Bethe-Salpeter Equation Formalism Research Articles

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.

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.

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

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

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.

Dynamic Structure Factor and Dielectric Function of Valence Electrons in Lithium Hydride: An Inelastic X‐Ray Scattering Study at Finite Momentum Transfer

The effects of electron–hole interaction in the dynamic structure factor and the complex dielectric function of valence electrons in lithium hydride at finite momentum transfer q are investigated by means of inelastic X‐ray scattering (IXS) spectroscopy and ab initio theoretical methods. Calculations of and are conducted within time‐dependent density functional theory (TDDFT) in adiabatic local density approximation (ALDA) and many‐body perturbation theory (MBPT) based on the Bethe–Salpeter equation (BSE). The findings reveal that for low q an explicit treatment of electron–hole interactions using BSE formalism slightly modifies the low‐energy structures in in comparison with ALDA results but affects strongly the macroscopic dielectric functions. A very good agreement between the experimental and theoretical and in all the range of investigated q values is achieved for calculations based on BSE after taking into account the full excitonic Hamiltonian. The results demonstrate the potential of approximations based on the BSE to accurately describe the valence excitation spectra, including the near‐onset region, and the dielectric response in insulating systems, where excitonic effects are relevant.

Large excitonic effect on van der Waals interaction between two-dimensional semiconductors.

An exceptionally large excitonic effect on the van der Waals (vdW) interaction between two-dimensional semiconductors is unraveled using the Lifshitz theory in conjunction with the ab initio GW plus Bethe-Salpeter equation formalism. Upon consideration of the electron-hole interaction, the vdW energy between two atomistic layers separated by 10 000 angstroms can be larger by a ratio of ∼30%, which is an order of magnitude greater than that seen for semi-infinite silicon surfaces. The large influence of the short-range electron-hole interaction on the long-range effect of quantum fluctuations is rooted in the ultra-thin nature of two-dimensional semiconductors which results in not only large exciton binding energy but also amplified roles of low-frequency dielectric responses.

Tunable optical and excitonic properties of phosphorene via oxidation

The optical properties and excitonic wave function of phosphorene oxides (PO) are studied using the first principle many-body Green function and the Bethe–Salpeter equation formalism. In this work, the optical properties are determined using ab initio calculations of the dielectric function. At the long wavelength limit q of EM wave (i.e. ), the dielectric function, the absorption spectrum, the lectivity, the electron energy loss spectra (EELS) and the wave function are calculated. The results show an excitonic binding energy of 818 meV with a bright exciton located in the armchair direction in pristine phosphorene. For PO, the arrangement of the oxygen atoms significantly influences the optical properties. In particular, the absorption spectrum is extended along the solar spectrum, with a high absorption coefficient observed in the dangling structures. The maximum lectivity values are observed for the high energies of the light spectrum. Moreover, the first EELS peak is located in the visible region in all the structures except for one configuration that exhibits the same behavior as pure phosphorene. Finally, the exciton effect reveals that all PO conformers have a dark exciton state, which is suitable for long-lived applications.

Quantitative first-principles calculations of valence and core excitation spectra of solid C60.

We present calculated valence and C 1s near-edge excitation spectra of solid C60 and experimental results measured with high-resolution electron energy-loss spectroscopy. The near-edge calculations are carried out using three different methods: solution of the Bethe-Salpeter equation (BSE) as implemented in the OCEAN suite (Obtaining Core Excitations with ab initio methods and the NIST BSE solver), the excited-electron core-hole approach (XCH), and the constrained-occupancy method using the Stockholm-Berlin core-excitation code, StoBe. The three methods give similar results and are in good agreement with experiment, though the BSE results are the most accurate. The BSE formalism is also used to carry out valence level calculations using the NIST Bethe-Salpeter Equation solver (NBSE). Theoretical results include self-energy corrections to the band gap and band widths, lifetime-damping effects, and Debye-Waller effects in the core-excitation case. A comparison of spectral features to those observed experimentally illustrates the sensitivity of certain features to computational details, such as self-energy corrections to the band structure and core-hole screening.

Optical properties of defected silicene: the many-body approach

The electronic structure and excitonic optical properties of pristine and defected silicene are investigated within many-body Green’s function and Bethe–Salpeter equation formalism. We show that compared with pristine one, defects can significantly alter band structure and form much better metallic characteristics. Also, it is shown that low defects in pristine silicene can considerably alter optical absorption peaks and change these peaks to the lower energies. Such effect is illustrated for 5 % defects in the pristine silicene and two main peaks in the optical spectrum are shown, one in low energies and another wider one in the higher energies. These peaks can be used as a tool to define the existence of defects and predict the amount of them in experiments.

Excitonic effects in GeC hybrid: Many-body Green's function calculations

Many-body effects on the electronic and optical absorption properties of a GeC sheet are studied by means of first principle many-body Green's function and Bethe–Salpeter equation formalism. The absence of soft modes in the phonon-spectrum indicates the stability of the system. The inclusion of quasiparticle corrections increases significantly the band gap. The local field effects induce significant change in the absorption spectra for the out-plane polarization rendering the GeC monolayer transparent below 7eV. The excitonic effects are significant on the optical absorption properties. A detailed analysis of the spectrum shows a strong binding energy of 1.82eV assigned to the lowest-energy bound excitons that is characterized by an effective mass of 1.68m0 and a Bohr radius of 2Å. The results of this study hold the promise for potential applications of the GeC hybrid in optoelectronics.

All-electronGW+Bethe-Salpeter calculations on small molecules

Accuracy of the first-principles $GW$+Bethe-Salpeter equation (BSE) method is examined for low-energy excited states of small molecules. The standard formalism, which is based on the one-shot $GW$ approximation and the Tamm-Dancoff approximation (TDA), is found to underestimate the optical gap of ${\mathrm{N}}_{2}$, CO, ${\mathrm{H}}_{2}\mathrm{O},\phantom{\rule{0.16em}{0ex}}{\mathrm{C}}_{2}{\mathrm{H}}_{4}$, and ${\mathrm{CH}}_{2}\mathrm{O}$ by about 1 eV. Possible origins are investigated separately for the effect of TDA and for the approximate schemes of the self-energy operator, which are known to cause overbinding of the electron-hole pair and overscreening of the interaction. By applying the known correction formula, we find the amount of the correction is too small to overcome the underestimated excitation energy. This result indicates a need for fundamental revision of the $GW$+BSE method rather than adjustment of the standard one. We expect that this study makes the problems in the current $GW$+BSE formalism clearer and provides useful information for further intrinsic development beyond the current framework.

Optical excitations and quasiparticle energies in the AlN monolayer honeycomb structure

We calculated the electronic structure and optical properties of the AlN monolayer based on first principle many-body Green’s function and Bethe–Salpeter equation formalism. We computed that the indirect (direct) band gap of the AlN monolayer honeycomb using quasiparticle correction is 5.36eV (6.07eV), but it has a value of 2.93eV (3.7eV) within the density functional theory. The calculations reveal that the optical absorption is sensitive to excitonic effects such as electron–hole interaction with binding energy of the first exciton of over 2.05eV within the GW+Bethe–Salpeter equation calculation. The enhanced excitonic effects in the AlN monolayer can be used to describe the optical properties in nano-optoelectronic devices.

The excitonic effects in single and double-walled boron nitride nanotubes.

The electronic structures and excitonic optical properties of single- and double-walled armchair boron nitride nanotubes (BNNTs) [e.g., (5,5) and (10,10), and (5,5)@(10,10)] are investigated within many-body Green's function and Bethe-Salpeter equation formalism. The first absorption peak of the double-walled nanotube has almost no shift compared with the single-walled (5,5) tube due to the strong optical transition in the double-walled tube that occurs within the inner (5,5) one. Dark and semi-dark excitonic states are detected in the lower energy region, stemming from the charge transfer between inner and outer tubes in the double-walled structure. Most interestingly, the charge transfer makes the electron and the hole reside in different tubes. Moreover, the excited electrons in the double-walled BNNT are able to transfer from the outer tube to the inner one, opposite to that which has been observed in double-walled carbon nanotubes.

Quasiparticle energies and optical excitations in the GaAs monolayer

Using first principles many-body theory methods (Green's function and Bethe Salpeter equation formalism) we calculated the electronic structure and optical properties of the GaAs monolayer. We computed that the indirect (direct) band gap of the GaAs Monolayer honeycomb using density functional theory is 0.21eV (0.97eV), but it has a value of 2.86eV (3.48eV) within the quasiparticle correction. The calculations reveal that the optical absorption is sensitive to excitonic effects such as electron–hole interaction with binding energy of the first exciton of over 2.37eV within the GW+Bethe Salpeter equation calculation. The enhanced excitonic effects in the GaAs Monolayer can be used to describe the optical properties in nano-optoelectronic devices.

Quasiparticle Energies and Optical Excitations in Chevron-Type Graphene Nanoribbon

The electronic structure and optical properties of very recently synthesized chevron-type graphene nanoribbon (CGNR) are investigated within many-body Green’s function and Bethe–Salpeter equation formalism. The CGNR can effectively confine both electrons and holes, leading to its exciton binding energy larger than that of regular GNRs. The excitonic peaks owing to electron–hole interactions dominate the optical spectra with a significant blue-shift and a different line shape in CGNR compared with those in regular GNRs. Moreover, the singlet–triplet exciton splitting of CGNR is also larger than that of the regular GNRs, which is expected to show high fluorescence luminescence efficiency. The enhanced excitonic effects in CGNR should be of great importance in optoelectronic applications.