Evaluating the GW Approximation with CCSD(T) for Charged Excitations Across the Oligoacenes

Polarizable Continuum Models and Green's Function GW Formalism: On the Dynamics of the Solvent Electrons

The many-body GW formalism, for the calculation of ionization potentials or electronic affinities, relies on the frequency-dependent dielectric function built from the electronic degrees of freedom. Considering the case of water as a solvent treated within the polarizable continuum model, we explore the impact of restricting the full frequency-dependence of the solvent electronic dielectric response to a frequency-independent (ϵ∞) optical dielectric constant. For solutes presenting small to large highest-occupied to lowest-unoccupied molecular orbital energy gaps, we show that such a restriction induces errors no larger than a few percent on the energy level shifts from the gas to the solvated phase. We further introduce a remarkably accurate single-pole model for mimicking the effect of the full frequency dependence of the water dielectric function in the visible-UV range. This allows a fully dynamical embedded GW calculation with the only knowledge of the cavity reaction field calculated for the ϵ∞ optical dielectric constant.

From Many-Body Ab Initio to Effective Excitonic Models: A Versatile Mapping Approach Including Environmental Embedding Effects

We present an original multistate projective diabatization scheme based on Green's function formalisms that allows the systematic mapping of many-body ab initio calculations onto effective excitonic models. This method inherits the ability of the Bethe-Salpeter equation to describe Frenkel molecular excitons and intermolecular charge-transfer states equally well, as well as the possibility for an effective description of environmental effects in a QM/MM framework. The latter is found to be a crucial element in order to obtain accurate model parameters for condensed phases and to ensure their transferability to excitonic models for extended systems. The method is presented through a series of examples illustrating its quality, robustness, and internal consistency.

Molecular Ionization Energies from GW and Hartree-Fock Theory: Polarizability, Screening, and Self-Energy Vertex Corrections

Accurate prediction of electron removal and addition energies is essential for reproducing neutral excitation spectra in molecules using Bethe-Salpeter equation methods. A Hartree-Fock starting point for GW/BSE calculations, combined with a random phase approximation (RPA) polarizability in the screened interaction, W, is well-known to overestimate neutral excitation energies. Using a Hartree-Fock starting point, we apply several different approximations for W to molecules in the Quest-3 database [Loos et al. J. Chem. Theory Comput. 2020, 16, 1711]. W is calculated using polarizabilities in RPA and time-dependent HF approximations. Inclusion of screened electron-hole attraction in the polarizability yields valence ionization energies in better agreement with experimental values and ADC(3) calculations than the more commonly applied RPA polarizability. Quasiparticle weights are also in better agreement with ADC(3) values when electron-hole attraction is included in W. Shake-up excitations for the 1π levels in benzene and azines are indicated only when electron-hole attraction is included. Ionization energies derived from HF eigenvalues via Koopmans theorem for molecules with nitrogen or oxygen lone pairs have the largest differences from experimental values in the molecules considered, leading to incorrect ordering of nonbonding and π bonding levels. Inclusion of electron-hole attraction in the polarizability results in correct ordering of ionization energies and marked improvement in agreement with experimental data. Vertex corrections to the self-energy further improve agreement with experimental ionization energies for these localized states.

Reference CC3 Excitation Energies for Organic Chromophores: Benchmarking TD-DFT, BSE/GW, and Wave Function Methods

To expand the QUEST database of highly accurate vertical transition energies, we consider a series of large organic chromogens ubiquitous in dye chemistry, such as anthraquinone, azobenzene, BODIPY, and naphthalimide. We compute, at the CC3 level of theory, the singlet and triplet vertical transition energies associated with the low-lying excited states. This leads to a collection of more than 120 new highly accurate excitation energies. For several singlet transitions, we have been able to determine CCSDT transition energies with a compact basis set, finding minimal deviations from the CC3 values for most states. Subsequently, we employ these reference values to benchmark a series of lower-order wave function approaches, including the popular ADC(2) and CC2 schemes, as well as time-dependent density-functional theory (TD-DFT), both with and without applying the Tamm-Dancoff approximation (TDA). At the TD-DFT level, we evaluate a large panel of global, range-separated, local, and double hybrid functionals. Additionally, we assess the performance of the Bethe-Salpeter equation (BSE) formalism relying on both G0W0 and evGW quasiparticle energies evaluated from various starting points. It turns out that CC2 and ADC(2.5) are the most accurate models among those with respective O ( N 5 ) and O ( N 6 ) scalings with system size. In contrast, CCSD does not outperform CC2. The best performing exchange-correlation functionals include BMK, M06-2X, M06-SX, CAM-B3LYP, ωB97X-D, and LH20t, with average deviations of approximately 0.20 eV or slightly below. Errors on vertical excitation energies can be further reduced by considering double hybrids. Both SOS-ωB88PP86 and SOS-ωPBEPP86 exhibit particularly attractive performances with overall quality on par with CC2, whereas PBE0-DH and PBE-QIDH are only slightly less efficient. BSE/evGW calculations based on Kohn-Sham starting points have been found to be particularly effective for singlet transitions, but much less for their triplet counterparts.

BSE@GW Prediction of Charge Transfer Exciton in Molecular Complexes: Assessment of Self-Energy and Exchange-Correlation Dependence

The Bethe-Salpeter equation using the GW approximation to the self-energy (BSE@GW) is a computationally attractive method for studying electronic excitation from first principles within the many-body Green's function theory framework. We examine its dependence on the underlying exchange-correlation (XC) approximation as well as on the GW approximation for predicting the charge transfer exciton formation at representative type-II interfaces between molecular systems of tetrachloro-1,2-benzoquinone (TCBQ) and acene derivatives. For the XC approximation, we consider several widely used generalized gradient approximation (GGA) and hybrid GGA functionals. For the GW self-energy approximation, we examine the recently proposed renormalized singles approach by Yang and coauthors [J. Phys. Chem. Lett. 2019, 10 (3), 447-452; J. Chem. Theory Comput. 2022, 18, 7570-7585] in addition to other commonly employed approximated GW schemes. We demonstrate a reliable prediction of the charge transfer exciton within the BSE@GW level of theory.

Theoretical study of short-range exchange interaction based on semiconductor dielectric function model toward time-dependent dielectric density functional theory

This study explores various models of semiconductor dielectric functions, with a specific emphasis on the large wavenumber spectrum and the derivation of the screened exchange interaction. Particularly, we discuss the short-range effect of the screened exchange potential. Our investigation reveals that the short-range effect originating from the high wavenumber spectrum is contingent upon the dielectric constant of the targeted system. To incorporate dielectric-dependent behaviors concerning the short-range aspect into the dielectric density functional theory (DFT) framework, we utilize the local Slater term and the Yukawa-type term, adjusting the ratio between these terms based on the dielectric constant. Additionally, we demonstrate the efficacy of the time-dependent dielectric DFT method in accurately characterizing the electronic structure of excited states in dyes and functional molecules. Several theoretical approaches have incorporated parameters dependent on the system to elucidate short-range exchange interactions. Our theoretical analysis and discussions will be useful for those studies.

The C1s core levels of polycyclic aromatic hydrocarbons and styrenic polymers: A first-principles study

Understanding core level shifts in aromatic compounds is crucial for the correct interpretation of x-ray photoelectron spectroscopy (XPS) of polycyclic aromatic hydrocarbons (PAHs), including acenes, as well as of styrenic polymers, which are increasingly relevant for the microelectronic industry, among other applications. The effect of delocalization through π aromatic systems on the stabilization of valence molecular orbitals has been widely investigated in the past. However, little has been reported on the impact on the deeper C1s core energy levels. In this work, we use first-principles calculations at the level of many body perturbation theory to compute the C1s binding energies of several aromatic systems. We report a C1s red shift in PAHs and acenes of increasing size, both in the gas phase and in the molecular crystal. C1s red shifts are also calculated for stacked benzene and naphthalene pairs at decreasing intermolecular distances. A C1s red shift is in addition found between oligomers of poly(p-hydroxystyrene) and polystyrene of increasing length, which we attribute to ring-ring interactions between the side-chains. The predicted shifts are larger than common instrumental errors and could, therefore, be detected in XPS experiments.

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.

Static versus dynamically polarizable environments within the many-body GW formalism

Continuum- or discrete-polarizable models for the study of optoelectronic processes in embedded subsystems rely mostly on the restriction of the surrounding electronic dielectric response to its low frequency limit. Such a description hinges on the assumption that the electrons in the surrounding medium react instantaneously to any excitation in the central subsystem, thus treating the environment in the adiabatic limit. Exploiting a recently developed embedded GW formalism with an environment described at the fully ab initio level, we assess the merits of the adiabatic limit with respect to an environment where the full dynamics of the dielectric response are considered. Furthermore, we show how to properly take the static limit of the environment's susceptibility by introducing the so-called Coulomb-hole and screened-exchange contributions to the reaction field. As a first application, we consider a C60 molecule at the surface of a C60 crystal, namely, a case where the dynamics of the embedded and embedding subsystems are similar. The common adiabatic assumption, when properly treated, generates errors below 10% on the polarization energy associated with frontier energy levels and associated energy gaps. Finally, we consider a water molecule inside a metallic nanotube, the worst case for the environment's adiabatic limit. The error on the gap polarization energy remains below 10%, even though the error on the frontier orbital polarization energies can reach a few tenths of an electronvolt.

Assessing the accuracy of TD-DFT excited-state geometries through optimal tuning with GW energy levels

We study the accuracy of excited state (ES) geometries using optimally tuned LC-PBE functionals with tuning based on GW quasiparticle energies. We compare the results obtained with the PBE, PBE0, non-tuned, and tuned LC-PBE functionals with available high-level CC reference values as well as experimental data. First, we compare ES geometrical parameters obtained for three different types of systems: molecules composed of a few atoms, 4-(dimethylamino)benzonitrile (DMABN), and conjugated dyes. To this end, we used wave-function results as benchmarks. Next, we evaluate the accuracy of the theoretically simulated spectra as compared to the experimental ones for five large dyes. Our results show that, besides small compact molecules for which tuning LC-PBE does not allow obtaining geometries more accurate than those computed with standard functionals, tuned range-separated functionals are clearly to be favored, not only for ES geometries but also for 0-0 energies, band shapes, and intensities for absorption and emission spectra. In particular, the results indicate that GW-tuned LC-PBE functionals provide improved matching with experimental spectra as compared to conventionally tuned functionals. It is an open question whether TD-DFT with GW-tuned functionals can qualitatively mimic the actual many-body Bethe-Salpeter (BSE/GW) formalism for which analytic ionic gradients remain to be developed.

Peeking into the Femtosecond Hot-Carrier Dynamics Reveals Unexpected Mechanisms in Plasmonic Photocatalysis

Plasmonic-driven photocatalysis may lead to reaction selectivity that cannot be otherwise achieved. A fundamental role is played by hot carriers, i.e., electrons and holes generated upon plasmonic decay within the metal nanostructure interacting with molecular species. Understanding the elusive microscopic mechanism behind such selectivity is a key step in the rational design of hot-carrier reactions. To accomplish that, we present state-of-the-art multiscale simulations, going beyond density functional theory, of hot-carrier injections for the rate-determining step of a photocatalytic reaction. We focus on carbon dioxide reduction, for which it was experimentally shown that the presence of a rhodium nanocube under illumination leads to the selective production of methane against carbon monoxide. We show that selectivity is due to a (predominantly) direct hole injection from rhodium to the reaction intermediate CHO. Unexpectedly, such an injection does not promote the selective reaction path by favoring proper bond breaking but rather by promoting bonding of the proper molecular fragment to the surface.

Excess and excited-state dipole moments of real-life dyes: a comparison between wave-function, BSE/GW, and TD-DFT values

In this work, we assess the accuracy of the Bethe-Salpeter equation (BSE) many-body Green's function formalism, adopting the eigenvalue-self-consistent evGW exchange-correlation kernel, for the calculation of the excited-state (μES) and excess dipole moments (Δμ), the latter ones being the changes of dipole amplitude between the ground and excited states (ES), in organic dyes. We compare the results obtained with wave-function methods [ADC(2), CC2, and CCSD], time-dependent density functional theory (TD-DFT), and BSE/evGW levels of theory. First, we compute the evolution of the dipole moments of the two lowest singlet excited states of 4-(dimethylamino)benzonitrile (DMABN) upon twisting of the amino group. Next, we use a set of 25 dyes having ES characters ranging from locally excited to charge transfer to determine both μES and Δμ. For DMABN our results show that BSE/evGW provides Δμ values closer to the CCSD reference and more consistent trends than TD-DFT. Moreover, a statistical analysis of both Δμ and μES for the set of 25 dyes shows that the BSE/evGW accuracy is comparable or sometimes slightly better than that of TD-M06-2X and TD-CAM-B3LYP, BSE/evGW outperforming TD-DFT in challenging cases (zwitterionic and cyanine transitions). Finally, the starting point dependency of BSE/evGW seems to be larger for Δμ, ES dipoles, and oscillator strengths than for transition energies.

Many-body GW calculations with very large scale polarizable environments made affordable: A fully ab initio QM/QM approach

We present a many-body GW formalism for quantum subsystems embedded in discrete polarizable environments containing up to several hundred thousand atoms described at a fully ab initio random phase approximation level. Our approach is based on a fragment approximation in the construction of the Green's function and independent-electron susceptibilities. Further, the environing fragments susceptibility matrices are reduced to a minimal but accurate representation preserving low order polarizability tensors through a constrained minimization scheme. This approach dramatically reduces the cost associated with inverting the Dyson equation for the screened Coulomb potential W, while preserving the description of short to long-range screening effects. The efficiency and accuracy of the present scheme is exemplified in the paradigmatic cases of fullerene bulk, surface, subsurface, and slabs with varying number of layers.

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.

Exploring Bethe-Salpeter Excited-State Dipoles: The Challenging Case of Increasingly Long Push-Pull Oligomers

The change of molecular dipole moment induced by photon absorption is key to interpret the measured optical spectra. Except for compact molecules, time-dependent density functional theory (TD-DFT) remains the only theory allowing to quickly predict excited-state dipoles (μ^ES), albeit with a strong dependency on the selected exchange-correlation functional. This Letter presents the first assessment of the performances of the many-body Green's function Bethe-Salpeter equation (BSE) formalism for the evaluation of the μ^ES. We explore increasingly long push-pull oligomers as they present an excited-state nature evolving with system size. This work shows that BSE's μ^ES do present the same evolution with oligomeric length as their CC2 and CCSD counterparts, with a dependency on the starting exchange-correlation functional that is strongly decreased as compared to TD-DFT. This Letter demonstrates that BSE is a valuable alternative to TD-DFT for properties related to the excited-state density and not only for transition energies and oscillator strengths.

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.

Assessing the Role of the Kohn-Sham Density in the Calculation of the Low-Lying Bethe-Salpeter Excitation Energies

We adopt the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation (BSE) to compute 57 excitation energies of a set of 37 molecules. By using the PBEh global hybrid functional and a self-consistent scheme on the eigenvalues in GW, we show a strong dependence of the BSE energy on the starting Kohn-Sham (KS) density functional. This arises from both the quasiparticle energies and the spatial localization of the frozen KS orbitals employed to compute the BSE. In order to address the arbitrariness in the mean field choice, we adopt an orbital-tuning scheme where the amount of Fock exchange, α, is tuned to impose that the KS HOMO matches the GW quasiparticle eigenvalue, thus fulfilling the ionization potential theorem in DFT. The performance of the proposed scheme yields excellent results and it is similar to M06-2X and PBEh with α = 75%, consistent with tuned values of α ranging between 60% and 80%.

Excited state potential energy surfaces of N-phenylpyrrole upon twisting: reference values and comparison between BSE/GW and TD-DFT

The puzzling case of the mixing between the charge transfer (CT) and local excited (LE) characters upon twisting of the geometry of N-phenylpyrrole (N-PP) is investigated considering the six low-lying singlet excited states (ES). The theoretical calculations of the potential energy surfaces (PES) have been performed for these states using a Coupled Cluster method accounting for the impact of the contributions from the triples, many-body Green's function GW and Bethe-Salpeter equation (BSE) formalisms, as well as Time-Dependent Density Functional Theory (TD-DFT) using various exchange-correlation functionals. Our findings confirm that the BSE formalism is more reliable than TD-DFT for close-lying ES with mixed CT/LE nature. More specifically, BSE/GW yields a more accurate evolution of the excited state PES than TD-DFT when compared to the reference coupled cluster values. BSE/GW PES curves also show negligible exchange-correlation functional starting point dependency in sharp contrast with their TD-DFT counterparts.

Connections between many-body perturbation and coupled-cluster theories

Here, we build on the works of Scuseria et al. [J. Chem. Phys. 129, 231101 (2008)] and Berkelbach [J. Chem. Phys. 149, 041103 (2018)] to show connections between the Bethe-Salpeter equation (BSE) formalism combined with the GW approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. In particular, we show how to recast the GW and Bethe-Salpeter equations as non-linear CC-like equations. Similitudes between BSE@GW and the similarity-transformed equation-of-motion CC method are also put forward. The present work allows us to easily transfer key developments and the general knowledge gathered in CC theory to many-body perturbation theory. In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the GW and BSE frameworks.

An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe–Salpeter equation calculations of molecules

The ab initio GW plus Bethe–Salpeter equation (GW-BSE, where G is the one particle Green's function and W is the screened Coulomb interaction) approach has emerged as a leading method for predicting excitations in both solids and molecules with a predictive power contingent upon several factors. Among these factors are the (1) generalized Kohn–Sham eigensystem used to construct the GW self-energy and to solve the BSE and (2) the efficacy and suitability of the Tamm–Dancoff approximation. Here, we present a detailed benchmark study of low-lying singlet excitations from a generalized Kohn–Sham (gKS) starting point based on an optimally tuned range-separated hybrid (OTRSH) functional. We show that the use of this gKS starting point with one-shot G0W0 and G0W0-BSE leads to the lowest mean absolute errors (MAEs) and mean signed errors (MSEs), with respect to high-accuracy reference values, demonstrated in the literature thus far for the ionization potentials of the GW100 benchmark set and for low-lying neutral excitations of Thiel’s set molecules in the gas phase, without the need for self-consistency. The MSEs and MAEs of one-shot G0W0-BSE@OTRSH excitation energies are comparable to or lower than those obtained with other functional starting points after self-consistency. Additionally, we compare these results with linear-response time-dependent density functional theory (TDDFT) calculations and find GW-BSE to be superior to TDDFT when calculations are based on the same exchange-correlation functional. This work demonstrates tuned range-separated hybrids used in combination with GW and GW-BSE can greatly suppress starting point dependence for molecules, leading to accuracy similar to that for higher-order wavefunction-based theories for molecules without the need for costlier iterations to self-consistency.

Unphysical discontinuities, intruder states and regularization in GW methods

By recasting the non-linear frequency-dependent GW quasiparticle equation into a linear eigenvalue problem, we explain the appearance of multiple solutions and unphysical discontinuities in various physical quantities computed within the GW approximation. Considering the GW self-energy as an effective Hamiltonian, it is shown that these issues are key signatures of strong correlation in the (N ± 1)-electron states and can be directly related to the intruder state problem. A simple and efficient regularization procedure inspired by the similarity renormalization group is proposed to avoid such issues and speed up the convergence of partially self-consistent GW calculations.

Static and dynamic Bethe-Salpeter equations in the T-matrix approximation

While the well-established GW approximation corresponds to a resummation of the direct ring diagrams and is particularly well suited for weakly correlated systems, the T-matrix approximation does sum ladder diagrams up to infinity and is supposedly more appropriate in the presence of strong correlation. Here, we derive and implement, for the first time, the static and dynamic Bethe-Salpeter equations when one considers T-matrix quasiparticle energies and a T-matrix-based kernel. The performance of the static scheme and its perturbative dynamical correction are assessed by computing the neutral excited states of molecular systems. A comparison with more conventional schemes as well as other wave function methods is also reported. Our results suggest that the T-matrix-based formalism performs best in few-electron systems where the electron density remains low.

The GW Miracle in Many-Body Perturbation Theory for the Ionization Potential of Molecules

We use the GW100 benchmark set to systematically judge the quality of several perturbation theories against high-level quantum chemistry methods. First of all, we revisit the reference CCSD(T) ionization potentials for this popular benchmark set and establish a revised set of CCSD(T) results. Then, for all of these 100 molecules, we calculate the HOMO energy within second and third-order perturbation theory (PT2 and PT3), and, GW as post-Hartree-Fock methods. We found GW to be the most accurate of these three approximations for the ionization potential, by far. Going beyond GW by adding more diagrams is a tedious and dangerous activity: We tried to complement GW with second-order exchange (SOX), with second-order screened exchange (SOSEX), with interacting electron-hole pairs (W_TDHF), and with a GW density-matrix (γ^GW). Only the γ^GW result has a positive impact. Finally using an improved hybrid functional for the non-interacting Green’s function, considering it as a cheap way to approximate self-consistency, the accuracy of the simplest GW approximation improves even more. We conclude that GW is a miracle: Its subtle balance makes GW both accurate and fast.

Scrutinizing GW-Based Methods Using the Hubbard Dimer

Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.

High-throughput virtual screening for organic electronics: a comparative study of alternative strategies

We present a review of the field of high-throughput virtual screening for organic electronics materials focusing on the sequence of methodological choices that determine each virtual screening protocol. These choices are present in all high-throughput virtual screenings and addressing them systematically will lead to optimised workflows and improve their applicability. We consider the range of properties that can be computed and illustrate how their accuracy can be determined depending on the quality and size of the experimental datasets. The approaches to generate candidates for virtual screening are also extremely varied and their relative strengths and weaknesses are discussed. The analysis of high-throughput virtual screening is almost never limited to the identification of top candidates and often new patterns and structure-property relations are the most interesting findings of such searches. The review reveals a very dynamic field constantly adapting to match an evolving landscape of applications, methodologies and datasets.

A theoretical study on solvatofluorochromic asymmetric thiazolothiazole (TTz) dyes using dielectric-dependent density functional theory

In this work, the excitation energies of asymmetric thiazolothizaole (TTz) dye molecules have been theoretically studied using dielectric-dependent density functional theory (DFT). In the dielectric-dependent DFT approach, the ratio (fraction) of the nonlocal Hartree exchange term incorporated into the DFT exchange-correlation functional is a system-dependent parameter, which is inversely proportional to the dielectric constant of the target material. The dielectric-dependent DFT method is closely related to the Coulomb hole and screened exchange (COHSEX) approximation in the GW method and therefore has been applied to crystalline systems with periodic boundary conditions, such as semiconductors and inorganic materials. By focusing on the solvatofluorochromic phenomena of asymmetric TTz dyes, we show that excitation energy calculations obtained from the dielectric-dependent DFT method can reproduce the corresponding experimental UV-vis absorption and emission spectra of dyes in solvents.

Exploring vibronic coupling in the benzene radical cation and anion with different levels of the GW approximation

The linear vibronic coupling constants of the benzene radical cation and anion have been obtained with different levels of the GW approximation, including G₀W₀, eigenvalue self-consistent GW, and quasiparticle self-consistent GW, as well as DFT with the following exchange-correlation functionals: BLYP, B3LYP, CAM-B3LYP, tuned CAM-B3LYP, and an IP-tuned CAM-B3LYP functional. The vibronic coupling constants were calculated numerically using the gradients of the eigenvalues of the degenerate HOMOs and LUMOs of the neutral benzene molecule for DFT, while the numerical gradients of the quasiparticle energies were used in the case of GW. The results were evaluated against those of high level wave function methods in the literature, and the approximate self-consistent GW methods and G₀W₀ with long-range corrected functionals were found to yield the best results on the whole.

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.

Spin-Conserved and Spin-Flip Optical Excitations from the Bethe-Salpeter Equation Formalism

Like adiabatic time-dependent density-functional theory (TD-DFT), the Bethe–Salpeter equation (BSE) formalism of many-body perturbation theory, in its static approximation, is “blind” to double (and higher) excitations, which are ubiquitous, for example, in conjugated molecules like polyenes. Here, we apply the spin-flip ansatz (which considers the lowest triplet state as the reference configuration instead of the singlet ground state) to the BSE formalism in order to access, in particular, double excitations. The present scheme is based on a spin-unrestricted version of the GW approximation employed to compute the charged excitations and screened Coulomb potential required for the BSE calculations. Dynamical corrections to the static BSE optical excitations are taken into account via an unrestricted generalization of our recently developed (renormalized) perturbative treatment. The performance of the present spin-flip BSE formalism is illustrated by computing excited-state energies of the beryllium atom, the hydrogen molecule at various bond lengths, and cyclobutadiene in its rectangular and square-planar geometries.

Cubic-Scaling All-Electron GW Calculations with a Separable Density-Fitting Space-Time Approach

We present an implementation of the GW space-time approach that allows cubic-scaling all-electron calculations with standard Gaussian basis sets without exploiting any localization or sparsity considerations. The independent-electron susceptibility is constructed in a time representation over a nonuniform distribution of real-space locations {r_k} optimized within a separable resolution-of-the-identity framework to reproduce standard Coulomb-fitting calculations with meV accuracy. The compactness of the obtained {r_k} distribution leads to a crossover with the standard Coulomb-fitting scheme for system sizes below a few hundred electrons. The needed analytic continuation follows a recent approach that requires the continuation of the screened Coulomb potential rather than the much more structured self-energy. The present scheme is benchmarked over large molecular sets, and scaling properties are demonstrated on a family of defected hexagonal boron-nitride flakes containing up to 6000 electrons.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Dynamical correction to the Bethe-Salpeter equation beyond the plasmon-pole approximation

The Bethe-Salpeter equation (BSE) formalism is a computationally affordable method for the calculation of accurate optical excitation energies in molecular systems. Similar to the ubiquitous adiabatic approximation of time-dependent density-functional theory, the static approximation, which substitutes a dynamical (i.e., frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the BSE formalism. Here, going beyond the static approximation, we compute the dynamical correction of the electron-hole screening for molecular excitation energies, thanks to a renormalized first-order perturbative correction to the static BSE excitation energies. The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation. Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing one to assess the clear improvement brought by the dynamical correction for both singlet and triplet optical transitions.

The Bethe-Salpeter Equation Formalism: From Physics to Chemistry

The Bethe-Salpeter equation (BSE) formalism is steadily asserting itself as a new efficient and accurate tool in the ensemble of computational methods available to chemists in order to predict optical excitations in molecular systems. In particular, the combination of the so-called GW approximation, giving access to reliable ionization energies and electron affinities, and the BSE formalism, able to model UV/vis spectra, has shown to provide accurate singlet excitation energies with a typical error of 0.1-0.3 eV. With a similar computational cost as time-dependent density-functional theory (TD-DFT), BSE is able to provide an accuracy on par with the most accurate global and range-separated hybrid functionals without the unsettling choice of the exchange-correlation functional, resolving further known issues (e.g., charge-transfer excitations). In this Perspective, we provide a historical overview of BSE, with a particular focus on its condensed-matter roots. We also propose a critical review of its strengths and weaknesses in different chemical situations.

Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies

The combination of the many-body Green's function GW approximation and the Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems. The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT). Here, we study the topology of the ground-state potential energy surfaces (PESs) of several diatomic molecules near their equilibrium bond length. Using comparisons with state-of-art computational approaches (CC3), we show that ACFDT@BSE is surprisingly accurate and can even compete with lower-order coupled cluster methods (CC2 and CCSD) in terms of total energies and equilibrium bond distances for the considered systems. However, we sometimes observe unphysical irregularities on the ground-state PES in relation with difficulties in the identification of a few GW quasiparticle energies.

Ab Initio Many-Body Perturbation Theory Calculations of the Electronic and Optical Properties of Cyclometalated Ir(III) Complexes

Cyclometalated Ir(III) compounds are the preferred choice as organic emitters in organic light-emitting diodes. In practice, the presence of the transition metal surrounded by carefully designed ligands allows fine-tuning of the emission frequency as well as good efficiency of the device. To support the development of new compounds, experimental measurements are generally compared with absorption and emission spectra obtained from ab initio calculations. The standard approach for these calculations is time-dependent density functional theory (TDDFT) with a hybrid exchange-correlation functional like B3LYP. Because of the size of these compounds, the application of more complex quantum chemistry approaches can be challenging. In this work, we used many-body perturbation theory approaches, in particular the GW approximation with the Bethe-Salpeter equation (BSE) implemented in Gaussian basis sets, to calculate the quasiparticle properties and the absorption spectra of six cyclometalated Ir(III) complexes, going beyond TDDFT. In the presented results, we compared standard TDDFT simulations with BSE calculations performed on top of perturbative G₀W₀ and accounting for eigenvalue self-consistency. Moreover, in order to investigate in detail the effect of the DFT starting point, we concentrated on Ir(ppy)₃ and performed GW-BSE simulations starting from different DFT exchange-correlation potentials.

Practical GW scheme for electronic structure of 3d-transition-metal monoxide anions: ScO-, TiO-, CuO-, and ZnO. J Chem Phys 2019;151:134305. [PMID: 31594362 DOI: 10.1063/1.5118671]

The GW approximation to many-body perturbation theory is a reliable tool for describing charged electronic excitations, and it has been successfully applied to a wide range of extended systems for several decades using a plane-wave basis. However, the GW approximation has been used to test limited spectral properties of a limited set of finite systems (e.g., frontier orbital energies of closed-shell sp molecules) only for about a decade using a local-orbital basis. Here, we calculate the quasiparticle spectra of closed- and open-shell molecular anions with partially and completely filled 3d shells (shallow and deep 3d states, respectively), ScO^-, TiO^-, CuO^-, and ZnO^-, using various levels of GW theory, and compare them to experiments to evaluate the performance of the GW approximation on the electronic structure of small molecules containing 3d transition metals. We find that the G-only eigenvalue self-consistent GW scheme with W fixed to the PBE level (G_nW₀@PBE), which gives the best compromise between accuracy and efficiency for solids, also gives good results for both localized (d) and delocalized (sp) states of 3d-transition-metal oxide molecules. The success of G_nW₀@PBE in predicting electronic excitations in these systems reasonably well is likely due to the fortuitous cancellation effect between the overscreening of the Coulomb interaction by PBE and the underscreening by the neglect of vertex corrections. Together with the absence of the self-consistent field convergence error (e.g., spin contamination in open-shell systems) and the GW multisolution issue, the G_nW₀@PBE scheme gives the possibility to predict the electronic structure of complex real systems (e.g., molecule-solid and sp-d hybrid systems) accurately and efficiently.

Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA

We explore a separable resolution-of-the-identity (RI) formalism built on quadratures over limited sets of real-space points designed for all-electron calculations. Our implementation preserves, in particular, the use of common atomic orbitals and their related auxiliary basis sets. The setup of the present density fitting scheme, i.e., the calculation of the system specific quadrature weights, scales cubically with respect to the system size. Extensive accuracy tests are presented for the Fock exchange and MP2 correlation energies. We finally demonstrate random phase approximation (RPA) correlation energy calculations with a scaling that is cubic in terms of operations, quadratic in memory, with a small crossover with respect to our standard RI-RPA implementation.

On-surface light-induced generation of higher acenes and elucidation of their open-shell character

Acenes are an important class of polycyclic aromatic hydrocarbons which have recently gained exceptional attention due to their potential as functional organic semiconductors. Fundamentally, they are important systems to study the convergence of physico-chemical properties of all-carbon sp2-frameworks in the one-dimensional limit; and by virtue of having a zigzag edge topology they also provide a fertile playground to explore magnetism in graphenic nanostructures. The study of larger acenes is thus imperative from both a fundamental and applied perspective, but their synthesis via traditional solution-chemistry route is hindered by their poor solubility and high reactivity. Here, we demonstrate the on-surface formation of heptacene and nonacene, via visible-light-induced photo-dissociation of α-bisdiketone precursors on an Au(111) substrate under ultra-high vacuum conditions. Through combined scanning tunneling microscopy/spectroscopy and non-contact atomic force microscopy investigations, together with state-of-the-art first principles calculations, we provide insight into the chemical and electronic structure of these elusive compounds.

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.

Screening mixing GW/Bethe-Salpeter approach for triplet states of organic molecules

We recently proposed a screening mixing many-body ansatz (Ziaei and Bredow 2017 Phys. Rev. B 96 195115) to decrease the typical overestimation of the Hartree-Fock based GW/Bethe-Salpeter equation (BSE) singlet-singlet excitation energies in molecular systems. Now we have evaluated the accuracy of the proposed scheme for triplet states of a set of 20 organic molecules known as the Thiel set. We show that by mixing different screenings into GW and BSE calculated within random phase approximation (in order to ensure best gap and an optimal exciton binding energy), the total mean absolute error of 0.59 eV in the standard Hartree-Fock based eigenvalue GW/BSE approach is reduced to 0.26 eV for 63 triplet states. We further demonstrate that the quasi-particle self-consistent GW/BSE approach in which orbitals and energies are updated in the Green's function and the dynamically screened interaction mostly and considerably underestimates the excitation energies as shown for a few molecules.

The Bethe-Salpeter formalism with polarisable continuum embedding: reconciling linear-response and state-specific features

The Bethe-Salpeter equation (BSE) formalism has been recently shown to be a valuable alternative to time-dependent density functional theory (TD-DFT) with the same computing time scaling with system size. In particular, problematic transitions for TD-DFT such as charge-transfer, Rydberg and cyanine-like excitations were shown to be accurately described with BSE. We demonstrate here that combining the BSE formalism with the polarisable continuum model (PCM) allows us to include simultaneously linear-response and state-specific contributions to solvatochromism. This is confirmed by exploring transitions of various natures (local, charge-transfer, etc.) in a series of solvated molecules (acrolein, indigo, p-nitro-aniline, donor-acceptor complexes, etc.) for which we compare BSE solvatochromic shifts to those obtained by linear-response and state-specific TD-DFT implementations. Such a remarkable and unique feature is particularly valuable for the study of solvent effects on excitations presenting a hybrid localised/charge-transfer character.