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.

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.

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.

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.

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.

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.

Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube.

New models of fluid transport are expected to emerge from the confinement of liquids at the nanoscale, with potential applications in ultrafiltration, desalination and energy conversion. Nevertheless, advancing our fundamental understanding of fluid transport on the smallest scales requires mass and ion dynamics to be ultimately characterized across an individual channel to avoid averaging over many pores. A major challenge for nanofluidics thus lies in building distinct and well-controlled nanochannels, amenable to the systematic exploration of their properties. Here we describe the fabrication and use of a hierarchical nanofluidic device made of a boron nitride nanotube that pierces an ultrathin membrane and connects two fluid reservoirs. Such a transmembrane geometry allows the detailed study of fluidic transport through a single nanotube under diverse forces, including electric fields, pressure drops and chemical gradients. Using this device, we discover very large, osmotically induced electric currents generated by salinity gradients, exceeding by two orders of magnitude their pressure-driven counterpart. We show that this result originates in the anomalously high surface charge carried by the nanotube's internal surface in water at large pH, which we independently quantify in conductance measurements. The nano-assembly route using nanostructures as building blocks opens the way to studying fluid, ionic and molecule transport on the nanoscale, and may lead to biomimetic functionalities. Our results furthermore suggest that boron nitride nanotubes could be used as membranes for osmotic power harvesting under salinity gradients.

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 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.

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.

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.

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.

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.

GW and Bethe-Salpeter study of small water clusters.

We study within the GW and Bethe-Salpeter many-body perturbation theories the electronic and optical properties of small (H2O)n water clusters (n = 1-6). Comparison with high-level CCSD(T) Coupled-Cluster at the Single Double (Triple) levels and ADC(3) Green's function third order algebraic diagrammatic construction calculations indicates that the standard non-self-consistent G0W0@PBE or G0W0@PBE0 approaches significantly underestimate the ionization energy by about 1.1 eV and 0.5 eV, respectively. Consequently, the related Bethe-Salpeter lowest optical excitations are found to be located much too low in energy when building transitions from a non-self-consistent G0W0 description of the quasiparticle spectrum. Simple self-consistent schemes, with update of the eigenvalues only, are shown to provide a weak dependence on the Kohn-Sham starting point and a much better agreement with reference calculations. The present findings rationalize the theory to experiment possible discrepancies observed in previous G0W0 and Bethe-Salpeter studies of bulk water. The increase of the optical gap with increasing cluster size is consistent with the evolution from gas to dense ice or water phases and results from an enhanced screening of the electron-hole interaction.

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.

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.

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.