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.

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.

Disentangling the multiorbital contributions of excitons by photoemission exciton tomography.

Excitons are realizations of a correlated many-particle wave function, specifically consisting of electrons and holes in an entangled state. Excitons occur widely in semiconductors and are dominant excitations in semiconducting organic and low-dimensional quantum materials. To efficiently harness the strong optical response and high tuneability of excitons in optoelectronics and in energy-transformation processes, access to the full wavefunction of the entangled state is critical, but has so far not been feasible. Here, we show how time-resolved photoemission momentum microscopy can be used to gain access to the entangled wavefunction and to unravel the exciton's multiorbital electron and hole contributions. For the prototypical organic semiconductor buckminsterfullerene (C60), we exemplify the capabilities of exciton tomography and achieve unprecedented access to key properties of the entangled exciton state including localization, charge-transfer character, and ultrafast exciton formation and relaxation dynamics.

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.

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.

Lagrangian Z-vector approach to Bethe-Salpeter analytic gradients: Assessing approximations.

We present an implementation of excited-state analytic gradients within the Bethe-Salpeter equation formalism using an adapted Lagrangian Z-vector approach with a cost independent of the number of perturbations. We focus on excited-state electronic dipole moments associated with the derivatives of the excited-state energy with respect to an electric field. In this framework, we assess the accuracy of neglecting the screened Coulomb potential derivatives, a common approximation in the Bethe-Salpeter community, as well as the impact of replacing the GW quasiparticle energy gradients by their Kohn-Sham analogs. The pros and cons of these approaches are benchmarked using both a set of small molecules for which very accurate reference data are available and the challenging case of increasingly extended push-pull oligomer chains. The resulting approximate Bethe-Salpeter analytic gradients are shown to compare well with the most accurate time-dependent density-functional theory (TD-DFT) data, curing in particular most of the pathological cases encountered with TD-DFT when a nonoptimal exchange-correlation functional is used.

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 (µES), 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 µES. We explore increasingly long push-pull oligomers as they present an excited-state nature evolving with system size. This work shows that BSE's µES 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.

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.

Modeling of excited state potential energy surfaces with the Bethe-Salpeter equation formalism: The 4-(dimethylamino)benzonitrile twist.

We present a benchmark study of excited state potential energy surfaces (PES) using the many-body Green's function GW and Bethe-Salpeter equation (BSE) formalisms, coupled cluster methods, as well as Time-Dependent Density Functional Theory (TD-DFT). More specifically, we investigate the evolution of the two lowest excited states of 4-(dimethylamino)benzonitrile (DMABN) upon the twisting of the amino group, a paradigmatic system for dual fluorescence and excited-state benchmarks. Our results demonstrate that the BSE/GW approach is able to reproduce the correct topology of excited state PES upon geometry changes in both gas and condensed phases. The vertical transition energies predicted by BSE/GW are indeed in good agreement with coupled cluster values, including triples. The BSE approach ability to include both linear response and state-specific solvent corrections further enables it to accurately describe the solvatochromism of both excited states during the twisting of DMABN. This contribution stands as one of the first proof-of-concept that BSE/GW PES should be accurate in cases for which TD-DFT struggles, including the central case of systems embedded in a dielectric environment.

Doping-Induced Dielectric Catastrophe Prompts Free-Carrier Release in Organic Semiconductors.

The control over material properties attainable through molecular doping is essential to many technological applications of organic semiconductors, such as organic light-emitting diodes or thermoelectrics. These excitonic semiconductors typically reach the degenerate limit only at impurity concentrations of 5-10%, a phenomenon that has been put in relation with the strong Coulomb binding between charge carriers and ionized dopants, and whose comprehension remained elusive so far. This study proposes a general mechanism for the release of carriers at finite doping in terms of collective screening phenomena. A multiscale model for the dielectric properties of doped organic semiconductor is set up by combining first principles and microelectrostatic calculations. The results predict a large nonlinear enhancement of the dielectric constant (tenfold at 8% load) as the system approaches a dielectric instability (catastrophe) upon increasing doping. This can be attributed to the presence of highly polarizable host-dopant complexes, plus a nontrivial leading contribution from dipolar interactions in the disordered and heterogeneous system. The enhanced screening in the material drastically reduces the (free) energy barriers for electron-hole separation, rationalizing the possibility for thermal charge release. The proposed mechanism is consistent with conductivity data and sets the basis for achieving higher conductivities at lower doping loads.

Reference Energies for Intramolecular Charge-Transfer Excitations.

With the aim of completing our previous efforts devoted to local and Rydberg transitions in organic compounds, we provide a series of highly accurate vertical transition energies for intramolecular charge-transfer transitions occurring in (π-conjugated) molecular compounds. To this end, we apply a composite protocol consisting of linear-response CCSDT excitation energies determined with Dunning's double-ζ basis set corrected by CC3/CCSDT-3 energies obtained with the corresponding triple-ζ basis. Further basis set corrections (up to aug-cc-pVQZ) are obtained at the CCSD and CC2 levels. We report 30 transitions obtained in 17 compounds (aminobenzonitrile, aniline, azulene, benzonitrile, benzothiadiazole, dimethylaminobenzonitrile, dimethylaniline, dipeptide, ß-dipeptide, hydrogen chloride, nitroaniline, nitrobenzene, nitrodimethylaniline, nitropyridine N-oxide, N-phenylpyrrole, phthalazine, and quinoxaline]. These reference values are then used to benchmark a series of wave functions [CIS(D), SOPPA, RPA(D), EOM-MP2, CC2, CCSD, CCSD(T)(a)*, CCSDR(3), CCSDT-3, CC3, ADC(2), ADC(3), and ADC(2.5)], the Green's function-based Bethe-Salpeter equation (BSE) formalism performed on top of the partially self-consistent evGW scheme considering two different starting points (BSE/evGW@HF and BSE/evGW@PBE0), and time-dependent density-functional theory (TD-DFT) combined with several exchange-correlation functionals (B3LYP, PBE0, M06-2X, CAM-B3LYP, LC-ωHPBE, ωB97X, ωB97X-D, and M11). It turns out that the CC methods including triples, namely, CCSD(T)(a)*, CCSDR(3), CCSDT-3, and CC3, provide rather small average deviations (≤0.10 eV), with CC3 emerging as the only chemically accurate approach. ADC(2.5) also performs nicely with a mean absolute error of 0.11 eV for a O(N6) formal scaling, whereas CC2 and BSE/evGW@PBE0 also deliver very satisfying results given their respective O(N5) and O(N4) computational scalings. In the TD-DFT context, the best performing functional is ωB97X-D, closely followed by CAM-B3LYP and M06-2X, all providing mean absolute errors around 0.15 eV relative to the theoretical best estimates.

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 {rk} optimized within a separable resolution-of-the-identity framework to reproduce standard Coulomb-fitting calculations with meV accuracy. The compactness of the obtained {rk} 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.

Strongly Bound Excitons in Metal-Organic Framework MOF-5: A Many-Body Perturbation Theory Study.

During the past years, one of the most iconic metal-organic frameworks (MOFs), MOF-5, has been characterized as a semiconductor by theory and experiments. Here we employ the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation to compute the electronic structure and optical properties of this MOF. The GW calculations show that MOF-5 is a wide-band-gap insulator with a fundamental gap of ∼8 eV. The strong excitonic effects, arising from highly localized states and low screening, result in an optical gap of 4.5 eV and in an optical absorption spectrum in excellent agreement with experiments. The origin of the incorrect conclusion reported by past studies and the implication of this result are also discussed.

Orientation dependent molecular electrostatics drives efficient charge generation in homojunction organic solar cells.

Organic solar cells usually utilise a heterojunction between electron-donating (D) and electron-accepting (A) materials to split excitons into charges. However, the use of D-A blends intrinsically limits the photovoltage and introduces morphological instability. Here, we demonstrate that polycrystalline films of chemically identical molecules offer a promising alternative and show that photoexcitation of α-sexithiophene (α-6T) films results in efficient charge generation. This leads to α-6T based homojunction organic solar cells with an external quantum efficiency reaching up to 44% and an open-circuit voltage of 1.61 V. Morphological, photoemission, and modelling studies show that boundaries between α-6T crystalline domains with different orientations generate an electrostatic landscape with an interfacial energy offset of 0.4 eV, which promotes the formation of hybridised exciton/charge-transfer states at the interface, dissociating efficiently into free charges. Our findings open new avenues for organic solar cell design where material energetics are tuned through molecular electrostatic engineering and mesoscale structural control.

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.

Accurate Prediction of the S1 Excitation Energy in Solvated Azobenzene Derivatives via Embedded Orbital-Tuned Bethe-Salpeter Calculations.

By employing the Bethe-Salpeter formalism coupled with a nonequilibrium embedding scheme, we demonstrate that the paradigmatic case of S1 band separation between cis and trans in azobenzene derivatives can be computed with excellent accuracy compared to experimental optical spectra. Besides embedding, we show that the choice of the Kohn-Sham exchange correlation functional for DFT is critical, despite the iterative convergence of GW quasiparticle energies. We address this by adopting an orbital-tuning approach via the global hybrid functional, PBEh, yielding an environment-consistent ionization potential. The vertical excitation energy of 20 azo molecules is predicted with a mean absolute error as low as 0.06 eV, up to three times smaller compared to standard functionals such as M06-2X and PBE0, and five times smaller compared to recent TDDFT results.