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Hedin's $GW$ approximation to the electronic self-energy has been impressively successful to calculate quasiparticle energies, such as ionization potentials, electron affinities, or electronic band structures. The success of this fairly simple approximation has been ascribed to the cancellation of the so-called vertex corrections that go beyond $GW$. This claim is mostly based on past calculations using vertex corrections within the crude local-density approximation. Here, we explore a wide variety of non-local vertex corrections in the polarizability and the self-energy, using first-order approximations or infinite summations to all orders. In particular, we use vertices based on statically screened interactions like in the Bethe-Salpeter equation. We demonstrate on realistic molecular systems that the two vertices in Hedin's equation essentially compensate. We further show that consistency between the two vertices is crucial to obtain realistic electronic properties. We finally consider increasingly large clusters and extrapolate that our conclusions would hold for extended systems., Comment: Update of original version: More detailed results including an additional figure, additional references, and another figure with diagrams

Reduced density matrix functional theory (RDMFT) and coupled cluster theory restricted to paired double excitations (pCCD) are emerging as efficient methodologies for accounting for the so-called non-dynamic electronic correlation effects. Up to now, molecular calculations have been performed with real-valued orbitals. However, before extending the applicability of these methodologies to extended systems, where Bloch states are employed, the subtleties of working with complex-valued orbitals and the consequences of imposing time-reversal symmetry must be carefully addressed. In this work, we describe the theoretical and practical implications of adopting time-reversal symmetry in RDMFT and pCCD when allowing for complex-valued orbital coefficients. The theoretical considerations primarily affect the optimization algorithms, while the practical implications raise fundamental questions about the stability of solutions. Specifically, we find that complex solutions lower the energy when non-dynamic electronic correlation effects are pronounced. We present numerical examples to illustrate and discuss these instabilities and possible problems introduced by N-representability violations.

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 GW can increase the magnitude of the electron-phonon coupling across this set of molecules by an average factor of 1.1-1.8 compared to 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 an accurate way of computing electron-phonon phenomena in molecules and beyond at a much lower computational cost than higher-end quantum chemistry techniques.

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, 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 $G_0W_0$ 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.

The accurate computation of non-linear optical properties (NLOPs) in large polymers requires accounting for electronic correlation effects with a reasonable computational cost. The Random Phase Approximation (RPA) used in the adiabatic connection fluctuation theorem is known to be a reliable and cost-effective method to render electronic correlation effects when combined with the density-fitting techniques and the integration over imaginary frequencies. We explore the ability of the RPA energy expression to predict NLOPs by evaluating RPA electronic energies in the presence of finite electric fields to obtain (using the finite difference method) static polarizabilities and hyper-polarizabilities. We show that RPA based on hybrid functional self-consistent field calculations yields as accurate NLOPs as the best-tuned double-hybrid functionals developed today, with the additional advantage that RPA avoids any system-specific adjustment.

The $GW$ approximation is a well-established method for calculating ionization potentials and electron affinities in solids and molecules. For numerous years, obtaining self-consistent $GW$ total energies in solids has been a challenging objective that is not accomplished yet. However, it was shown recently that the linearized $GW$ density matrix permits a reliable prediction of the self-consistent $GW$ total energy for molecules [F. Bruneval et. al. J. Chem. Theory Comput. 17, 2126 (2021)] for which self-consistent $GW$ energies are available. Here we implement, test, and benchmark the linearized $GW$ density matrix for several solids. We focus on the total energy, lattice constant, and bulk modulus obtained from the $GW$ density matrix and compare our findings to more traditional results obtained within the random phase approximation (RPA). We conclude on the improved stability of the total energy obtained from the linearized $GW$ density matrix with respect to the mean-field starting point. We bring compelling clues that the RPA and the $GW$ density matrix total energies are certainly close to the self-consistent $GW$ total energy in solids if we use hybrid functionals with enriched exchange as a starting point., Comment: 16 pages, 8 figures

The assessment of myocardial fiber disarray is of major interest for the study of the progression of myocardial disease. However, time-resolved imaging of the myocardial structure remains unavailable in clinical practice. In this study, we introduce 3D Backscatter Tensor Imaging (3D-BTI), an entirely novel ultrasound-based imaging technique that can map the myocardial fibers orientation and its dynamics with a temporal resolution of 10 ms during a single cardiac cycle, non-invasively and in vivo in entire volumes. 3D-BTI is based on ultrafast volumetric ultrasound acquisitions, which are used to quantify the spatial coherence of backscattered echoes at each point of the volume. The capability of 3D-BTI to map the fibers orientation was evaluated in vitro in 5 myocardial samples. The helicoidal transmural variation of fiber angles was in good agreement with the one obtained by histological analysis. 3D-BTI was then performed to map the fiber orientation dynamics in vivo in the beating heart of an open-chest sheep at a volume rate of 90 volumes/s. Finally, the clinical feasibility of 3D-BTI was shown on a healthy volunteer. These initial results indicate that 3D-BTI could become a fully non-invasive technique to assess myocardial disarray at the bedside of patients.

A large collaboration carefully benchmarks 20 first principles many-body electronic structure methods on a test set of 7 transition metal atoms, and their ions and monoxides. Good agreement is attained between the 3 systematically converged methods, resulting in experiment-free reference values. These reference values are used to assess the accuracy of modern emerging and scalable approaches to the many-electron problem. The most accurate methods obtain energies indistinguishable from experimental results, with the agreement mainly limited by the experimental uncertainties. Comparison between methods enables a unique perspective on calculations of many-body systems of electrons., Comment: Simons collaboration on the many-electron problem

Ab initio many-body perturbation theory within the $GW$ approximation is a Green's function formalism widely used in the calculation of quasiparticle excitation energies of solids. In what has become an increasingly standard approach, Kohn-Sham eigenenergies, generated from a DFT calculation with a strategically-chosen exchange correlation functional ``starting point'', are used to construct $G$ and $W$, and then perturbatively corrected by the resultant $GW$ self-energy. In practice, there are several ways to construct the $GW$ self-energy, and these can lead to variations in predicted quasiparticle energies. For example, for ZnO and TiO$_2$, reported $GW$ fundamental gaps can vary by more than 1 eV. In this work, we address the convergence and key approximations in contemporary $G_0W_0$ calculations, including frequency-integration schemes and the treatment of the Coulomb divergence in the exact-exchange term. We study several systems,and compare three different $GW$ codes: BerkeleyGW, Abinit and Yambo. We demonstrate, for the first time, that the same quasiparticle energies for systems in the condensed phase can be obtained with different codes, and we provide a comprehensive assessment of implementations of the $GW$ approximation.

The accurate prediction of singlet and triplet excitation energies is of significant fundamental interest and is critical for many applications. An area of intense research, most calculations of singlet and triplet energies use time-dependent density functional theory (TDDFT) in conjunction with an approximate exchange-correlation functional. In this work, we examine and critically assess an alternative method for predicting low-lying neutral excitations with similar computational cost, the ab initio Bethe-Salpeter equation (BSE) approach, and compare results against high-accuracy wavefunction-based methods. We consider singlet and triplet excitations of 27 prototypical organic molecules, including members of Thiel's set, the acene series, and several aromatic hydrocarbons exhibiting charge-transfer-like excitations. Analogous to its impact in TDDFT, we find that the Tamm-Dancoff approximation (TDA) overcomes triplet instabilities in the BSE approach, improving both triplet and singlet energetics relatively to higher level theories. Finally, we find that BSE-TDA calculations built on good DFT starting points, such as those utilizing optimally-tuned range-separated hybrid functionals, can yield accurate singlet and triplet excitation energies for gas-phase organic molecules.

Charged excitations of the oligoacene family of molecules, relevant for astrophysics and technological applications, are widely studied and therefore provide an excellent system for benchmarking theoretical methods. In this work, we evaluate the performance of many-body perturbation theory within the GW approximation relative to new high-quality CCSD(T) reference data for charged excitations of the acenes. We compare GW calculations with a number of hybrid density functional theory starting points and with eigenvalue self-consistency. Special focus is given to elucidating the trend of GW-predicted excitations with molecule length increasing from benzene to hexacene. We find that GW calculations with starting points based on an optimally tuned range-separated hybrid (OTRSH) density functional and eigenvalue self-consistency can yield quantitative ionization potentials for the acenes. However, for larger acenes, the predicted electron affinities can deviate considerably from reference values. Our work paves the way for predictive and cost-effective GW calculations of charged excitations of molecules and identifies certain limitations of current GW methods used in practice for larger molecules., Comment: J. Chem. Theory Comput. (2016)

The Ruddlesden-Popper (RP) homologous series Sr$_{n+1}$Ti$_{n}$O$_{3n+1}$ provides a useful template for the study and control of the effects of dimensionality and quantum confinement on the excited state properties of the complex oxide SrTiO$_3$. We use ab initio many-body perturbation theory within the $GW$ approximation and the Bethe-Salpeter equation approach to calculate quasiparticle energies and absorption spectrum of Sr$_{n+1}$Ti$_{n}$O$_{3n+1}$ for $n=1-5$ and $\infty$. Our computed direct and indirect optical gaps are in excellent agreement with spectroscopic measurements. The calculated optical spectra reproduce the main experimental features and reveal excitonic structure near the gap edge. We find that electron-hole interactions are important across the series, leading to significant exciton binding energies that increase for small $n$ and reach a value of 330~meV for $n=1$, a trend attributed to increased quantum confinement. We find that the lowest-energy singlet exciton of Sr$_2$TiO$_4$ ($n=1$) localizes in the 2D plane defined by the TiO$_2$ layer, and explain the origin of its localization.

The predictive power of the ab initio Bethe-Salpeter equation (BSE) approach, rigorously based on many-body Green's function theory but incorporating information from density functional theory, has already been demonstrated for the optical gaps and spectra of solid-state systems. Interest in photoactive hybrid organic/inorganic systems has recently increased, and so has the use of the BSE for computing neutral excitations of organic molecules. However, no systematic benchmarks of the BSE for neutral electronic excitations of organic molecules exist. Here, we study the performance of the BSE for the 28 small molecules in Thiel's widely-used time-dependent density functional theory benchmark set [M. Schreiber et al. J. Chem. Phys. 128, 134110 (2008)]. We observe that the BSE produces results that depend critically on the mean-field starting point employed in the perturbative approach. We find that this starting point dependence is mainly introduced through the quasiparticle energies obtained at the intermediate GW step, and that with a judicious choice of starting mean-field, singlet excitation energies obtained from BSE are in excellent quantitative agreement with higher-level wavefunction methods. The quality of the triplet excitations is slightly less satisfactory.

We report an implementation of the constrained random phase approximation (cRPA) method within the projector augmented-wave framework. It allows for the calculation of the screened interaction in the same Wannier orbitals as our recent DFT+U and DFT+DMFT implementations. We present calculations of the dynamical Coulomb screened interaction in uranium dioxide and alpha and gamma cerium on Wannier functions. We show that a self-consistent calculation of the static screened interaction in DFT+U together with a consistent Wannier basis is mandatory for gamma cerium and uranium dioxide. We emphasize that a static approximation for the screened interaction in alpha cerium is too drastic., Comment: 12 pages, 5 figures

Despite the lack of reproducible experimental confirmation, group-V elements have been considered as possible sources of p-type doping in ZnO in the form of simple and complex defects. Using ab initio calculations, based on state-of-the-art hybrid exchange-correlation functional, we study a wide range of defects and defect complexes related with N, P, As, and Sb impurities. We show that none of the candidates for p-type doping can be considered a good source of holes in the valence band due to deep acceptor levels and low formation energies of compensating donor defects. In addition, we discuss the stability of complexes in different regimes., Comment: 7 pages, 7 figures

Modeling point defects at an atomic scale requires careful treatment of the long-range atomic relaxations. This elastic field can strongly affect point defect properties calculated in atomistic simulations because of the finite size of the system under study. This is an important restriction for ab initio methods which are limited to a few hundred atoms. We propose an original approach coupling ab initio calculations and linear elasticity theory to obtain the properties of an isolated point defect for reduced supercell sizes. The reliability and benefit of our approach are demonstrated for three problematic cases: the self-interstitial in zirconium, clusters of self-interstitials in iron, and the neutral vacancy in silicon.

The silicon vacancy is a prominent intrinsic defect of cubic SiC (3C-SiC) to which much effort has been devoted so far, experimentally and theoretically. We calculate its properties using the GW approximation that does not suffer from the band gap problem. The obtained formation and transition energies deviate significantly from the usual density functional theory evaluations and now compare favorably with experiment. A new assignment for the main line of photoluminescence is then proposed. We further perform GW calculations for the saddle point of reaction paths. The resulting barrier energies explain the thermal annealing experiments thanks to an original mechanism mediated by a minority charge configuration.

We use hybrid functionals and restricted self-consistent GW, state-of-the-art theoretical approaches for quasiparticle band structures, to study the electronic states of delafossite Cu(Al,In)O$_2$, the first p-type and bipolar transparent conductive oxides. We show that self-consistent GW gives remarkably wider band gaps than all the other approaches used so far. Accounting for polaronic effects in the GW scheme we recover a very nice agreement with experiments. Furthermore, the modifications with respect to the Kohn-Sham bands are strongly k-dependent, which makes questionable the common practice of using a scissor operator. Finally, our results support the view that the low energy structures found in optical experiments, and initially attributed to an indirect transition, are due to intrinsic defects in the samples., Comment: published

In the description of the interaction between electrons beyond the classical Hartree picture, bare exchange often yields a leading contribution. Here we discuss its effect on optical spectra of solids, comparing three different frameworks: time-dependent Hartree-Fock, a recently introduced combined density-functional and Green's functions approach applied to the bare exchange self-energy, and time-dependent exact-exchange within time-dependent density-functional theory (TD-EXX). We show that these three approximations give rise to identical excitonic effects in solids; these effects are drastically overestimated for semiconductors. They are partially compensated by the usual overestimation of the quasiparticle band gap within Hartree-Fock. The physics that lacks in these approaches can be formulated as screening. We show that the introduction of screening in TD-EXX indeed leads to a formulation that is equivalent to previously proposed functionals derived from Many-Body Perturbation Theory. It can be simulated by reducing the long-range part of the Coulomb interaction: this produces absorption spectra of semiconductors in good agreement with experiment., Comment: 12 pages, 3 figures, 1 table