Coupled cluster Green function: Model involving single and double excitations

Restoring translational symmetry in periodic all-orbital dynamical mean-field theory simulations

Dynamical mean-field theory (DMFT) and its cluster extensions provide an efficient Green's function formalism to simulate spectral properties of periodic systems at the quantum many-body level. However, traditional cluster DMFT breaks translational invariance in solid-state materials, and the best strategy to capture non-local correlation effects within cluster DMFT remains elusive. In this work, we investigate the use of overlapping atom-centered impurity fragments in recently-developed ab initio all-orbital DMFT, where all local orbitals within the impurity are treated with high-level quantum chemistry impurity solvers. We demonstrate how the translational symmetry of the lattice self-energy can be restored by designing symmetry-adapted embedding problems, which results in an improved description of spectral functions in two-dimensional boron nitride monolayers and graphene at the levels of many-body perturbation theory (GW) and coupled-cluster theory. Furthermore, we study the convergence of self-energy and density of states as the embedding size is systematically expanded in one-shot and self-consistent DMFT calculations.

Machine Learning Many-Body Green's Functions for Molecular Excitation Spectra

We present a machine learning (ML) framework for predicting Green's functions of molecular systems, from which photoemission spectra and quasiparticle energies at quantum many-body level can be obtained. Kernel ridge regression is adopted to predict self-energy matrix elements on compact imaginary frequency grids from static and dynamical mean-field electronic features, which gives direct access to real-frequency many-body Green's functions through analytic continuation and Dyson's equation. Feature and self-energy matrices are represented in a symmetry-adapted intrinsic atomic orbital plus projected atomic orbital basis to enforce rotational invariance. We demonstrate good transferability and high data efficiency of the proposed ML method across molecular sizes and chemical species by showing accurate predictions of density of states (DOS) and quasiparticle energies at the level of many-body perturbation theory (GW) or full configuration interaction. For the ML model trained on 48 out of 1995 molecules randomly sampled from the QM7 and QM9 data sets, we report the mean absolute errors of ML-predicted highest occupied and lowest unoccupied molecular orbital energies to be 0.13 and 0.10 eV, respectively, compared to GW@PBE0. We further showcase the capability of this method by applying the same ML model to predict DOS for significantly larger organic molecules with up to 44 heavy atoms.

Algebraic Diagrammatic Construction Theory for Simulating Charged Excited States and Photoelectron Spectra

Charged excitations are electronic transitions that involve a change in the total charge of a molecule or material. Understanding the properties and reactivity of charged species requires insights from theoretical calculations that can accurately describe orbital relaxation and electron correlation effects in open-shell electronic states. In this Review, we describe the current state of algebraic diagrammatic construction (ADC) theory for simulating charged excitations and its recent developments. We start with a short overview of ADC formalism for the one-particle Green's function, including its single- and multireference formulations and extension to periodic systems. Next, we focus on the capabilities of ADC methods and discuss recent findings about their accuracy for calculating a wide range of excited-state properties. We conclude our Review by outlining possible directions for future developments of this theoretical approach.

Real-Time Equation-of-Motion Coupled-Cluster Cumulant Green's Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-Body Methods Infrastructure

We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function method [ J. Chem. Phys. 2020, 152, 174113] within the Tensor Algebra for Many-body Methods (TAMM) infrastructure. TAMM is a massively parallel heterogeneous tensor library designed for utilizing forthcoming exascale computing resources. The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin-explicit forms of the various operators when evaluating the tensor contractions. Unlike our previous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation supports fully complex algebra. The RT-EOM-CC singles (S) and doubles (D) time-dependent amplitudes are propagated using a first-order Adams-Moulton method. This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis functions, with parallel efficiencies above 90% up to 400 GPUs. The TAMM RT-EOM-CCSD was used to study core photoemission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules. Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals. The relative quasiparticle ionization energies and overall spectral functions agree well with available experimental results.

Periodic Coupled-Cluster Green's Function for Photoemission Spectra of Realistic Solids

We present an efficient implementation of the coupled-cluster Green's function (CCGF) method for simulating photoemission spectra of periodic systems. We formulate the periodic CCGF approach with Brillouin zone sampling in the Gaussian basis at the coupled-cluster singles and doubles (CCSD) level. To enable CCGF calculations of realistic solids, we propose an active-space self-energy correction scheme by combining CCGF with the cheaper many-body perturbation theory (GW) and implement the model order reduction (MOR) frequency interpolation technique. We find that the active-space self-energy correction and MOR techniques significantly reduce the computational cost of CCGF while maintaining the high accuracy. We apply the developed CCGF approaches to compute spectral properties and band structure of silicon (Si) and zinc oxide (ZnO) crystals using triple-ζ Gaussian basis sets and medium-size k-point sampling and find good agreement with experimental measurements.

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.

Equation-of-Motion Coupled-Cluster Cumulant Green's Function for Excited States and X-Ray Spectra

Green’s function methods provide a robust, general framework within many-body theory for treating electron correlation in both excited states and x-ray spectra. Conventional methods using the Dyson equation or the cumulant expansion are typically based on the GW self-energy approximation. In order to extend this approximation in molecular systems, a non-perturbative real-time coupled-cluster cumulant Green’s function approach has been introduced, where the cumulant is obtained as the solution to a system of coupled first order, non-linear differential equations. This approach naturally includes non-linear corrections to conventional cumulant Green’s function techniques where the cumulant is linear in the GW self-energy. The method yields the spectral function for the core Green’s function, which is directly related to the x-ray photoemission spectra (XPS) of molecular systems. The approach also yields very good results for binding energies and satellite excitations. The x-ray absorption spectrum (XAS) is then calculated using a convolution of the core spectral function and an effective, one-body XAS. Here this approach is extended to include the full coupled-cluster-singles (CCS) core Green’s function by including the complete form of the non-linear contributions to the cumulant as well as all single, double, and triple cluster excitations in the CC amplitude equations. This approach naturally builds in orthogonality and shake-up effects analogous to those in the Mahan-Noizeres-de Dominicis edge singularity corrections that enhance the XAS near the edge. The method is illustrated for the XPS and XAS of NH₃.

Efficient implementation of the single-reference algebraic diagrammatic construction theory for charged excitations: Applications to the TEMPO radical and DNA base pairs

We present an efficient implementation of the second- and third-order single-reference algebraic diagrammatic construction (ADC) theory for electron attachment and ionization energies and spectra [EA/IP-ADC(n), n = 2, 3]. Our new EA/IP-ADC program features spin adaptation for closed-shell systems, density fitting for efficient handling of the two-electron integral tensors, and vectorized and parallel implementation of tensor contractions. We demonstrate capabilities of our efficient implementation by applying the EA/IP-ADC(n) (n = 2, 3) methods to compute the photoelectron spectrum of the (2,2,6,6-tetramethylpiperidin-1-yl)oxyl (TEMPO) radical, as well as the vertical and adiabatic electron affinities of TEMPO and two DNA base pairs (guanine-cytosine and adenine-thymine). The spectra and electron affinities computed using large diffuse basis sets with up to 1028 molecular orbitals are found to be in good agreement with the best available results from the experiment and theoretical simulations.

Approximate Green's Function Coupled Cluster Method Employing Effective Dimension Reduction

The Green's function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structure of molecular and periodic systems, especially when electrons of the system are strongly correlated. However, in order for the GFCC to become a method that may be routinely used in the electronic structure calculations, robust numerical techniques and approximations must be employed to reduce its extremely high computational overhead. In our recent studies, it has been demonstrated that the GFCC equations can be solved directly in the frequency domain using iterative linear solvers, which can be easily distributed in a massively parallel environment. In the present work, we demonstrate a successful application of model-order-reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires high-resolution descriptions of spectral function, instead of solving the GFCC linear equation of full dimension for every single frequency point of interest, an efficiently solvable linear system model of a reduced dimension may be built upon projecting the original GFCC linear system onto a subspace. From this reduced order model is obtained a reasonable approximation to the full dimensional GFCC linear equations in both interpolative and extrapolative spectral regions. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some selected frequencies within the spectral region of interest. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The method is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems such as carbon monoxide, 1,3-butadiene, benzene, and adenine. To reach the same level of accuracy as that of the original GFCC method, the application of MOR in the GFCC method is able to significantly lower the original computational cost for the aforementioned molecules in designated frequency regimes. As a byproduct, the reduced order model obtained by this method is found to provide a high-quality initial guess, which improves the convergence rate for the existing iterative linear solver.

Wannier interpolation of one-particle Green's functions from coupled-cluster singles and doubles (CCSD)

We propose two schemes for interpolation of the one-particle Green's function (GF) calculated within a coupled-cluster singles and doubles (CCSD) method for a periodic system. These schemes use Wannier orbitals for circumventing huge cost for a large number of sampled k points. One of the schemes is the direct interpolation, which obtains the GF straightforwardly by using Fourier transformation. The other is the self-energy-mediated interpolation, which obtains the GF via the Dyson equation. We apply the schemes to a LiH chain and trans-polyacetylene and examine their validity in detail. It is demonstrated that the direct-interpolated GFs suffer from numerical artifacts stemming from slow convergence of CCSD GFs in real space, while the self-energy-mediated interpolation provides more physically appropriate GFs due to the localized nature of CCSD self-energies. Our schemes are also applicable to other correlated methods capable of providing GFs.

Green's function coupled cluster formulations utilizing extended inner excitations

In this paper, we analyze new approximations of the Green's function coupled cluster (GFCC) method where locations of poles are improved by extending the excitation level of inner auxiliary operators. These new GFCC approximations can be categorized as the GFCC-i(n, m) method, where the excitation level of the inner auxiliary operators (m) used to describe the ionization potential and electron affinity effects in the N - 1 and N + 1 particle spaces is higher than the excitation level (n) used to correlate the ground-state coupled cluster wave function for the N-electron system. Furthermore, we reveal the so-called "n + 1" rule in this category [or the GFCC-i(n, n + 1) method], which states that in order to maintain size-extensivity of the Green's function matrix elements, the excitation level of inner auxiliary operators X _p (ω) and Y _q (ω) cannot exceed n + 1. We also discuss the role of the moments of coupled cluster equations that in a natural way assures these properties. Our implementation in the present study is focused on the first approximation in this GFCC category, i.e., the GFCC-i(2,3) method. As our first practice, we use the GFCC-i(2,3) method to compute the spectral functions for the N₂ and CO molecules in the inner and outer valence regimes. In comparison with the Green's function coupled cluster singles, doubles results, the computed spectral functions from the GFCC-i(2,3) method exhibit better agreement with the experimental results and other theoretical results, particularly in terms of providing higher resolution of satellite peaks and more accurate relative positions of these satellite peaks with respect to the main peak positions.

Green's Function Coupled-Cluster Approach: Simulating Photoelectron Spectra for Realistic Molecular Systems

In this paper, we present an efficient implementation for the analytical energy-dependent Green's function coupled-cluster with singles and doubles (GFCCSD) approach with our first practice being computing spectral functions of realistic molecular systems. Because of its algebraic structure, the presented method is highly scalable and is capable of computing spectral function for a given molecular system in any energy region. Several typical examples have been given to demonstrate its capability of computing spectral functions not only in the valence band but also in the core-level energy region. Satellite peaks have been observed in the inner valence band and core-level energy region where a many-body effect becomes significant and the single particle picture of ionization often breaks down. The accuracy test has been carried out by extensively comparing the computed spectral functions by our GFCCSD method with experimental photoelectron spectra as well as the theoretical ionization potentials obtained from other methods. It turns out the GFCCSD method is able to provide a qualitative or semiquantitative level of description of ionization processes in both the core and valence regimes. To significantly improve the GFCCSD results for the main ionic states, a larger basis set can usually be employed, whereas the improvement of the GFCCSD results for the satellite states needs higher-order many-body terms to be included in the GFCC implementation.

Development and application of a 2-electron reduced density matrix approach to electron transport via molecular junctions

Can an electronic device be constructed using only a single molecule? Since this question was first asked by Aviram and Ratner in the 1970s [Chem. Phys. Lett. 29, 277 (1974)], the field of molecular electronics has exploded with significant experimental advancements in the understanding of the charge transport properties of single molecule devices. Efforts to explain the results of these experiments and identify promising new candidate molecules for molecular devices have led to the development of numerous new theoretical methods including the current standard theoretical approach for studying single molecule charge transport, i.e., the non-equilibrium Green's function formalism (NEGF). By pairing this formalism with density functional theory (DFT), a wide variety of transport problems in molecular junctions have been successfully treated. For some systems though, the conductance and current-voltage curves predicted by common DFT functionals can be several orders of magnitude above experimental results. In addition, since density functional theory relies on approximations to the exact exchange-correlation functional, the predicted transport properties can show significant variation depending on the functional chosen. As a first step to addressing this issue, the authors have replaced density functional theory in the NEGF formalism with a 2-electron reduced density matrix (2-RDM) method, creating a new approach known as the NEGF-RDM method. 2-RDM methods provide a more accurate description of electron correlation compared to density functional theory, and they have lower computational scaling compared to wavefunction based methods of similar accuracy. Additionally, 2-RDM methods are capable of capturing static electron correlation which is untreatable by existing NEGF-DFT methods. When studying dithiol alkane chains and dithiol benzene in model junctions, the authors found that the NEGF-RDM predicts conductances and currents that are 1-2 orders of magnitude below those of B3LYP and M06 DFT functionals. This suggests that the NEGF-RDM method could be a viable alternative to NEGF-DFT for molecular junction calculations.