Electronic Excitations in Push–Pull Oligomers and Their Complexes with Fullerene from Many-Body Green’s Functions Theory with Polarizable Embedding

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.

Quantum-Quantum and Quantum-Quantum-Classical Schemes for Near-Gap Excitations with Projection-Based-Embedded GW-Bethe-Salpeter Equation

We present quantum-quantum and quantum-quantum-classical schemes based on many-body Green's functions theory in the GW approximation with the Bethe-Salpeter equation (GW-BSE) employing projection-based-embedding (PbE). Such approaches allow defining active and inactive subsystems of larger, complex molecular systems, with only the smaller active subsystem being explicitly treated by GW-BSE offering significant computational advantages. However, as PbE can modify the single-particle states in the Kohn-Sham (KS) ground state calculation and screening effects from the inactive region are not automatically included in GW-BSE, results from such PbE-GW-BSE calculations can deviate from a full-system reference. Here, we scrutinize in detail, e.g., the individual and combined effects of different choices of active regions, the influence of omitting the screening from the inactive region, and strategies for basis set truncation on frontier orbital and near-gap electron-hole excitation energies. As prototypical systems, we consider a diketopyrrolopyrrole bicyclic ring including side-chains, a polarity-sensitive dye (prodan) in aqueous environment, and a π-stacked dimer of benzene and tetracyanoethylene in water, respectively, covering a variety of excitation characters in molecular systems with complex chemical environments and photoinduced processes. Our results suggest that to obtain agreement of approximately 0.1 eV between near-gap excitation energies from embedded and full calculations, the active region should be chosen based on the Mulliken population of the full highest-occupied molecular orbital and that careful benchmarking should be done on the KS level before the actual GW-BSE steps when basis set truncation is used. We find that PbE-GW-BSE offers significant reductions in computation times and, more importantly, memory requirements, making calculations for considerably larger systems tractable.

Electronic Couplings and Conversion Dynamics between Localized and Charge Transfer Excitations from Many-Body Green's Functions Theory

We investigate the determination of electronic coupling between localized excitations (LEs) and charge-transfer (CT) excitations based on many-body Green's functions theory in the GW approximation with the Bethe-Salpeter equation (GW-BSE). Using a small molecule dimer system, we first study the influence of different diabatization methods, as well as different model choices within GW-BSE, such as the self-energy models or different levels of self-consistency, and find that these choices affect the LE-CT couplings only minimally. We then consider a large-scale low-donor morphology formed from rubrene and fullerene and evaluate the LE-CT couplings based on coupled GW-BSE-molecular mechanics calculations. For these disordered systems of bulky molecules, we observe differences in the couplings based on the Edmiston-Ruedenberg diabatization compared to the more approximate Generalize Mulliken-Hush and fragment charge difference diabatization formalisms. In a kinetic model for the conversion between LE and CT states, these differences affect the details of state populations in an intermediate time scale but not the final populations.

Elucidating Solvatochromic Shifts in Two-Dimensional Photocatalysts by Solving the Bethe-Salpeter Equation Coupled with Implicit Solvation Method

Many studies have focused on tailoring the photophysical properties of two-dimensional (2D) materials for photocatalytic (PC) or photoelectrochemical (PEC) applications. To understand the optical properties of 2D materials in solution, we established a computational method that combined the Bethe-Salpeter equation (BSE) calculations with our GW-GPE method, allowing for GW/BSE-level calculations with implicit solvation described using the generalized Poisson equation (GPE). We applied this method to MoS2, phosphorene (PP), and g-C3N4 and found that when the solvent dielectric increased, it reduced the exciton binding energy and quasiparticle bandgap, resulting in almost no solvatochromic shift in the excitonic peaks of MoS2 and PP, which is consistent with previous experiments. However, our calculations predicted that the solvent dielectric had a significant impact on the excitonic properties of g-C3N4, exhibiting a large solvatochromic shift. We expect that our GW/BSE-GPE method will offer insights into the design of 2D materials for PC and PEC applications.

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.

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.

Quasiparticle Self-Consistent GW-Bethe-Salpeter Equation Calculations for Large Chromophoric Systems

The GW-Bethe–Salpeter equation (BSE) method is promising for calculating the low-lying excitonic states of molecular systems. However, so far it has only been applied to rather small molecules and in the commonly implemented diagonal approximations to the electronic self-energy, it depends on a mean-field starting point. We describe here an implementation of the self-consistent and starting-point-independent quasiparticle self-consistent (qsGW)-BSE approach, which is suitable for calculations on large molecules. We herein show that eigenvalue-only self-consistency can lead to an unfaithful description of some excitonic states for chlorophyll dimers while the qsGW-BSE vertical excitation energies (VEEs) are in excellent agreement with spectroscopic experiments for chlorophyll monomers and dimers measured in the gas phase. Furthermore, VEEs from time-dependent density functional theory calculations tend to disagree with experimental values and using different range-separated hybrid (RSH) kernels does change the VEEs by up to 0.5 eV. We use the new qsGW-BSE implementation to calculate the lowest excitation energies of the six chromophores of the photosystem II (PSII) reaction center (RC) with nearly 2000 correlated electrons. Using more than 11,000 (6000) basis functions, the calculation could be completed in less than 5 (2) days on a single modern compute node. In agreement with previous TD-DFT calculations using RSH kernels on models that also do not include environmental effects, our qsGW-BSE calculations only yield states with local characters in the low-energy spectrum of the hexameric complex. Earlier works with RSH kernels have demonstrated that the protein environment facilitates the experimentally observed interchromophoric charge transfer. Therefore, future research will need to combine correlation effects beyond TD-DFT with an explicit treatment of environmental electrostatics.

Implicit Solvation Methods for Catalysis at Electrified Interfaces

Implicit solvation is an effective, highly coarse-grained approach in atomic-scale simulations to account for a surrounding liquid electrolyte on the level of a continuous polarizable medium. Originating in molecular chemistry with finite solutes, implicit solvation techniques are now increasingly used in the context of first-principles modeling of electrochemistry and electrocatalysis at extended (often metallic) electrodes. The prevalent ansatz to model the latter electrodes and the reactive surface chemistry at them through slabs in periodic boundary condition supercells brings its specific challenges. Foremost this concerns the difficulty of describing the entire double layer forming at the electrified solid–liquid interface (SLI) within supercell sizes tractable by commonly employed density functional theory (DFT). We review liquid solvation methodology from this specific application angle, highlighting in particular its use in the widespread ab initio thermodynamics approach to surface catalysis. Notably, implicit solvation can be employed to mimic a polarization of the electrode’s electronic density under the applied potential and the concomitant capacitive charging of the entire double layer beyond the limitations of the employed DFT supercell. Most critical for continuing advances of this effective methodology for the SLI context is the lack of pertinent (experimental or high-level theoretical) reference data needed for parametrization.

Fragment-Based Excited-State Calculations Using the GW Approximation and the Bethe-Salpeter Equation

Herein, we present a fragment-based approach for calculating the charged and neutral excited states in molecular systems, based on the many-body Green's function method within the GW approximation and the Bethe-Salpeter equation (BSE). The implementation relies on the many-body expansion of the total irreducible polarizability on the basis of fragment molecular orbitals. The GW quasi-particle energies in complex molecular environments are obtained by the GW calculation for the target fragment plus induced polarization contributions of the surrounding fragments at the static Coulomb-hole plus screened exchange level. In addition, we develop a large-scale GW/BSE method for calculating the delocalized excited states of molecular aggregates, based on the fragment molecular orbital method and the exciton model. The accuracy of fragment-based GW and GW/BSE methods was evaluated on molecular clusters and molecular crystals. We found that the accuracy of the total irreducible polarizability can be improved systematically by including two-body correction terms, and the fragment-based calculations can reasonably reproduce the results of the corresponding unfragmented calculations with a relative error of less than 100 meV. The proposed approach enables efficient excited-state calculations for large molecular systems with reasonable accuracy.

Atomistic Polarizable Embeddings: Energy, Dynamics, Spectroscopy, and Reactivity

The computational modeling of realistic extended systems, relevant in, e.g., Chemistry and Biophysics, is a fundamental problem of paramount importance in contemporary research. Enzymatic catalysis and photoinduced processes in pigment-protein complexes are typical problems targeted by computer-aided approaches, to complement experiments as interpretative tools at a molecular scale. The daunting complexity of this task lies in between the opposite stringent requirements of results' reliability for structural/dynamical properties and related intermolecular interactions, and a mandatory principle of realism in the modeling strategy. Therefore, in practice, a truly realistic computational model of a biologically relevant system can easily fail to meet the accuracy requirement, in order to balance the excessive computational cost necessary to reach the desired precision.To address such an "accuracy vs reality" dualistic requirement, mixed quantum mechanics/classical mechanics approaches within Atomistic (i.e., preserving the discrete particle configuration) Polarizable Embeddings (QM/APEs) methods have been proposed over the years. In this Account, we review recent developments in the design and application of general QM/APE methods, targeting situations where a local intrinsically quantum behavior is coupled to a large molecular system (i.e., an environment), often involving processes with different dynamical time scales, in order to avoid brute-force, unpractical quantum chemistry calculations on the complete system.In the first place, our interest is devoted to the available APEs models presently implemented in computational software, highlighting the quantum chemistry methods that can be used to treat the QM subsystem. We review the coupling strategy between the QM subsystem and the APE, which requires to examine the way the QM/MM mutual interactions are accounted for and how the polarization of the classical environment is considered with respect to (wrt) the quantum variables. Because of the need of reliable molecular and macromolecular structures, a pivotal aspect to address here is the handling of the system dynamics (i.e., gradients wrt nuclear positions are required), especially for large molecular assemblies composed by an overwhelming number of atoms, exploring many conformations on a complex energy landscape.Alongside, we highlight our views on the necessary steps to take toward more accurate general-purposes and transferable explicit embeddings. The main objective to achieve here is to design a more physically grounded multiscale approach. To do so, one should apply advanced new generation classical models to account for refined induction effects that are able to (i) improve the quality of QM/MM interaction energies; (ii) enhance transferability by avoiding the compulsory partial (or total) reparameterization of the classical model. Moreover, the extension of recent developments originating from the field of advanced classical molecular dynamics (MD) to the realm of QM/APE methods is a key direction to improve both speed and efficiency for the phase space exploration of systems of growing size and complexity.Lastly, we point out specific research topics where an advanced QM/APE dynamics can certainly shed some light. For example, we discuss chemical reactions in "harsh" environments and the case of spectroscopic theoretical modeling where the inclusion of refined environment effects is often mandatory.

Excited-State Geometry Optimization of Small Molecules with Many-Body Green's Functions Theory

We present a benchmark study of gas phase geometry optimizations in the excited states of carbon monoxide, acetone, acrolein, and methylenecyclopropene using many-body Green’s functions theory within the GW approximation and the Bethe–Salpeter equation (BSE) employing numerical gradients. We scrutinize the influence of several typical approximations in the GW-BSE framework; we used one-shot G₀W₀ or eigenvalue self-consistent evGW, employing a fully analytic approach or plasmon-pole model for the frequency dependence of the electron self-energy, or performing the BSE step within the Tamm–Dancoff approximation. The obtained geometries are compared to reference results from multireference perturbation theory (CASPT2), variational Monte Carlo (VMC) method, second-order approximate coupled cluster (CC2) method, and time-dependent density-functional theory (TDDFT). We find overall a good agreement of the structural parameters optimized with the GW-BSE calculations with CASPT2, with an average relative error of around 1% for the G₀W₀ and 1.5% for the evGW variants based on a PBE0 ground state, respectively, while the other approximations have negligible influence. The relative errors are also smaller than those for CC2 and TDDFT with different functionals and only larger than VMC, indicating that the GW-BSE method does not only yield excitation energies but also geometries in good agreement with established higher-order wave function methods.

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.

Excited-state electronic structure of molecules using many-body Green's functions: Quasiparticles and electron-hole excitations with VOTCA-XTP

We present the open-source VOTCA-XTP software for the calculation of the excited-state electronic structure of molecules using many-body Green's function theory in the GW approximation with the Bethe-Salpeter equation (BSE). This work provides a summary of the underlying theory and discusses the details of its implementation based on Gaussian orbitals, including resolution-of-identity techniques and different approaches to the frequency integration of the self-energy or acceleration by offloading compute-intensive matrix operations using graphics processing units in a hybrid OpenMP/Cuda scheme. A distinctive feature of VOTCA-XTP is the capability to couple the calculation of electronic excitations to a classical polarizable environment on an atomistic level in a coupled quantum- and molecular-mechanics (QM/MM) scheme, where a complex morphology can be imported from Molecular Dynamics simulations. The capabilities and limitations of the GW-BSE implementation are illustrated with two examples. First, we study the dependence of optically active electron-hole excitations in a series of diketopyrrolopyrrole-based oligomers on molecular-architecture modifications and the number of repeat units. Second, we use the GW-BSE/MM setup to investigate the effect of polarization on localized and intermolecular charge-transfer excited states in morphologies of low-donor content rubrene-fullerene mixtures. These showcases demonstrate that our implementation currently allows us to treat systems with up to 2500 basis functions on regular shared-memory workstations, providing accurate descriptions of quasiparticle and coupled electron-hole excited states of various characters on an equal footing.

Charge-transfer excited states in the donor/acceptor interface from large-scale GW calculations

Predicting the charge-transfer (CT) excited states across the donor/acceptor (D/A) interface is essential for understanding the charge photogeneration process in an organic solar cell. Here, we present a fragment-based GW implementation that can be applied to a D/A interface structure and thus enables accurate determination of the CT states. The implementation is based on the fragmentation approximation of the polarization function and the combined GW and Coulomb-hole plus screened exchange approximations for self-energies. The fragment-based GW is demonstrated by application to the pentacene/C₆₀ interface structure containing more than 2000 atoms. The CT excitation energies were estimated from the quasiparticle energies and electron-hole screened Coulomb interactions; the computed energies are in reasonable agreement with experimental estimates from the external quantum efficiency measurements. We highlight the impact of the induced polarization effects on the electron-hole energetics. The proposed fragment-based GW method offers a first-principles tool to compute the quasiparticle energies and electronic excitation energies of organic materials.

Quantifying the Plasmonic Character of Optical Excitations in a Molecular J-Aggregate

The definition of plasmon at the microscopic scale is far from being understood. Yet, it is very important to recognize plasmonic features in optical excitations, as they can inspire new applications and trigger new discoveries by analogy with the rich phenomenology of metal nanoparticle plasmons. Recently, the concepts of plasmonicity index and the generalized plasmonicity index (GPI) have been devised as computational tools to quantify the plasmonic nature of optical excitations. The question may arise whether any strong absorption band, possibly with some sort of collective character in its microscopic origin, shares the status of plasmon. Here we demonstrate that this is not always the case, by considering a well-known class of systems represented by J-aggregates molecular crystals, characterized by the intense J band of absorption. By means of first-principles simulations, based on a many-body perturbation theory formalism, we investigate the optical properties of a J-aggregate made of push–pull organic dyes. We show that the effect of aggregation is to lower the GPI associated with the J-band with respect to the isolated dye one, which corresponds to a nonplasmonic character of the electronic excitations. In order to rationalize our finding, we then propose a simplified one-dimensional theoretical model of the J-aggregate. A useful microscopic picture of what discriminates a collective molecular crystal excitation from a plasmon is eventually obtained.

Electronic Excitations in Complex Molecular Environments: Many-Body Green's Functions Theory in VOTCA-XTP

Many-body Green's functions theory within the GW approximation and the Bethe-Salpeter Equation (BSE) is implemented in the open-source VOTCA-XTP software, aiming at the calculation of electronically excited states in complex molecular environments. Based on Gaussian-type atomic orbitals and making use of resolution of identity techniques, the code is designed specifically for nonperiodic systems. Application to a small molecule reference set successfully validates the methodology and its implementation for a variety of excitation types covering an energy range from 2 to 8 eV in single molecules. Further, embedding each GW-BSE calculation into an atomistically resolved surrounding, typically obtained from Molecular Dynamics, accounts for effects originating from local fields and polarization. Using aqueous DNA as a prototypical system, different levels of electrostatic coupling between the regions in this GW-BSE/MM setup are demonstrated. Particular attention is paid to charge-transfer (CT) excitations in adenine base pairs. It is found that their energy is extremely sensitive to the specific environment and to polarization effects. The calculated redshift of the CT excitation energy compared to a nucelobase dimer treated in vacuum is of the order of 1 eV, which matches expectations from experimental data. Predicted lowest CT energies are below that of a single nucleobase excitation, indicating the possibility of an initial (fast) decay of such an UV excited state into a binucleobase CT exciton. The results show that VOTCA-XTP's GW-BSE/MM is a powerful tool to study a wide range of types of electronic excitations in complex molecular environments.

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.

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.

Molecular Model of a Quantum Dot Beyond the Constant Interaction Approximation

We present a physically intuitive model of molecular quantum dots beyond the constant interaction approximation. It accurately describes their charging behavior and allows the extraction of important molecular properties that are otherwise experimentally inaccessible. The model is applied to data recorded with a noncontact atomic force microscope on three different molecules that act as a quantum dot when attached to the microscope tip. The results are in excellent agreement with first-principles simulations.

Getting excited: challenges in quantum-classical studies of excitons in polymeric systems

A combination of classical molecular dynamics (MM/MD) and quantum chemical calculations based on the density functional theory (DFT) was performed to describe the conformational properties of diphenylethyne (DPE), methylated-DPE and poly para phenylene ethynylene (PPE). DFT calculations were employed to improve and develop force field parameters for MM/MD simulations. Many-body Green's function theory within the GW approximation and the Bethe-Salpeter (GW-BSE) equation were utilized to describe the excited states of the systems. The reliability of the excitation energies based on the MM/MD conformations was examined and compared to the excitation energies from DFT conformations. The results show an overall agreement between the optical excitations based on MM/MD conformations and DFT conformations. This allows for the calculation of excitation energies based on MM/MD conformations.

The Bethe–Salpeter equation in chemistry: relations with TD-DFT, applications and challenges

We review the Bethe–Salpeter formalism and analyze its performances for the calculation of the excited state properties of molecular systems.

Hot and Cold Charge-Transfer Mechanisms in Organic Photovoltaics: Insights into the Excited States of Donor/Acceptor Interfaces

The evolution of the excited-state manifold in organic D/A aggregates (e.g., the prototypical P3HT/PCBM) is investigated through a bottom-up approach via first-principles calculations. We show how the excited-state energies, the charge transfer (CT) states, and the electron-hole density distributions are strongly influenced by the size, the orientation, and the position (i.e., on-top versus on-edge phases) of P3HT/PCBM domains. We discuss how the structural order influences the excited-state electronic structure, providing an atomistic interpretation of the photophysics of organic blends. We show how the simultaneous presence of on-top and on-edge phases does not alter the optical absorption spectrum of the blend but does affect the photophysics. Photovoltaic processes such as (i) the simultaneous charge generation obtained from hot and cold excitations, (ii) the instantaneous and delayed charge separation, and (iii) the pump-push-probe charge generation can be interpreted based on our study.

Intermolecular Singlet and Triplet Exciton Transfer Integrals from Many-Body Green's Functions Theory

A general approach to determine orientation and distance-dependent effective intermolecular exciton transfer integrals from many-body Green's functions theory is presented. On the basis of the GW approximation and the Bethe-Salpeter equation (BSE), a projection technique is employed to obtain the excitonic coupling by forming the expectation value of a supramolecular BSE Hamiltonian with electron-hole wave functions for excitations localized on two separated chromophores. Within this approach, accounting for the effects of coupling mediated by intermolecular charge transfer (CT) excitations is possible via perturbation theory or a reduction technique. Application to model configurations of pyrene dimers shows an accurate description of short-range exchange and long-range Coulomb interactions for the coupling of singlet and triplet excitons. Computational parameters, such as the choice of the exchange-correlation functional in the density-functional theory (DFT) calculations that underly the GW-BSE steps and the convergence with the number of included CT excitations, are scrutinized. Finally, an optimal strategy is derived for simulations of full large-scale morphologies by benchmarking various approximations using pairs of dicyanovinyl end-capped oligothiophenes (DCV5T), which are used as donor material in state-of-the-art organic solar cells.

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.

Combining the GW formalism with the polarizable continuum model: A state-specific non-equilibrium approach

We have implemented the polarizable continuum model within the framework of the many-body Green's function GW formalism for the calculation of electron addition and removal energies in solution. The present formalism includes both ground-state and non-equilibrium polarization effects. In addition, the polarization energies are state-specific, allowing to obtain the bath-induced renormalisation energy of all occupied and virtual energy levels. Our implementation is validated by comparisons with ΔSCF calculations performed at both the density functional theory and coupled-cluster single and double levels for solvated nucleobases. The present study opens the way to GW and Bethe-Salpeter calculations in disordered condensed phases of interest in organic optoelectronics, wet chemistry, and biology.

Benchmark of Bethe-Salpeter for Triplet Excited-States

We have evaluated the accuracy of the Bethe-Salpeter singlet-triplet transition energies as well as singlet-triplet and triplet-triplet splittings for 20 organic molecules, using as reference the CC3 values determined by Thiel and co-workers with both the TZVP and aug-cc-pVTZ atomic basis sets. Our excitation energies are obtained on the basis of GW quasiparticle energy levels that are self-consistently converged with respect to the starting DFT eigenvalues. In its current form, BSE/GW is often unable to provide a balanced description of both singlet and triplet excited-states. While the singlet-singlet and triplet-triplet energy separations are obtained accurately, triplets are located too close in energy from the ground-state, by typically -0.55 eV when using standard functionals to generate the starting eigenstates. Applying the Tamm-Dancoff approximation upshifts the BSE triplet energies and allows reducing this error to ca. -0.40 eV, while using M06-HF eigenstates allows a further increase and hence a reduction of the error for triplet states, but at the cost of larger errors for the singlet excited-states. At this stage, the most accurate TD-DFT estimates therefore remain competitive for computing singlet-triplet transition energies. Indeed, with M06-2X, irrespective of the application or not of the Tamm-Dancoff approximation and of the selected atomic basis set, the deviations obtained with TD-DFT are rather small.

Bethe-Salpeter study of cationic dyes: Comparisons with ADC(2) and TD-DFT

We present a theoretical investigation of the excited-state properties of a large series of structurally diverse arylcarbonium derivatives that are known to be challenging for theoretical models. More specifically, we compare the pros and cons of TD-DFT (TD-M06-2X), ADC(2), and BSE/GW approaches for a large panel of compounds, using two different solvent models. Both 0-0 and vertical transition energies are considered and compared to the experimental values. All approaches reasonably reproduce the auxochromic and acidochromic shifts, although in most cases both TD-DFT and BSE/GW return larger correlation with experimental values than ADC(2) for these shifts. In contrast, the absolute transition energies obtained with ADC(2) tend to be closer to the measurements, TD-DFT using the M06-2X functional largely overestimating the experimental references (by ca. 0.5 eV), and BSE/GW providing intermediate values. In addition, we show that the selected solvent model has a significant impact on the results, the corrected linear-response approach providing larger transition energies than its linear-response counterpart.

Theoretical description of protein field effects on electronic excitations of biological chromophores

Photoinitiated phenomena play a crucial role in many living organisms. Plants, algae, and bacteria absorb sunlight to perform photosynthesis, and convert water and carbon dioxide into molecular oxygen and carbohydrates, thus forming the basis for life on Earth. The vision of vertebrates is accomplished in the eye by a protein called rhodopsin, which upon photon absorption performs an ultrafast isomerisation of the retinal chromophore, triggering the signal cascade. Many other biological functions start with the photoexcitation of a protein-embedded pigment, followed by complex processes comprising, for example, electron or excitation energy transfer in photosynthetic complexes. The optical properties of chromophores in living systems are strongly dependent on the interaction with the surrounding environment (nearby protein residues, membrane, water), and the complexity of such interplay is, in most cases, at the origin of the functional diversity of the photoactive proteins. The specific interactions with the environment often lead to a significant shift of the chromophore excitation energies, compared with their absorption in solution or gas phase. The investigation of the optical response of chromophores is generally not straightforward, from both experimental and theoretical standpoints; this is due to the difficulty in understanding diverse behaviours and effects, occurring at different scales, with a single technique. In particular, the role played by ab initio calculations in assisting and guiding experiments, as well as in understanding the physics of photoactive proteins, is fundamental. At the same time, owing to the large size of the systems, more approximate strategies which take into account the environmental effects on the absorption spectra are also of paramount importance. Here we review the recent advances in the first-principle description of electronic and optical properties of biological chromophores embedded in a protein environment. We show their applications on paradigmatic systems, such as the light-harvesting complexes, rhodopsin and green fluorescent protein, emphasising the theoretical frameworks which are of common use in solid state physics, and emerging as promising tools for biomolecular systems.

Combining the Many-Body GW Formalism with Classical Polarizable Models: Insights on the Electronic Structure of Molecular Solids

We present an original hybrid QM/MM scheme merging the many-body Green's function GW formalism with classical discrete polarizable models and its application to the paradigmatic case of a pentacene crystal. Our calculated transport gap is found to be in excellent agreement with reference periodic bulk GW calculations, together with properly parametrized classical microelectrostatic calculations, and with photoionization measurements at crystal surfaces. More importantly, we prove that the gap is insensitive to the partitioning of pentacene molecules in QM and MM subsystems, as a result of the mutual compensation of quantum and classical polarizabilities, clarifying the relation between polarization energy and delocalization. The proposed hybrid method offers a computationally attractive strategy to compute the full spectrum of charged excitations in complex molecular environments, accounting for both QM and MM contributions to the polarization energy, a crucial requirement in the limit of large QM subsystems.

0-0 Energies Using Hybrid Schemes: Benchmarks of TD-DFT, CIS(D), ADC(2), CC2, and BSE/GW formalisms for 80 Real-Life Compounds

The 0-0 energies of 80 medium and large molecules have been computed with a large panel of theoretical formalisms. We have used an approach computationally tractable for large molecules, that is, the structural and vibrational parameters are obtained with TD-DFT, the solvent effects are accounted for with the PCM model, whereas the total and transition energies have been determined with TD-DFT and with five wave function approaches accounting for contributions from double excitations, namely, CIS(D), ADC(2), CC2, SCS-CC2, and SOS-CC2, as well as Green's function based BSE/GW approach. Atomic basis sets including diffuse functions have been systematically applied, and several variations of the PCM have been evaluated. Using solvent corrections obtained with corrected linear-response approach, we found that three schemes, namely, ADC(2), CC2, and BSE/GW allow one to reach a mean absolute deviation smaller than 0.15 eV compared to the measurements, the two former yielding slightly better correlation with experiments than the latter. CIS(D), SCS-CC2, and SOS-CC2 provide significantly larger deviations, though the latter approach delivers highly consistent transition energies. In addition, we show that (i) ADC(2) and CC2 values are extremely close to each other but for systems absorbing at low energies; (ii) the linear-response PCM scheme tends to overestimate solvation effects; and that (iii) the average impact of nonequilibrium correction on 0-0 energies is negligible.

Benchmarking the Bethe-Salpeter Formalism on a Standard Organic Molecular Set

We perform benchmark calculations of the Bethe-Salpeter vertical excitation energies for the set of 28 molecules constituting the well-known Thiel’s set, complemented by a series of small molecules representative of the dye chemistry field. We show that Bethe-Salpeter calculations based on a molecular orbital energy spectrum obtained with non-self-consistent G0W0 calculations starting from semilocal DFT functionals dramatically underestimate the transition energies. Starting from the popular PBE0 hybrid functional significantly improves the results even though this leads to an average -0.59 eV redshift compared to reference calculations for Thiel’s set. It is shown, however, that a simple self-consistent scheme at the GW level, with an update of the quasiparticle energies, not only leads to a much better agreement with reference values, but also significantly reduces the impact of the starting DFT functional. On average, the Bethe-Salpeter scheme based on self-consistent GW calculations comes close to the best time-dependent DFT calculations with the PBE0 functional with a 0.98 correlation coefficient and a 0.18 (0.25) eV mean absolute deviation compared to TD-PBE0 (theoretical best estimates) with a tendency to be red-shifted. We also observe that TD-DFT and the standard adiabatic Bethe-Salpeter implementation may differ significantly for states implying a large multiple excitation character.