Perspective on Coupled-cluster Theory. The evolution toward simplicity in quantum chemistry.

Coupled-cluster theory has revolutionized quantum chemistry. It has provided the framework to effectively solve the problem of electron correlation, the main focus of the field for over 60 years. This has enabled ab initio quantum chemistry to provide predictive quality results for most quantities of interest that are obtainable from first-principle calculations. The best that one can do in a basis is the 'full CI,' the exact solution of the non-relativistic Schrödinger equation or, if need be, the relativistic Dirac equation. With due regard to converging the basis set and adequate consideration of higher clusters and relativity in a calculation, virtually predictive results can be obtained. But in addition to its numerical performance, coupled-cluster theory also offers a conceptually new, many-body foundation for the theory that should be appreciated by all practitioners. The latter is emphasized in this perspective, leading to the 'evolution toward simplicity' in the title. The ultimate theory will benefit from the several features that are uniquely exact in coupled-cluster theory and its equation-of-motion (EOM-CC) extensions.

A new "gold standard": Perturbative triples corrections in unitary coupled cluster theory and prospects for quantum computing.

A major difficulty in quantum simulation is the adequate treatment of a large collection of entangled particles, synonymous with electron correlation in electronic structure theory, with coupled cluster (CC) theory being the leading framework for dealing with this problem. Augmenting computationally affordable low-rank approximations in CC theory with a perturbative account of higher-rank excitations is a tractable and effective way of accounting for the missing electron correlation in those approximations. This is perhaps best exemplified by the "gold standard" CCSD(T) method, which bolsters the baseline CCSD with the effects of triple excitations using considerations from many-body perturbation theory (MBPT). Despite this established success, such a synergy between MBPT and the unitary analog of CC theory (UCC) has not been explored. In this work, we propose a similar approach wherein converged UCCSD amplitudes are leveraged to evaluate energy corrections associated with triple excitations, leading to the UCCSD[T] method. In terms of quantum computing, this correction represents an entirely classical post-processing step that improves the energy estimate by accounting for triple excitation effects without necessitating new quantum algorithm developments or increasing demand for quantum resources. The rationale behind this choice is shown to be rigorous by studying the properties of finite-order UCC energy functionals, and our efforts do not support the addition of the fifth-order contributions as in the (T) correction. We assess the performance of these approaches on a collection of small molecules and demonstrate the benefits of harnessing the inherent synergy between MBPT and UCC theories.

A comparison of QTP functionals against coupled-cluster methods for EAs of small organic molecules.

EA-EOM-CCSD electron affinities and LUMO energies of various Kohn-Sham density functional theory (DFT) methods are calculated for an a priori IP benchmark set of 64 small, closed-shell molecules. The purpose of these calculations was to investigate whether the QTP KS-DFT functionals can emulate EA-EOM-CC with only a mean-field approximation. We show that the accuracy of DFT-relative to CCSD-improves significantly when elements of correlated orbital theory are introduced into the parameterization to define the QTP family of functionals. In particular, QTP(02), which has only a single range separation parameter, provides results accurate to a MAD of <0.15 eV for the whole set of 64 molecules compared to EA-EOM-CCSD, far exceeding the results from the non-QTP family of density functionals.

Dark Doubly Excited States with Modified Coupled Cluster Models: A Reliable Compromise between Cost and Accuracy?

To treat doubly excited states, the treatment of triple excitations is considered necessary in the framework of equation-of-motion coupled cluster (EOM-CC) methods. We investigate models without explicit triples and seek quantitative measure for the performance of EOM based on CC with singles and doubles (CCSD) or modified CCSD (Distinguishable Cluster Approximation) approaches for states with predominant double excitation character. We also test the efficacy of including triples in perturbative manner through EOM-CCSD(T) and in an iterative way through EOM-CCSDT-3 method. Extended similarity transformed EOM-CCSD(EXT-STEOM-CCSD) method is also tested and provides superior quality results at comparatively low cost. We use the QUEST2 benchmark set of double excitations proposed by Loos et al. [ J. Chem. Theory Comput.2019, 15, 1939] to investigate the performance of methods such as EOM-CCSD, EOM-DCSD, EXT-STEOM-CCSD, ΔCCSD, and ΔDCSD. We also test a tailored CC approach, ΔpairCCD-TCCSD.

On the iterative diagonalization of matrices in quantum chemistry: Reconciling preconditioner design with Brillouin-Wigner perturbation theory.

Iterative diagonalization of large matrices to search for a subset of eigenvalues that may be of interest has become routine throughout the field of quantum chemistry. Lanczos and Davidson algorithms hold a monopoly, in particular, owing to their excellent performance on diagonally dominant matrices. However, if the eigenvalues happen to be clustered inside overlapping Gershgorin disks, the convergence rate of both strategies can be noticeably degraded. In this work, we show how Davidson, Jacobi-Davidson, Lanczos, and preconditioned Lanczos correction vectors can be formulated using the reduced partitioning procedure, which takes advantage of the inherent flexibility promoted by Brillouin-Wigner perturbation (BW-PT) theory's resolvent operator. In doing so, we establish a connection between various preconditioning definitions and the BW-PT resolvent operator. Using Natural Localized Molecular Orbitals (NLMOs) to construct Configuration Interaction Singles (CIS) matrices, we study the impact the preconditioner choice has on the convergence rate for these comparatively dense matrices. We find that an attractive by-product of preconditioning the Lanczos algorithm is that the preconditioned variant only needs 21%-35% and 54%-61% of matrix-vector operations to extract the lowest energy solution of several Hartree-Fock- and NLMO-based CIS matrices, respectively. On the other hand, the standard Davidson preconditioning definition seems to be generally optimal in terms of requisite matrix-vector operations.

Excited states with pair coupled cluster doubles tailored coupled cluster theory.

It is known that some non-dynamic effects of electron correlation can be included in coupled cluster theory using a tailoring technique that separates the effects of non-dynamic and dynamic correlations. Recently, the simple pCCD (pair coupled cluster doubles) wavefunction was shown to provide good results for some non-dynamic correlation problems, such as bond-breaking, in a spin-adapted way with no active space selection. In this paper, we report a study of excited states using "tailored coupled cluster singles and doubles," to attempt to use pCCD as a kernel for more complete coupled-cluster singles and doubles (CCSD) results for excited states. Several excited states are explored from those primarily due to single excitations to those dominated by doubly excited states and from singlet-triplet splittings for some diradical states. For the first two situations, tailored pCCD-TCCSD offers no improvement over equation of motion-CCSD. However, when we explore the singlet-triplet gap of diradical molecules that are manifestly multi-reference, a pCCD kernel provides improved results, particularly with generalized valence bond orbitals.

Examining fundamental and excitation gaps at the thermodynamic limit: A combined (QTP) DFT and coupled cluster study on trans-polyacetylene and polyacene.

Interest in ab initio property prediction of π-conjugated polymers for technological applications places significant demand on "cost-effective" and conceptual computational methods, particularly effective, one-particle theories. This is particularly relevant in the case of Kohn-Sham Density Functional Theory (KS-DFT) and its new competitors that arise from correlated orbital theory, the latter defining the QTP family of DFT functionals. This study presents large, ab initio equation of motion-coupled cluster calculations using the massively parallel ACESIII to target the fundamental bandgap of two prototypical organic polymers, trans-polyacetylene (tPA) and polyacene (Ac), and provides an assessment of the new quantum theory project (QTP) functionals for this problem. Further results focusing on the 1Ag (1Ag), 1Bu (1B2u), and 3Bu (3B2u) excited states of tPA (Ac) are also presented. By performing calculations on oligomers of increasing size, extrapolations to the thermodynamic limit for the fundamental and all excitation gaps, as well as estimations of the exciton binding energy, are made. Thermodynamic-limit results for a combination of "optimal" and model geometries are presented. Calculated results for excitations that are adequately described using a single-particle model illustrate the benefits of requiring a KS-DFT functional to satisfy the Bartlett ionization potential theorem.

Benchmarking isotropic hyperfine coupling constants using (QTP) DFT functionals and coupled cluster theory.

Significant effort has been devoted to benchmarking isotropic hyperfine coupling constants for both wavefunction and density-based approaches in recent years, as accurate theoretical predictions aid the fitting of experimental model Hamiltonians. However, literature examining the predictive quality of a Density Functional Theory (DFT) functional abiding by the Bartlett IP condition is absent. In an attempt to rectify this, we report isotropic hyperfine coupling constant predictions of 24 commonly used DFT functionals on a total of 56 radicals, with the intent of exploring the successes and failures of the Quantum Theory Project (QTP) line of DFT functionals (i.e., CAM-QTP00, CAM-QTP01, CAM-QTP02, and QTP17) for this property. Included in this benchmark study are both small and large organic radicals as well as transition metal complexes, all of which have been studied to some extent in prior work. Subsequent coupled-cluster singles and doubles (CCSD) and CCSD withperturbative triples [CCSD(T)] calculations on small and large organic radicals show modest improvement as compared to prior work and offer an additional avenue for evaluation of DFT functional performance. We find that the QTP17 and CAM-QTP00 functionals consistently underperform, despite being parameterized to satisfy an IP eigenvalue condition primarily focused on inner shell electrons. On the other hand, the CAM-QTP01 functional is the most accurate functional in both organic radical datasets. Furthermore, both CAM-QTP01 and CAM-QTP02 are the most accurate functionals tested on the transition metal dataset. A significant portion of functionals were found to have comparable errors (within 5-15 MHz), but the hybrid class of DFT functionals maintains a consistently optimal balance between accuracy and precision across all datasets.

Density functionals for core excitations.

The core excitation energies and related principal ionization energies are obtained for selected molecules using several density functionals and compared with benchmark equation-of-motion coupled cluster (EOM-CC) results. Both time-dependent and time-independent formulations of excitation spectra in the time-dependent density functional theory and the EOM-CC are employed to obtain excited states that are not always easily accessible with the time-independent method. Among those functionals, we find that the QTP(00) functional, which is only parameterized to reproduce the five IPs of water, provides excellent core IPs and core excitation energies, consistently yielding better excitation and ionization energies. We show that orbital eigenvalues of KS density functional theory play an important role in determining the accuracy of the excitation and photoelectron spectra.

The intermediate state approach for doubly excited dark states in EOM-coupled-cluster theory.

Solution of dark, doubly excited states using equation-of-motion coupled-cluster (EOM-CC) usually equires at least triple excitations or even quadruples beyond the standard singles and doubles (EOM-CCSD) for an appropriate treatment. A new route to obtain these doubly excited states using EOM-CCSD is demonstrated. Traditionally, EOM-CC is performed on a closed shell reference state that has a well-described single reference CC wavefunction. In this Communication, we attempt to use low spin open-shell states such as the MS = 0 triplet and open-shell singlet as a reference state. Using this intermediate excited state as a reference state provides us with the benefit of obtaining a doubly excited state, as a single excitation at the cost of EOM-CCSD.

Equation of motion coupled-cluster study of core excitation spectra II: Beyond the dipole approximation.

We present the time-independent (TI) and time-dependent (TD) equation of motion coupled-cluster (EOM-CC) oscillator strengths not limited to those obtained by the dipole approximation. For the conventional TI-EOM-CC, we implement all the terms in the multipole expansion through second order that contributes to the oscillator strength. These include contributions such as magnetic dipole, electric quadrupole, electric octupole, and magnetic quadrupole. In TD-EOM-CC, we only include the quadrupole moment contributions. This augments our previous work [Y. C. Park, A. Perera, and R. J. Bartlett, J. Chem. Phys. 151, 164117 (2019)]. The inclusion of the quadrupole contributions (and all the other contributions through second order in the case of TI-EOM-CCSD) enables us to obtain the intensities for the pre-edge transitions in the metal K-edge spectra, which are dipole inactive. The TI-EOM-CCSD and TD-EOM-CCSD spectra of Ti4+ atoms are used to showcase the implementation of the second-order oscillator strengths. The origin of 1s → e and 1s → t2 in core spectra from iron tetrachloride and titanium tetrachloride is discussed and compared with the experiment.

The Devil's Triangle of Kohn-Sham density functional theory and excited states.

Exchange-correlation (XC) functionals from Density Functional Theory (DFT) developed under the rigorous arguments of Correlated Orbital Theory (COT) address the Devil's Triangle of prominent errors in Kohn-Sham (KS) DFT. At the foundation of this triangle lie the incorrect one-particle spectrum, the lack of integer discontinuity, and the self-interaction error. At the top level, these failures manifest themselves in incorrect charge transfer and Rydberg excitation energies, along with poor activation barriers. Accordingly, the Quantum Theory Project (QTP) XC functionals have been created to address several of the long-term issues encountered in KS theory and its Time Dependent DFT (TDDFT) variant for electronic excitations. Recognizing that COT starts with a correct one-particle spectrum, a condition imposed on the QTP functionals by means of minimum parameterization, the question that arises is how does this affect the electronically excited states? Among up to 28 XC functionals considered, the QTP family provides one of the smallest mean absolute deviations for charge-transfer excitations while also showing excellent results for Rydberg states. However, there is some room for improvement in the case of excitation energies to valence states, which are systematically underestimated by all functionals investigated. An alternative path for better treatment of excitation energies, mainly for valence states, is offered by using orbital energies from QTP functionals, especially by CAM-QTP-02 and LC-QTP. In this case, the deviations from the reference data can be reduced approximately by half.

A route to improving RPA excitation energies through its connection to equation-of-motion coupled cluster theory.

We revisit the connection between equation-of-motion coupled cluster (EOM-CC) and random phase approximation (RPA) explored recently by Berkelbach [J. Chem. Phys. 149, 041103 (2018)] and unify various methodological aspects of these diverse treatments of ground and excited states. The identity of RPA and EOM-CC based on the ring coupled cluster doubles is established with numerical results, which was proved previously on theoretical grounds. We then introduce new approximations in EOM-CC and RPA family of methods, assess their numerical performance, and explore a way to reap the benefits of such a connection to improve on excitation energies. Our results suggest that addition of perturbative corrections to account for double excitations and missing exchange effects could result in significantly improved estimates.

Advanced concepts in electronic structure (ACES) software programs.

The advanced concepts in electronic structure (ACES) programs are products of the Bartlett research group at the University of Florida. They consist of ACES II, which is serial, and ACES III and Aces4, which are massively parallel. All three programs are publically available free of charge. The focus of the ACES implementations is coupled cluster theory and many-body-perturbation theory. We give an overview of the ACES programs, discuss the many features of the program systems, and document the number of benchmarks.

Index of multi-determinantal and multi-reference character in coupled-cluster theory.

A full configuration interaction calculation (FCI) ultimately defines the innate molecular orbital description of a molecule. Its density matrix and the natural orbitals obtained from it quantify the difference between having N-dominantly occupied orbitals in a reference determinant for a wavefunction to describe N-correlated electrons and how many of those N-electrons are left to the remaining virtual orbitals. The latter provides a measure of the multi-determinantal character (MDC) required to be in a wavefunction. MDC is further split into a weak correlation part and a part that indicates stronger correlation often called multi-reference character (MRC). If several virtual orbitals have high occupation numbers, then one might argue that these additional orbitals should be allowed to have a larger role in the calculation, as in MR methods, such as MCSCF, MR-CI, or MR-coupled-cluster (MR-CC), to provide adequate approximations toward the FCI. However, there are problems with any of these MR methods that complicate the calculations compared to the uniformity and ease of application of single-reference CC calculations (SR-CC) and their operationally single-reference equation-of-motion (EOM-CC) extensions. As SR-CC theory is used in most of today's "predictive" calculations, an assessment of the accuracy of SR-CC at some truncation of the cluster operator would help to quantify how large an issue MRC actually is in a calculation, and how it might be alleviated while retaining the convenient SR computational character of CC/EOM-CC. This paper defines indices that identify MRC situations and help assess how reliable a given calculation is.

Perturbation Improved Natural Linear-Scaled Coupled-Cluster Method and Its Application to Conformational Analysis.

The fragment-based coupled-cluster (CC) theory utilizing the transferable functional groups through natural localized molecular orbital (NLMO), that is, the natural linear-scaled coupled-cluster (NLSCC) has been further developed to take the extra-fragment interactions into account. The correction to the interaction energies sacrificed during the fragmentation process for the previous NLSCC method is computed by a computationally efficient perturbation theory that maintains the original scaling. The new linear-scaled coupled-cluster for the singles and doubles (CCSD) method is applied to the analysis of relative energies of delicate conformational problems of polypeptides. By adding a perturbation correction, results accurate to less than a kcal/mol are obtained for the alanine tetramer.

Adventures in DFT by a wavefunction theorist.

The attraction density functional theory (DFT) has for electronic structure theory is that it is easier to do computationally than ab initio, correlated wavefunction methods, due to its effective one-particle structure. On the contrary, ab initio theorists insist on the ability to converge to the right answer in appropriate limits, but this requires a treatment of the reduced two-particle density matrix. DFT avoids that by appealing to an "existence" theorem (not a constructive one) that all its effects are subsummed into a DFT functional of the one-particle density. However, the existence of thousands of DFT functionals emphasizes that there is no satisfactory way to systematically improve the Kohn-Sham (KS) version as most changes in parameterization or formulation seldom lead to a new functional that is genuinely better than others. Some researchers in the DFT community try to address this issue by imposing conditions rigorously derived from exact DFT considerations, but to date, no one has shown how this route will ever lead to converged results even for the ground state, much less for all the other electronic states obtained from time-dependent DFT that are critically important for chemistry. On the contrary, coupled-cluster (CC) theory and its equation-of-motion extensions provide rigorous results for both that KS-DFT methods are attempting to emulate. How to use them and their exact formal properties to tie CC theory to an effective one-particle form is the target of this perspective. This route addresses the devil's triangle of KS-DFT problems: the one-particle spectrum, self-interaction, and the integer discontinuity.

Equation of motion coupled-cluster for core excitation spectra: Two complementary approaches.

This paper presents core excitation spectra from coupled-cluster (CC) theory obtained from both a time-independent and a new time-dependent formalism. The conventional time-independent CC formulation for excited states is the equation-of-motion (EOM-CC) method whose eigenvalues and eigenvectors describe the core excited states. An alternative computational procedure is offered by a time-dependent CC description. In that case, the dipole transition operator is expressed in the CC effective Hamiltonian form and propagated with respect to time. The absorption spectrum is obtained from the CC dipole autocorrelation function via a Fourier transformation. Comparisons are made among the time-dependent results obtained from second-order perturbation theory, to coupled cluster doubles and their linearized forms (CCD and LCCD), to CC singles and doubles (CCSD) and the linearized form (LCCSD). In the time-independent case, considerations of triples (EOM-CCSDT) and quadruples (EOM-CCSDTQ) are used to approach sub-electron volt accuracy. A particular target is the allyl radical, as an example of an open-shell molecule. As the results have to ultimately be the same, the two procedures offer a complementary approach toward analyzing experimental results.

Similarity-transformed equation-of-motion coupled-cluster singles and doubles method with spin-orbit effects for excited states.

The similarity transformed equation-of-motion coupled-cluster method (STEOM-CCSD) for excited states is extended to treat spin-orbit coupling interactions (SOIs) using the spin-orbit mean field approximation of the Breit-Pauli Hamiltonian. Two possible schemes to include the spin-orbit splittings of excited states within the STEOM-CCSD model are formulated. They are identified as "diagonalize-then-perturb" and "perturb-then-diagonalize" approaches. The second approach is more suited for cases where SOI is larger, and the first approach breaks down. With the aid of the standard many-body diagrammatic techniques, expressions for all the necessary matrix elements can be derived. These new formulations are implemented in the ACES III suite of parallel coupled cluster programs, and benchmark studies are performed. Numerical tests for several atoms and molecules show a good agreement of calculated spin-orbit splittings to experiment, while also documenting the numerical differences between the two approaches.

Vertical valence ionization potential benchmarks from equation-of-motion coupled cluster theory and QTP functionals.

The ionization potential (IP) of a molecule quantifies the energy required to remove an electron from the system. As such, it is a fundamental quantity in the context of redox chemistry, charge transfer, and molecular electronics. The accurate theoretical prediction of this property is therefore highly desirable for virtual materials design. Furthermore, vertical IPs are of interest in the development of many-body Green's function methods like the GW formalism, as well as density functionals and semiempirical methods. In this contribution, we report over 1468 vertical valence IPs calculated with the IP variant of equation-of-motion coupled cluster theory with singles and doubles (IP-EOM-CCSD) covering 155 molecules. The purpose of this is two-fold: First, the quality of the predicted IPs is compared with respect to experiments and higher-order coupled cluster theory. This confirms the overall high accuracy and robustness of this method, with some outliers which are discussed in detail. Second, a large set of consistent theoretical reference values for vertical valence IPs are generated. This addresses a lack of reliable reference data for lower-lying valence IPs, where experimental data are often unavailable or of dubious quality. The benchmark set is then used to assess the quality of the eigenvalues predicted by different density functional approximations (via Bartlett's IP-eigenvalue theorem) and the extended Koopmans' theorem approach. The QTP family of functionals are found to be remarkably accurate, low-cost alternatives to IP-EOM-CCSD.