects (e.g. oxygen vacancies), or dopants for band engineering. In this review, we present an account of the recent efforts in predicting and understanding the optoelectronic properties of oxides using ab initio theoretical methods. In particular, we discuss the performance of recently developed dielectric-dependent hybrid functionals, providing a comparison against the results of many-body GW calculations, including G ₀ W ₀ as well as more refined approaches, such as quasiparticle self-consistent GW. We summarize results in the recent literature for the band gap, the band level alignment at surfaces, and optical transition energies in defective oxides, including wide gap oxide semiconductors and transition metal oxides. Correlated transition metal oxides are also discussed. For each method, we describe successes and drawbacks, emphasizing the challenges faced by the development of improved theoretical approaches. The theoretical section is preceded by a critical overview of the main experimental techniques needed to characterize the optoelectronic properties of semiconductors, including absorption and reflection spectroscopy, photoemission, and scanning tunneling spectroscopy (STS).

Accurate description of the electronic structure of organic semiconductors by GW methods

Electronic properties associated with charged excitations, such as the ionization potential (IP), the electron affinity (EA), and the energy level alignment at interfaces, are critical parameters for the performance of organic electronic devices. To computationally design organic semiconductors and functional interfaces with tailored properties for target applications it is necessary to accurately predict these properties from first principles. Many-body perturbation theory is often used for this purpose within the GW approximation, where G is the one particle Green's function and W is the dynamically screened Coulomb interaction. Here, the formalism of GW methods at different levels of self-consistency is briefly introduced and some recent applications to organic semiconductors and interfaces are reviewed.

Ab initio Electronic Structure of Liquid Water

Self-consistent GW calculations with efficient vertex corrections are employed to determine the electronic structure of liquid water. Nuclear quantum effects are taken into account through ab initio path-integral molecular dynamics simulations. We reveal a sizable band-gap renormalization of up to 0.7 eV due to hydrogen-bond quantum fluctuations. Our calculations lead to a band gap of 8.9 eV, in accord with the experimental estimate. We further resolve the ambiguities in the band-edge positions of liquid water. The valence-band maximum and the conduction-band minimum are found at -9.4 and -0.5  eV with respect to the vacuum level, respectively.

Synthesis and characterization of metastable transition metal oxides and oxide nitrides

New routes to vanadium sesquioxide and tantalum oxide nitride (γ- and δ-phase) are presented. Phase pure V2O3 with bixbyite-type structure, a metastable polymorph, was obtained from vanadium fluoride hydrates at ~750 K. It crystallizes in the cubic crystal system in space group I a 3 ¯ $Ia\bar 3$ with lattice parameter a=939.30(5) pm. The catalytical properties of the corresponding oxide nitride phases and their oxidation and reduction solid-state kinetics were investigated. The preparation of γ-TaON as a phase pure sample can be realized by ammonolysis of X-ray amorphous tantalum oxide precursors at 1073 K. This metastable tantalum oxide nitride crystallizes in the monoclinic VO2(B)-type structure in space group C2/m. The same precursors can be used to synthesize the δ-modification with an anatase-type structure at 1023 K. It crystallizes in the tetragonal crystal system in space group I41/amd. A maximum yield of 82 m % could be obtained. The fundamental band gaps of the synthesized and of other metastable TaON polymorphs were calculated from first principles using the GW method. The present results are compared to experimental data and to previous calculations at hybrid DFT level.