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.

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.

Computing photoionization spectra in Gaussian basis sets.

We present a method to compute the photoionization spectra of atoms and molecules in linear-response, time-dependent density functional theory. The electronic orbital variations corresponding to ionized electrons are expanded on a basis set of delocalized functions, obtained as the solution of the inhomogeneous Helmholtz equation, with gaussian basis set functions as the right-hand side. The resulting scheme is able to reproduce the photoionization spectra without any need for artificial regularization or localization. We demonstrate that this Green's function-based approach is able to produce accurate spectra for semilocal exchange-correlation functionals, even using relatively small standard gaussian basis sets.

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.

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.

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.

Robust Analytic-Continuation Approach to Many-Body GW Calculations.

The analytic continuation of the GW self-energy from the imaginary to the real-energy axis is a central difficulty for approaches exploiting the favorable properties of response functions at imaginary frequencies. Within a scheme merging contour-deformation and analytic-continuation techniques, we show on the basis of extensive calculations for large molecular sets that it is preferable to perform an analytic continuation of the dynamically screened-Coulomb potential W rather than the much more structured self-energy operator. The case of states lying far away from the gap, including core states, is addressed by generalizing the analytic-continuation scheme, accounting further for quasiparticle lifetimes.

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.

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.

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

We review the many-body Green's function Bethe-Salpeter equation (BSE) formalism that is rapidly gaining importance for the study of the optical properties of molecular organic systems. We emphasize in particular its similarities and differences with time-dependent density functional theory (TD-DFT), both methods sharing the same formal O(N4) computing time scaling with system size. By comparison with higher level wavefunction based methods and experimental results, the advantages of BSE over TD-DFT are presented, including an accurate description of charge-transfer states and an improved accuracy for the challenging cyanine dyes. We further discuss the models that have been developed for including environmental effects. Finally, we summarize the challenges to be faced so that BSE reaches the same popularity as TD-DFT.

Calculations of n→π* Transition Energies: Comparisons Between TD-DFT, ADC, CC, CASPT2, and BSE/GW Descriptions.

Using a large panel of theoretical approaches, namely, CC2, CCSD, CCSDR(3), CC3, ADC(2), ADC(3), CASPT2, time-dependent density functional theory (TD-DFT), and BSE/evGW, the two latter combined with different exchange-correlation functionals, we investigate the lowest singlet transition in 23 n→π* compounds based on the nitroso, thiocarbonyl, carbonyl, and diazo chromophores. First, for 16 small derivatives we compare the transition energies provided by the different wave function approaches to define theoretical best estimates. For this set, it surprisingly turned out that ADC(2) offers a better match with CC3 than ADC(3). Next, we use 10 functionals belonging to the "LYP" and "M06" families and compare the TD-DFT and the BSE/evGW descriptions. The BSE/evGW results are less sensitive than their TD-DFT counterparts to the selected functional, especially in the M06 series. Nevertheless, BSE/evGW delivers larger errors than TD-CAM-B3LYP, which provides extremely accurate results in the present case, especially when the Tamm-Dancoff approximation is applied. In addition, we show that, among the different starting points for BSE/evGW calculations, M06-2X eigenstates stand as the most appropriate. Finally, we confirm that the trends observed on the small compounds pertain in larger molecules.

Helium Atom Excitations by the GW and Bethe-Salpeter Many-Body Formalism.

The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body GW approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.

Is the Bethe-Salpeter Formalism Accurate for Excitation Energies? Comparisons with TD-DFT, CASPT2, and EOM-CCSD.

Developing ab initio approaches able to provide accurate excited-state energies at a reasonable computational cost is one of the biggest challenges in theoretical chemistry. In that framework, the Bethe-Salpeter equation approach, combined with the GW exchange-correlation self-energy, which maintains the same scaling with system size as TD-DFT, has recently been the focus of a rapidly increasing number of applications in molecular chemistry. Using a recently proposed set encompassing excitation energies of many kinds [J. Phys. Chem. Lett. 2016, 7, 586-591], we investigate here the performances of BSE/GW. We compare these results to CASPT2, EOM-CCSD, and TD-DFT data and show that BSE/GW provides an accuracy comparable to the two wave function methods. It is particularly remarkable that the BSE/GW is equally efficient for valence, Rydberg, and charge-transfer excitations. In contrast, it provides a poor description of triplet excited states, for which EOM-CCSD and CASPT2 clearly outperform BSE/GW. This contribution therefore supports the use of the Bethe-Salpeter approach for spin-conserving transitions.