Are multi-quasiparticle interactions important in molecular ionization?

Reference Energies for Valence Ionizations and Satellite Transitions

Upon ionization of an atom or a molecule, another electron (or more) can be simultaneously excited. These concurrently generated states are called "satellites" (or shakeup transitions) as they appear in ionization spectra as higher-energy peaks with weaker intensity and larger width than the main peaks associated with single-particle ionizations. Satellites, which correspond to electronically excited states of the cationic species, are notoriously challenging to model using conventional single-reference methods due to their high excitation degree compared to the neutral reference state. This work reports 42 satellite transition energies and 58 valence ionization potentials (IPs) of full configuration interaction quality computed in small molecular systems. Following the protocol developed for the quest database [Véril, M.; Scemama, A.; Caffarel, M.; Lipparini, F.; Boggio-Pasqua, M.; Jacquemin, D.; and Loos, P.-F. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2021, 11, e1517], these reference energies are computed using the configuration interaction using a perturbative selection made iteratively (CIPSI) method. In addition, the accuracy of the well-known coupled-cluster (CC) hierarchy (CC2, CCSD, CC3, CCSDT, CC4, and CCSDTQ) is gauged against these new accurate references. The performances of various approximations based on many-body Green's functions (GW, GF2, and T-matrix) for IPs are also analyzed. Their limitations in correctly modeling satellite transitions are discussed.

Going Beyond the GW Approximation Using the Time-Dependent Hartree-Fock Vertex

The time-dependent Hartree-Fock (TDHF) vertex of many-body perturbation theory (MBPT) makes it possible to extend TDHF theory to charged excitations. Here we assess its performance by applying it to spherical atoms in their neutral electronic configuration. On a theoretical level, we recast the TDHF vertex as a reducible vertex, highlighting the emergence of a self-energy expansion purely in orders of the bare Coulomb interaction; then, on a numerical level, we present results for polarizabilities, ionization energies (IEs), and photoemission satellites. We confirm the superiority of THDF over simpler methods such as the random phase approximation for the prediction of atomic polarizabilities. We then find that the TDHF vertex reliably provides better IEs than GW and low-order self-energies do in the light-atom, few-electron regime; its performance degrades in heavier, many-electron atoms instead, where an expansion in orders of an unscreened Coulomb interaction becomes less justified. New relevant features are introduced in the satellite spectrum by the TDHF vertex, but the experimental spectra are not fully reproduced due to a missing account of nonlinear effects connected to hole relaxation. We also explore various truncations of the self-energy given by the TDHF vertex, but do not find them to be more convenient than low-order approximations such as GW and second Born (2B), suggesting that vertex corrections should be carried out consistently both in the self-energy and in the polarizability.

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.

Comparing Self-Consistent GW and Vertex-Corrected G0W0 (G0W0Γ) Accuracy for Molecular Ionization Potentials

We test the performance of self-consistent GW and several representative implementations of vertex-corrected G0W0 (G0W0Γ). These approaches are tested on benchmark data sets covering full valence spectra (first ionization potentials and some inner valence shell excitations). For small molecules, when comparing against state-of-the-art wave function techniques, our results show that full self-consistency in the GW scheme either systematically outperforms vertex-corrected G0W0 or gives results of at least comparative quality. Moreover, G0W0Γ results in additional computational cost when compared to G0W0 or self-consistent GW. The dependency of G0W0Γ on the starting mean-field solution is frequently more dominant than the magnitude of the vertex correction itself. Consequently, for molecular systems, self-consistent GW performed on the imaginary axis (and then followed by modern analytical continuation techniques) offers a more reliable approach to make predictions of ionization potentials.

RT-EOM-CCSD Calculations of Inner and Outer Valence Ionization Energies and Spectral Functions

Photoelectron spectroscopy (PES) is a standard experimental method for material characterization, but its interpretation can be hampered by its reliance on standard materials. To facilitate the study of unknown systems, theoretical methods are desirable. Here, we present a real-time equation-of-motion coupled cluster (RT-EOM-CC) approach for valence PES, extending our core-level development. We demonstrate that RT-EOM-CC yields ionization energies and spectral functions in good agreement with experimental and CI-based results, even for some more correlated cases.

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.

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.

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.

Real-time equation-of-motion CC cumulant and CC Green's function simulations of photoemission spectra of water and water dimer

Newly developed coupled-cluster (CC) methods enable simulations of ionization potentials and spectral functions of molecular systems in a wide range of energy scales ranging from core-binding to valence. This paper discusses results obtained with the real-time equation-of-motion CC cumulant approach (RT-EOM-CC), and CC Green's function (CCGF) approaches in applications to the water and water dimer molecules. We compare the ionization potentials obtained with these methods for the valence region with the results obtained with the CCSD(T) formulation as a difference of energies for N and N-1 electron systems. All methods show good agreement with each other. They also agree well with experiment, with errors usually below 0.1 eV for the ionization potentials.We also analyze unique features of the spectral functions, associated with the position of satellite peaks, obtained with the RT-EOM-CC and CCGF methods employing single and double excitations, as a function of the monomer OH bond length and the proton transfer coordinate in the dimer. Finally, we analyze the impact of the basis set effects on the quality of calculated ionization potentials and find that the basis set effects are less pronounced for the augmented-type sets.

Numerically Precise Benchmark of Many-Body Self-Energies on Spherical Atoms

We investigate the performance of beyond-GW approaches in many-body perturbation theory by addressing atoms described within the spherical approximation via a dedicated numerical treatment based on B-splines and spherical harmonics. We consider the GW, second Born (2B), and GW + second order screened exchange (GW+SOSEX) self-energies and use them to obtain ionization potentials from the quasi-particle equation (QPE) solved perturbatively on top of independent-particle calculations. We also solve the linearized Sham–Schlüter equation (LSSE) and compare the resulting xc potentials against exact data. We find that the LSSE provides consistent starting points for the QPE but does not present any practical advantage in the present context. Still, the features of the xc potentials obtained with it shed light on possible strategies for the inclusion of beyond-GW diagrams in the many-body self-energy. Our findings show that solving the QPE with the GW+SOSEX self-energy on top of a PBE or PBE0 solution is a viable scheme to go beyond GW in finite systems, even in the atomic limit. However, GW shows a comparable performance if one agrees to use a hybrid starting point. We also obtain promising results with the 2B self-energy on top of Hartree–Fock, suggesting that the full time-dependent Hartree–Fock vertex may be another viable beyond-GW scheme for finite systems.

Real-Time Equation-of-Motion CCSD Cumulant Green's Function

Many-body excitations in X-ray photoemission spectra have been difficult to simulate from first principles. We have recently developed a cumulant-based one-electron Green's function method using the real-time coupled-cluster-singles equation-of-motion approach (RT-EOM-CCS) that provides a general framework for treating these problems. Here we extend this approach to include double excitations in the ground-state energy and in the coupled cluster amplitudes, which have been implemented using subroutines generated by the Tensor Contraction Engine (TCE). As in the case of the singles approximation, RT-EOM-CCSD yields a nonperturbative cumulant form of the Green's function in terms of the time-dependent cluster amplitudes, adding nonlinear corrections to the traditional cumulant forms. The extended approach is applied to the core-hole spectral function for small molecular systems. We find that, when core-optimized basis sets are used, the doubles contributions reduce the mean absolute errors in the core binding energies of the 10e systems from 0.8 to 0.3 eV. They also significantly improve the quasiparticle-satellite gap by reducing its overestimation from about 3-5 to about 0-1 eV in CH₄, NH₃, and H₂O, and also improving the overall shape of the satellite features. Finally, we demonstrate the application of the new implementation to the larger, classical XPS ESCA series of molecules and show that the singles approximation can be paired with a modest basis set to study carbon speciation.

The Effect of Geometry, Spin, and Orbital Optimization in Achieving Accurate, Correlated Results for Iron-Sulfur Cubanes

Iron-sulfur clusters comprise an important functional motif in the catalytic centers of biological systems, capable of enabling important chemical transformations at ambient conditions. This remarkable capability derives from a notoriously complex electronic structure that is characterized by a high density of states that is sensitive to geometric changes. The spectral sensitivity to subtle geometric changes has received little attention from correlated, large active space calculations, owing partly to the exceptional computational complexity for treating these large and correlated systems accurately. To provide insight into this aspect, we report the first Complete Active Space Self Consistent Field (CASSCF) calculations for different geometries of the [Fe(II/III)₄S₄(SMe)₄]^-2 clusters using two complementary, correlated solvers: spin-pure Adaptive Sampling Configuration Interaction (ASCI) and Density Matrix Renormalization Group (DMRG). We find that the previously established picture of a double-exchange driven magnetic structure, with minute energy gaps (<1 mHa) between consecutive spin states, has a weak dependence on the underlying geometry. However, the spin gap between the singlet and the spin state 2S + 1 = 19, corresponding to a maximal number of Fe-d electrons being unpaired and of parallel spin, is strongly geometry dependent, changing by a factor of 3 upon slight deformations that are still within biologically relevant parameters. The CASSCF orbital optimization procedure, using active spaces as large as 86 electrons in 52 orbitals, was found to reduce this gap compared to typical mean-field orbital approaches. Our results show the need for performing large active space calculations to unveil the challenging electronic structure of these complex catalytic centers and should serve as accurate starting points for fully correlated treatments upon inclusion of dynamical correlation outside the active space.

Extended quasiparticle approach to non-resonant and resonant X-ray emission spectroscopy

The initial state of X-ray emission spectroscopy (XES) and resonant inelastic X-ray emission spectroscopy (RIXS) is a highly excited eigenstate with a deep core hole after a X-ray photoelectron spectroscopy...

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.