Embedding vertex corrections in GW self-energy: Theory, implementation, and outlook.

The vertex function (Γ) within the Green's function formalism encapsulates information about all higher-order electron-electron interaction beyond those mediated by density fluctuations. Herein, we present an efficient approach that embeds vertex corrections in the one-shot GW correlation self-energy for isolated and periodic systems. The vertex-corrected self-energy is constructed through the proposed separation-propagation-recombination procedure: the electronic Hilbert space is separated into an active space and its orthogonal complement denoted as the "rest;" the active component is propagated by a space-specific effective Hamiltonian different from the rest. The vertex corrections are introduced by a rescaled time-dependent nonlocal exchange interaction. The direct Γ correction to the self-energy is further updated by adjusting the rescaling factor in a self-consistent post-processing cycle. Our embedding method is tested mainly on donor-acceptor charge-transfer systems. The embedded vertex effects consistently and significantly correct the quasiparticle energies of the gap-edge states. The fundamental gap is generally improved by 1-3 eV upon the one-shot GW approximation. Furthermore, we provide an outlook for applications of (embedded) vertex corrections in calculations of extended solids.

Efficient treatment of molecular excitations in the liquid phase environment via stochastic many-body theory.

Accurate predictions of charge excitation energies of molecules in the disordered condensed phase are central to the chemical reactivity, stability, and optoelectronic properties of molecules and critically depend on the specific environment. Herein, we develop a stochastic GW method for calculating these charge excitation energies. The approach employs maximally localized electronic states to define the electronic subspace of a molecule and the rest of the system, both of which are randomly sampled. We test the method on three solute-solvent systems: phenol, thymine, and phenylalanine in water. The results are in excellent agreement with the previous high-level calculations and available experimental data. The stochastic calculations for supercells containing up to 1000 electrons representing the solvated systems are inexpensive and require ≤1000 central processing unit hrs. We find that the coupling with the environment accounts for ∼40% of the total correlation energy. The solvent-to-solute feedback mechanism incorporated in the molecular correlation term causes up to 0.6 eV destabilization of the quasiparticle energy. Simulated photo-emission spectra exhibit red shifts, state-degeneracy lifting, and lifetime shortening. Our method provides an efficient approach for an accurate study of excitations of large molecules in realistic condensed phase environments.

Are multi-quasiparticle interactions important in molecular ionization?

Photo-emission spectroscopy directly probes individual electronic states, ranging from single excitations to high-energy satellites, which simultaneously represent multiple quasiparticles (QPs) and encode information about electronic correlation. The first-principles description of the spectra requires an efficient and accurate treatment of all many-body effects. This is especially challenging for inner valence excitations where the single QP picture breaks down. Here, we provide the full valence spectra of small closed-shell molecules, exploring the independent and interacting quasiparticle regimes, computed with the fully correlated adaptive sampling configuration interaction method. We critically compare these results to calculations with the many-body perturbation theory, based on the GW and vertex corrected GWΓ approaches. The latter explicitly accounts for two-QP quantum interactions, which have often been neglected. We demonstrate that for molecular systems, the vertex correction universally improves the theoretical spectra, and it is crucial for the accurate prediction of QPs as well as capturing the rich satellite structures of high-energy excitations. GWΓ offers a unified description across all relevant energy scales. Our results suggest that the multi-QP regime corresponds to dynamical correlations, which can be described via perturbation theory.

Decomposition and embedding in the stochastic GW self-energy.

We present two new developments for computing excited state energies within the GW approximation. First, calculations of the Green's function and the screened Coulomb interaction are decomposed into two parts: one is deterministic, while the other relies on stochastic sampling. Second, this separation allows constructing a subspace self-energy, which contains dynamic correlation from only a particular (spatial or energetic) region of interest. The methodology is exemplified on large-scale simulations of nitrogen-vacancy states in a periodic hBN monolayer and hBN-graphene heterostructure. We demonstrate that the deterministic embedding of strongly localized states significantly reduces statistical errors, and the computational cost decreases by more than an order of magnitude. The computed subspace self-energy unveils how interfacial couplings affect electronic correlations and identifies contributions to excited-state lifetimes. While the embedding is necessary for the proper treatment of impurity states, the decomposition yields new physical insight into quantum phenomena in heterogeneous systems.

Stochastic time-dependent DFT with optimally tuned range-separated hybrids: Application to excitonic effects in large phosphorene sheets.

We develop a stochastic approach to time-dependent density functional theory with optimally tuned range-separated hybrids containing nonlocal exchange, for calculating optical spectra. The attractive electron-hole interaction, which leads to the formation of excitons, is included through a time-dependent linear-response technique with a nonlocal exchange interaction which is computed very efficiently through a stochastic scheme. The method is inexpensive and scales quadratically with the number of electrons, at almost the same (low) cost of time dependent Kohn-Sham with local functionals. Our results are in excellent agreement with experimental data, and the efficiency of the approach is demonstrated on large finite phosphorene sheets containing up to 1958 valence electrons.

Effects of symmetry breaking on the translation-rotation eigenstates of H2, HF, and H2O inside the fullerene C60.

Splittings of the translation-rotation (TR) eigenstates of the solid light-molecule endofullerenes M@C60 (M = H2, H2O, HF) attributed to the symmetry breaking have been observed in the infrared (IR) and inelastic neutron scattering spectra of these species in the past couple of years. In a recent paper [Felker et al., Phys. Chem. Chem. Phys., 2017, 19, 31274], we established that the electrostatic, quadrupolar interaction between the guest molecule M and the twelve nearest-neighbor C60 cages of the solid is the main source of the symmetry breaking. The splittings of the three-fold degenerate ground states of the endohedral ortho-H2, ortho-H2O and the j = 1 level of HF calculated using this model were found to be in excellent agreement with the experimental results. Utilizing the same electrostatic model, this theoretical study investigates the effects of the symmetry breaking on the excited TR eigenstates of the three species, and how they manifest in their simulated low-temperature (5-6 K) near-IR (NIR) and far-IR (FIR) spectra. The TR eigenstates are calculated variationally for both the major P and minor H crystal orientations. For the H orientation, the calculated splittings of all of the TR levels of these species are less than 0.1 cm-1. For the dominant P orientation, the splittings vary strongly depending on the character of the excitations involved. In all of the species, the splittings of the higher rotationally excited levels are comparable in magnitude to those for the j = 1 levels. For the levels corresponding to purely translational excitations, the calculated splittings are about an order of magnitude smaller than those of the purely rotational eigenstates. Based on the computed TR eigenstates, the low-temperature NIR (for M = H2) and FIR (for M = HF and H2O) spectra are simulated for both the P and H orientations, and also combined as their weighted sum (0.15H + 0.85P). The weighted sum spectra computed for M = H2 and HF match quantitatively the corresponding measured spectra, while for M = H2O, the weighted sum FIR spectrum predicts features that can potentially be observed experimentally.

Simple eigenvalue-self-consistent Δ ¯ G W 0 .

We show that a rigid scissors-like GW self-consistency approach, labeled here Δ ¯ G W 0 , can be trivially implemented at zero additional cost for large scale one-shot G 0 W 0 calculations. The method significantly improves one-shot G 0 W 0 and for large systems is very accurate. Δ ¯ G W 0 is similar in spirit to evGW 0 where the self-consistency is only applied on the eigenvalues entering Green's function, while both W and the eigenvectors of Green's function are held fixed. Δ ¯ G W 0 further assumes that the shift of the eigenvalues is rigid scissors-like so that all occupied states are shifted by the same amount and analogously for all the unoccupied states. We show that this results in a trivial modification of the time-dependent G 0 W 0 self-energy, enabling an a posteriori self-consistency cycle. The method is applicable for our recent stochastic-GW approach, thereby enabling self-consistent calculations for giant systems with thousands of electrons. The accuracy of Δ ¯ G W 0 increases with the system size. For molecules, it is up to 0.4-0.5 eV away from coupled-cluster single double triple (CCSD(T)), but for tetracene and hexacene, it matches the ionization energies from both CCSD(T) and evGW 0 to better than 0.05 eV. For solids, as exemplified here by periodic supercells of semiconductors and insulators with 6192 valence electrons, the method matches evGW 0 quite well and both methods are in good agreement with the experiment.

Explaining the symmetry breaking observed in the endofullerenes H2@C60, HF@C60, and H2O@C60.

Symmetry breaking has been recently observed in the endofullerenes M@C60 (M = H2, HF, H2O), manifesting in the splittings of the three-fold degenerate ground states of the endohedral ortho-H2, ortho-H2O and the j = 1 level of HF. The nature of the interaction causing the symmetry breaking is established in this study. A fragment of the solid C60 is considered, comprised of the central C60 molecule surrounded by twelve nearest-neighbor (NN) C60 molecules. The fullerenes have either P (major) or H (minor) orientational orderings, and are assumed to be rigid with Ih symmetry. Only the central C60 is occupied by the guest molecule M, while the NN fullerenes are all empty. The key proposition of the study is that the electrostatic interactions between the charge densities on the NN C60 molecules and that on M inside the central C60 give rise to the symmetry breaking responsible for the measured level splittings. Using this model, the M@C60 level splittings of interest are calculated variationally and using perturbation theory, for both the P and H orientations. Those obtained for the dominant P orientation are in excellent agreement with the experimental results, with respect to the splitting magnitudes and patterns, for all three M@C60 systems considered, pointing strongly to the quadrupolar M-NN interactions as the main cause of the symmetry breaking. The level splittings calculated for the H orientation are about 30 times smaller than the ones in the P orientation.

Spontaneous Charge Carrier Localization in Extended One-Dimensional Systems.

Charge carrier localization in extended atomic systems has been described previously as being driven by disorder, point defects, or distortions of the ionic lattice. Here we show for the first time by means of first-principles computations that charge carriers can spontaneously localize due to a purely electronic effect in otherwise perfectly ordered structures. Optimally tuned range-separated density functional theory and many-body perturbation calculations within the GW approximation reveal that in trans-polyacetylene and polythiophene the hole density localizes on a length scale of several nanometers. This is due to exchange-induced translational symmetry breaking of the charge density. Ionization potentials, optical absorption peaks, excitonic binding energies, and the optimally tuned range parameter itself all become independent of polymer length as it exceeds the critical localization length. Moreover, we find that lattice disorder and the formation of a polaron result from the charge localization in contrast to the traditional view that lattice distortions precede charge localization. Our results can explain experimental findings that polarons in conjugated polymers form instantaneously after exposure to ultrafast light pulses.

Deviations from piecewise linearity in the solid-state limit with approximate density functionals.

In exact density functional theory, the total ground-state energy is a series of linear segments between integer electron points, a condition known as "piecewise linearity." Deviation from this condition is indicative of poor predictive capabilities for electronic structure, in particular of ionization energies, fundamental gaps, and charge transfer. In this article, we take a new look at the deviation from linearity (i.e., curvature) in the solid-state limit by considering two different ways of approaching it: a large finite system of increasing size and a crystal represented by an increasingly large reference cell with periodic boundary conditions. We show that the curvature approaches vanishing values in both limits, even for functionals which yield poor predictions of electronic structure, and therefore cannot be used as a diagnostic or constructive tool in solids. We find that the approach towards zero curvature is different in each of the two limits, owing to the presence of a compensating background charge in the periodic case. Based on these findings, we present a new criterion for functional construction and evaluation, derived from the size-dependence of the curvature, along with a practical method for evaluating this criterion. For large finite systems, we further show that the curvature is dominated by the self-interaction of the highest occupied eigenstate. These findings are illustrated by computational studies of various solids, semiconductor nanocrystals, and long alkane chains.

Electrical resistivity and thermal conductivity of liquid Fe alloys at high P and T, and heat flux in Earth's core.

Earth's magnetic field is sustained by magnetohydrodynamic convection within the metallic liquid core. In a thermally advecting core, the fraction of heat available to drive the geodynamo is reduced by heat conducted along the core geotherm, which depends sensitively on the thermal conductivity of liquid iron and its alloys with candidate light elements. The thermal conductivity for Earth's core is very poorly constrained, with current estimates based on a set of scaling relations that were not previously tested at high pressures. We perform first-principles electronic structure computations to determine the thermal conductivity and electrical resistivity for Fe, Fe-Si, and Fe-O liquid alloys. Computed resistivity agrees very well with existing shock compression measurements and shows strong dependence on light element concentration and type. Thermal conductivity at pressure and temperature conditions characteristic of Earth's core is higher than previous extrapolations. Conductive heat flux near the core-mantle boundary is comparable to estimates of the total heat flux from the core but decreases with depth, so that thermally driven flow would be constrained to greater depths in the absence of an inner core.

Efficient Quasiparticle Determination beyond the Diagonal Approximation via Random Compression.

Calculations of excited states in the Green's function formalism often invokes the diagonal approximation, in which the quasiparticle states are taken from a mean-field calculation. In this paper, we extend the stochastic approaches applied in the many-body perturbation theory and overcome this limitation for large systems in which we are interested in a small subset of states. We separate the problem into a core subspace whose coupling to the remainder of the system environment is stochastically sampled. This method is exemplified on computing hole injection energies into CO2 on an extended gold surface with nearly 3000 electrons. We find that in the extended system the size of the problem can be compressed up to 95% using stochastic sampling. This result provides a way forward for self-consistent stochastic methods and determination of Dyson orbitals in large systems.

Polyurethane Foam Chemical Recycling: Fast Acidolysis with Maleic Acid and Full Recovery of Polyol.

Chemical recycling of polyurethane (PU) waste is essential to displace the need for virgin polyol production and enable sustainable PU production. Currently, less than 20% of PU waste is downcycled through rebinding to lower value products than the original PU. Chemical recycling of PU waste often requires significant input of materials like solvents and slow reaction rates. Here, we report the fast (<10 min) and solvent-free acidolysis of a model toluene diisocyanate (TDI)-based flexible polyurethane foam (PUF) at <200 °C using maleic acid (MA) with a recovery of recycled polyol (repolyol) in 95% isolated yield. After workup (hydrolysis of repolyl ester and separations), the repolyol exhibits favorable physical properties that are comparable to the virgin polyol; these include 54.1 mg KOH/g OH number and 624 cSt viscosity. Overall, 80% by weight of the input PUF is isolated into two clean-cut fractions containing the repolyol and toluene diamine (TDA). Finally, end-of-life (EOL) mattress PUF waste is recycled successfully with high recovery of repolyol using MA acidolysis. The solvent-free and fast acidolysis with MA demonstrated in this work with both model and EOL PUF provides a potential pathway for sustainable and closed-loop PU production.

Vapor-Phase Dicarboxylic Acids and Anhydrides Drive Depolymerization of Polyurethanes.

Polyurethane (PU) is the sixth most used plastic in the world. Because many PU derived materials are thermosets and the monomers are valuable, chemical recycling to recover the polyol component is the most viable pathway to utilizing postconsumer PU waste in a closed-loop fashion. Acidolysis is an effective method to recover polyol from PU waste. Previous studies of PU acidolysis rely on the use of dicarboxylic acid (DCA) in high temperature reactions (>200 °C) in the liquid phase and result in unwanted byproducts, high energy consumption, complex separations of excess organic acid, and an overall process that is difficult to scale up. In this work, we demonstrate selective PU acidolysis with DCA vapor to release polyol at temperatures below the melting points of the DCAs (<150 °C). Notably, acidolysis with DCA vapor adheres to the principles of green chemistry and prevents in part esterification of the polyol product, eliminating the need for additional hydrolysis/processing to obtain the desired product. The methodology was successfully applied to a commercial PU foam (PUF) postconsumer waste.

Spatial Decay and Limits of Quantum Solute-Solvent Interactions.

Molecular excitations in the liquid-phase environment are renormalized by the surrounding solvent molecules. Herein, we employ the GW approximation to investigate the solvation effects on the ionization energy of phenol in various solvent environments. The electronic effects differ by up to 0.4 eV among the five investigated solvents. This difference depends on both the macroscopic solvent polarizability and the spatial decay of the solvation effects. The latter is probed by separating the electronic subspace and the GW correlation self-energy into fragments. The fragment correlation energy decays with increasing intermolecular distance and vanishes at ∼9 Å, and this pattern is independent of the type of solvent environment. The 9 Å cutoff defines an effective interacting volume within which the ionization energy shift per solvent molecule is proportional to the macroscopic solvent polarizability. Finally, we propose a simple model for computing the ionization energies of molecules in an arbitrary solvent environment.

Exceptional Spatial Variation of Charge Injection Energies on Plasmonic Surfaces.

Charge injection into a molecule on a metallic interface is a key step in many photoactivated reactions. We employ the many-body perturbation theory and compute the hole and electron injection energies for CO2 molecule on an Au nanoparticle with ∼3,000 electrons and compare it to results for idealized infinite surfaces. We demonstrate a surprisingly large variation of the injection energy barrier depending on the precise molecular position on the surface. Multiple "hot spots," characterized by low energy barriers, arise due to the competition between the plasmonic coupling and the degree of hybridization between the molecule and the substrate. The charge injection barrier to the adsorbate on the nanoparticle surface decreases from the facet edge to the facet center. This finding contrasts with the typical picture in which the electric field enhancement on the nanoparticle edges is considered the most critical factor.

Reduced Scaling of Optimal Regional Orbital Localization via Sequential Exhaustion of the Single-Particle Space.

Wannier functions have become a powerful tool in the electronic structure calculations of extended systems. The generalized Pipek-Mezey Wannier functions exhibit appealing characteristics (e.g., reaching an optimal localization and the separation of the σ-π orbitals) compared with other schemes. However, when applied to giant nanoscale systems, the orbital localization suffers from a large computational cost overhead if one is interested in localized states in a small fragment of the system. Herein, we present a swift, efficient, and robust approach for obtaining regionally localized orbitals of a subsystem within the generalized Pipek-Mezey scheme. The proposed algorithm introduces a reduced work space and sequentially exhausts the entire orbital space until the convergence of the localization functional. It tackles systems with ∼10000 electrons within 0.5 h with no loss in localization quality compared to the traditional approach. Regionally localized orbitals with a higher extent of localization are obtained via judiciously extending the subsystem's size. Exemplifying on large bulk and a 4 nm wide slab of diamond with an NV- center, we demonstrate the methodology and discuss how the choice of the localization region affects the excitation energy of the defect. Furthermore, we show how the sequential algorithm is easily extended to stochastic methodologies that do not provide individual single-particle eigenstates. It is thus a promising tool to obtain regionally localized states for solving the electronic structure problems of a subsystem embedded in giant condensed systems.

Quasiparticles and Band Structures in Organized Nanostructures of Donor-Acceptor Copolymers.

The performance of organic semiconductor devices is linked to highly ordered nanostructures of self-assembled molecules and polymers. Many-body perturbation theory is employed to study the excited states in bulk copolymers. The results show that acceptors in the polymer scaffold introduce a, hitherto unrecognized, conduction impurity band that leads to electron localization. The donor states are responsible for the formation of conjugated bands, which are only mildly perturbed by the presence of the acceptors. Along the polymer axis, the nonlocal electronic correlations among copolymer strands hinder efficient band transport, which is, however, strongly enhanced across individual chains. Holes are most effectively transported along the π-π stacking, while electrons in the impurity band follow the edge-to-edge directions. The copolymers exhibit regions with inverted transport polarity, in which electrons and holes are efficiently transported in mutually orthogonal directions.

Stochastic many-body perturbation theory for Moiré states in twisted bilayer phosphorene.

We implement stochastic many-body perturbation theory for systems with 2D periodic boundary conditions. The method is used to compute quasiparticle excitations in twisted bilayer phosphorene. Excitation energies are studied using stochastic [Formula: see text] and partially self-consistent [Formula: see text] approaches. The approach is inexpensive; it is used to study twisted systems with unit cells containing >2700 atoms (>13 500 valence electrons), which corresponds to a minimum twisting angle of [Formula: see text] [Formula: see text]. Twisted bilayers exhibit band splitting, increased localization and formation of localized Moiré impurity states, as documented by band-structure unfolding. Structural changes in twisted structures lift band degeneracies. Energies of the impurity states vary with the twisting angle due to an interplay between non-local exchange and polarization effects. The mechanisms of quasiparticle energy (de)stabilization due to twisting are likely applicable to a wide range of low-dimensional Moiré superstructures.

Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies.

New stochastic approaches for the computation of electronic excitations are developed within the many-body perturbation theory. Three approximations to the electronic self-energy are considered: G0W0, G0W0tc, and G0W0tcΓX. All three methods are formulated in the time domain, and the latter two incorporate nonlocal vertex corrections. In the case of G0W0tcΓX, the vertex corrections are included both in the screened Coulomb interaction and in the expression for the self-energy. The implementation of the three approximations is verified by comparison to deterministic results for a set of small molecules. The performance of the fully stochastic implementation is tested on acene molecules, C60 and PC60BM. The vertex correction appears crucial for the description of unoccupied states. Unlike conventional (deterministic) approaches, all three stochastic methods scale linearly with the number of electrons.