Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies

First-Principles Investigation of the Optical Properties of Eumelanin Protomolecules

Using first-principles calculations, we investigate the absorption spectra (in the near-infrared, visible, and first UV range) of the two most probable eumelanin tetrameric molecules exhibiting either a linear open-chain or a cyclic porphyrine-like configuration. In order to simulate a realistic molecular system, an implicit solvent model is used in our calculations to mimic the effect of the solvated environment around the eumelanin molecule. Although the presence of solvent is found not to significantly affect the absorption pattern of both molecules, the onset of the spectra are shifted toward higher energies, especially for the linear tetramer. Interestingly, the absorption spectra and optical onsets of the two molecules differ significantly both in a vacuum and in ethanol. However, the two predicted spectra do not allow us to definitely discriminate between the two configurations when comparing the theoretical predictions with the available experimental spectrum. In addition, a mix of the two eumelanin configurations (close to fifty-fifty) leads to a maximum overlap between theoretical and experimental spectra. Consequently, this theoretical research shows that deeper insight can be gained using beyond DFT techniques on the real form of eumelanin protomolecules present in living systems as well as on their possible use in hybrid solar cells.

A Similarity Renormalization Group Approach to Green's Function Methods

The family of Green's function methods based on the GW approximation has gained popularity in the electronic structure theory thanks to its accuracy in weakly correlated systems combined with its cost-effectiveness. Despite this, self-consistent versions still pose challenges in terms of convergence. A recent study [Monino and Loos J. Chem. Phys. 2022, 156, 231101.] has linked these convergence issues to the intruder-state problem. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. The SRG formalism enables us to derive, from first-principles, the expression of a naturally static and Hermitian form of the self-energy that can be employed in quasiparticle self-consistent GW (qsGW) calculations. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qsGW calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.

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

Connections between many-body perturbation and coupled-cluster theories

Here, we build on the works of Scuseria et al. [J. Chem. Phys. 129, 231101 (2008)] and Berkelbach [J. Chem. Phys. 149, 041103 (2018)] to show connections between the Bethe-Salpeter equation (BSE) formalism combined with the GW approximation from many-body perturbation theory and coupled-cluster (CC) theory at the ground- and excited-state levels. In particular, we show how to recast the GW and Bethe-Salpeter equations as non-linear CC-like equations. Similitudes between BSE@GW and the similarity-transformed equation-of-motion CC method are also put forward. The present work allows us to easily transfer key developments and the general knowledge gathered in CC theory to many-body perturbation theory. In particular, it may provide a path for the computation of ground- and excited-state properties (such as nuclear gradients) within the GW and BSE frameworks.

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.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Unphysical discontinuities, intruder states and regularization in GW methods

By recasting the non-linear frequency-dependent GW quasiparticle equation into a linear eigenvalue problem, we explain the appearance of multiple solutions and unphysical discontinuities in various physical quantities computed within the GW approximation. Considering the GW self-energy as an effective Hamiltonian, it is shown that these issues are key signatures of strong correlation in the (N ± 1)-electron states and can be directly related to the intruder state problem. A simple and efficient regularization procedure inspired by the similarity renormalization group is proposed to avoid such issues and speed up the convergence of partially self-consistent GW calculations.

Static and dynamic Bethe-Salpeter equations in the T-matrix approximation

While the well-established GW approximation corresponds to a resummation of the direct ring diagrams and is particularly well suited for weakly correlated systems, the T-matrix approximation does sum ladder diagrams up to infinity and is supposedly more appropriate in the presence of strong correlation. Here, we derive and implement, for the first time, the static and dynamic Bethe-Salpeter equations when one considers T-matrix quasiparticle energies and a T-matrix-based kernel. The performance of the static scheme and its perturbative dynamical correction are assessed by computing the neutral excited states of molecular systems. A comparison with more conventional schemes as well as other wave function methods is also reported. Our results suggest that the T-matrix-based formalism performs best in few-electron systems where the electron density remains low.

NMR Coupling Constants Based on the Bethe-Salpeter Equation in the GW Approximation

We present the first steps to extend the Green's function GW method and the Bethe-Salpeter equation (BSE) to molecular response properties such as nuclear magnetic resonance (NMR) indirect spin-spin coupling constants. We discuss both a nonrelativistic one-component and a quasi-relativistic two-component formalism. The latter describes scalar-relativistic and spin-orbit effects and allows us to study heavy-element systems with reasonable accuracy. Efficiency is maintained by the application of the resolution of the identity approximation throughout. The performance is demonstrated using conventional central processing units (CPUs) and modern graphics processing units (GPUs) for molecules involving several thousand basis functions. Our results show that a large amount of Hartree-Fock exchange is vital to provide a sufficient Kohn-Sham starting point to compute the GW quasi-particle energies. As the GW-BSE approach is generally less accurate for triplet excitations or related properties such as the Fermi-contact interaction, the admixture of the Kohn-Sham correlation kernel through the contracted BSE (cBSE) method improves the results for NMR coupling constants. This leads to remarkable results when combined with the eigenvalue-only self-consistent variant (evGW) and Becke's half and half functional (BH&HLYP) or the CAM-QTP family. The developed methodology is used to calculate the Karplus curve of tin molecules, illustrating its applicability to extended chemically relevant molecules. Here, the GW-cBSE method improves upon the chosen BH&HLYP Kohn-Sham starting points.

Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals

We develop a new family of electronic structure methods for capturing at the same time the dynamic and nondynamic correlation effects. We combine the natural orbital functional theory (NOFT) and many-body perturbation theory (MBPT) through a canonicalization procedure applied to the natural orbitals to gain access to any MBPT approximation. We study three different scenarios: corrections based on second-order Møller-Plesset (MP2), random-phase approximation (RPA), and coupled-cluster singles doubles (CCSD). Several chemical problems involving different types of electron correlation in singlet and multiplet spin states have been considered. Our numerical tests reveal that RPA-based and CCSD-based corrections provide similar relative errors in molecular dissociation energies (D_e) to the results obtained using a MP2 correction. With respect to the MP2 case, the CCSD-based correction improves the prediction, while the RPA-based correction reduces the computational cost.

Scrutinizing GW-Based Methods Using the Hubbard Dimer

Using the simple (symmetric) Hubbard dimer, we analyze some important features of the GW approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot GW method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard GW approximation and find, in particular, that 1) some neutral excitation energies become complex when the electron-electron interaction U increases, which can be traced back to the approximate nature of the GW quasiparticle energies; 2) the BSE formalism yields accurate correlation energies over a wide range of U when the trace (or plasmon) formula is employed; 3) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); 4) the trace formula has the correct behavior for weak (i.e., small U) interaction, unlike the ACFDT expression.

Exploring vibronic coupling in the benzene radical cation and anion with different levels of the GW approximation

The linear vibronic coupling constants of the benzene radical cation and anion have been obtained with different levels of the GW approximation, including G₀W₀, eigenvalue self-consistent GW, and quasiparticle self-consistent GW, as well as DFT with the following exchange-correlation functionals: BLYP, B3LYP, CAM-B3LYP, tuned CAM-B3LYP, and an IP-tuned CAM-B3LYP functional. The vibronic coupling constants were calculated numerically using the gradients of the eigenvalues of the degenerate HOMOs and LUMOs of the neutral benzene molecule for DFT, while the numerical gradients of the quasiparticle energies were used in the case of GW. The results were evaluated against those of high level wave function methods in the literature, and the approximate self-consistent GW methods and G₀W₀ with long-range corrected functionals were found to yield the best results on the whole.

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.

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.

Improved One-Shot Total Energies from the Linearized GW Density Matrix

The linearized GW density matrix (γ^GW) is an efficient method to improve the static portion of the self-energy compared to that of ordinary perturbative GW while keeping the single-shot simplicity of the calculation. Previous work has shown that γ^GW gives an improved Fock operator and total energy components that approach the self-consistent GW quality. Here, we test γ^GW for dimer dissociation for the first time by studying N₂, LiH, and Be₂. We also calculate a set of self-consistent GW results in identical basis sets for a direct and consistent comparison. γ^GW approaches self-consistent GW total energies for a starting point based on a high amount of exact exchange. We also compare the accuracy of different total energy functionals, which differ when evaluated with a non-self-consistent density or density matrix. While the errors in total energies among different functionals and starting points are small, the individual energy components show noticeable errors when compared to reference data. The energy component errors of γ^GW are smaller than functionals of the density and we suggest that the linearized GW density matrix is a route to improving total energy evaluations in the adiabatic connection framework.

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.