An optimally tuned range-separated hybrid starting point for ab initio GW plus Bethe–Salpeter equation calculations of molecules

Influence of Electronic Correlations on Electron-Phonon Interactions of Molecular Systems with the GW and Coupled Cluster Methods

Electron-phonon interactions are of great importance to a variety of physical phenomena, and their accurate description is an important goal for first-principles calculations. Isolated examples of materials and molecular systems have emerged where electron-phonon coupling is enhanced over density functional theory (DFT) when using the Green's-function-based ab initio GW method, which provides a more accurate description of electronic correlations. It is, however, unclear how general this enhancement is and how employing high-end quantum chemistry methods, which further improve the description of electronic correlations, might further alter electron-phonon interactions over GW or DFT. Here, we address these questions by computing the renormalization of the highest occupied molecular orbital energies of Thiel's set of organic molecules by harmonic vibrations using DFT, GW, and equation-of-motion coupled-cluster calculations. We find that, depending on the amount of exact exchange included in the DFT starting point, GW can increase the magnitude of the electron-phonon coupling across Thiel's set of molecules by an average factor of 1.1-1.8 compared to the underlying DFT, while equation-of-motion coupled-cluster leads to an increase of 1.4-2. The electron-phonon coupling predicted with the ab initio GW method is generally in much closer agreement to coupled cluster values compared to DFT, establishing GW as a promising route for accurately computing electron-phonon phenomena in molecules and beyond at a much lower computational cost than higher-end quantum chemistry techniques.

Fully Dynamic G3W2 Self-Energy for Finite Systems: Formulas and Benchmark

Over the years, Hedin's GW self-energy has been proven to be a rather accurate and simple approximation to evaluate electronic quasiparticle energies in solids and in molecules. Attempts to improve over the simple GW approximation, the so-called vertex corrections, have been constantly proposed in the literature. Here, we derive, analyze, and benchmark the complete second-order term in the screened Coulomb interaction W for finite systems. This self-energy named G3W2 contains all the possible time orderings that combine 3 Green's functions G and 2 dynamic W. We present the analytic formula and its imaginary frequency counterpart, with the latter allowing us to treat larger molecules. The accuracy of the G3W2 self-energy is evaluated on well-established benchmarks (GW100, Acceptor 24, and Core 65) for valence and core quasiparticle energies. Its link with the simpler static approximation, named SOSEX for static screened second-order exchange, is analyzed, which leads us to propose a more consistent approximation named 2SOSEX. In the end, we find that neither the G3W2 self-energy nor any of the investigated approximations to it improve over one-shot G0W0 with a good starting point. Only quasi-particle self-consistent GW HOMO energies are slightly improved by addition of the G3W2 self-energy correction. We show that this is due to the self-consistent update of the screened Coulomb interaction, leading to an overall sign change of the vertex correction to the frontier quasiparticle energies.

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.

Deterministic/Fragmented-Stochastic Exchange for Large-Scale Hybrid DFT Calculations

We develop an efficient approach to evaluate range-separated exact exchange for grid- or plane-wave-based representations within the generalized Kohn-Sham-density functional theory (GKS-DFT) framework. The Coulomb kernel is fragmented in reciprocal space, and we employ a mixed deterministic-stochastic representation, retaining long-wavelength (low-k) contributions deterministically and using a sparse ("fragmented") stochastic basis for the high-k part. Coupled with a projection of the Hamiltonian onto a subspace of valence and conduction states from a prior local-DFT calculation, this method allows for the calculation of the long-range exchange of large molecular systems with hundreds and potentially thousands of coupled valence states delocalized over millions of grid points. We find that even a small number of valence and conduction states is sufficient for converging the HOMO and LUMO energies of the GKS-DFT. Excellent tuning of long-range separated hybrids (RSH) is easily obtained in the method for very large systems, as exemplified here for the chlorophyll hexamer of Photosystem II with 1320 electrons.

Linear Scaling Calculations of Excitation Energies with Active-Space Particle-Particle Random-Phase Approximation

We developed an efficient active-space particle-particle random-phase approximation (ppRPA) approach to calculate accurate charge-neutral excitation energies of molecular systems. The active-space ppRPA approach constrains both indexes in particle and hole pairs in the ppRPA matrix, which only selects frontier orbitals with dominant contributions to low-lying excitation energies. It employs the truncation in both orbital indexes in the particle-particle and the hole-hole spaces. The resulting matrix, whose eigenvalues are excitation energies, has a dimension that is independent of the size of the systems. The computational effort for the excitation energy calculation, therefore, scales linearly with system size and is negligible compared with the ground-state calculation of the (N - 2)-electron system, where N is the electron number of the molecule. With the active space consisting of 30 occupied and 30 virtual orbitals, the active-space ppRPA approach predicts the excitation energies of valence, charge-transfer, Rydberg, double, and diradical excitations with the mean absolute errors (MAEs) smaller than 0.03 eV compared with the full-space ppRPA results. As a side product, we also applied the active-space ppRPA approach in the renormalized singles (RS) T-matrix approach. Combining the non-interacting pair approximation that approximates the contribution to the self-energy outside the active space, the active-space GRSTRS@PBE approach predicts accurate absolute and relative core-level binding energies with the MAEs around 1.58 and 0.3 eV, respectively. The developed linear scaling calculation of excitation energies is promising for applications to large and complex systems.

Robust relativistic many-body Green's function based approaches for assessing core ionized and excited states

A two-component contour deformation (CD) based GW method that employs frequency sampling to drastically reduce the computational effort when assessing quasiparticle states far away from the Fermi level is outlined. Compared to the canonical CD-GW method, computational scaling is reduced by an order of magnitude without sacrificing accuracy. This allows for an efficient calculation of core ionization energies. The improved computational efficiency is used to provide benchmarks for core ionized states, comparing the performance of 15 density functional approximations as Kohn-Sham starting points for GW calculations on a set of 65 core ionization energies of 32 small molecules. Contrary to valence states, GW calculations on core states prefer functionals with only a moderate amount of Hartree-Fock exchange. Moreover, modern ab initio local hybrid functionals are also shown to provide excellent generalized Kohn-Sham references for core GW calculations. Furthermore, the core-valence separated Bethe-Salpeter equation (CVS-BSE) is outlined. CVS-BSE is a convenient tool to probe core excited states. The latter is tested on a set of 40 core excitations of eight small inorganic molecules. Results from the CVS-BSE method for excitation energies and the corresponding absorption cross sections are found to be in excellent agreement with those of reference damped response BSE calculations.

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.

Optimized attenuated interaction: Enabling stochastic Bethe-Salpeter spectra for large systems

We develop an improved stochastic formalism for the Bethe-Salpeter equation (BSE), based on an exact separation of the effective-interaction W into two parts, W = (W - vW) + vW, where the latter is formally any translationally invariant interaction, vW(r - r'). When optimizing the fit of the exchange kernel vW to W, using a stochastic sampling W, the difference W - vW becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling W.

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.

Assessing the Role of the Kohn-Sham Density in the Calculation of the Low-Lying Bethe-Salpeter Excitation Energies

We adopt the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation (BSE) to compute 57 excitation energies of a set of 37 molecules. By using the PBEh global hybrid functional and a self-consistent scheme on the eigenvalues in GW, we show a strong dependence of the BSE energy on the starting Kohn-Sham (KS) density functional. This arises from both the quasiparticle energies and the spatial localization of the frozen KS orbitals employed to compute the BSE. In order to address the arbitrariness in the mean field choice, we adopt an orbital-tuning scheme where the amount of Fock exchange, α, is tuned to impose that the KS HOMO matches the GW quasiparticle eigenvalue, thus fulfilling the ionization potential theorem in DFT. The performance of the proposed scheme yields excellent results and it is similar to M06-2X and PBEh with α = 75%, consistent with tuned values of α ranging between 60% and 80%.

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.

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.

Combining Renormalized Singles GW Methods with the Bethe-Salpeter Equation for Accurate Neutral Excitation Energies

We apply the renormalized singles (RS) Green's function in the Bethe-Salpeter equation (BSE)/GW approach to predict accurate neutral excitation energies of molecular systems. The BSE calculations are performed on top of the G_RSW_RS method, which uses the RS Green's function also for the computation of the screened Coulomb interaction W. We show that the BSE/G_RSW_RS approach significantly outperforms BSE/G₀W₀ for predicting excitation energies of valence, Rydberg, and charge-transfer (CT) excitations by benchmarking the Truhlar-Gagliardi set, Stein CT set, and an atomic Rydberg test set. For the Truhlar-Gagliardi test set, BSE/G_RSW_RS provides comparable accuracy to time-dependent density functional theory (TDDFT) and is slightly better than BSE starting from eigenvalue self-consistent GW (evGW). For the Stein CT test set, BSE/G_RSW_RS significantly outperforms BSE/G₀W₀ and TDDFT with the accuracy comparable to BSE/evGW. We also show that BSE/G_RSW_RS predicts Rydberg excitation energies of atomic systems well. Besides the excellent accuracy, BSE/G_RSW_RS largely eliminates the dependence on the choice of the density functional approximation. This work demonstrates that the BSE/G_RSW_RS approach is accurate and efficient for predicting excitation energies for a broad range of systems, which expands the applicability of the BSE/GW approach.