Equation-of-motion Coupled-cluster Research Articles

Cholesky Decomposition-Based Implementation of Relativistic Two-Component Coupled-Cluster Methods for Medium-Sized Molecules.

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field spin-orbit integrals (the X2CAMF scheme) is reported. The present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. Our CD-based implementation extends the applicability of X2CAMF-CC and EOM-CC methods to medium-sized molecules with the possibility to correlate around 1000 spinors. Benchmark calculations for uranium-containing small molecules were performed to assess the dependence of the CC results on the Cholesky threshold. A Cholesky threshold of 10-4 is shown to be sufficient to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond-dissociation energy of the uranium hexafluoride molecule, UF6, with up to quadruple-ζ basis sets, and the lowest excitation energy in the solvated uranyl ion [UO22+(H2O)12].

Site-Specific Ionization Potentials and Electron Affinities in Large Molecular Systems at Coupled Cluster Level.

A many-body expansion of ionization potentials and electron affinities is developed based on a combination of the fragment molecular orbital method and equation-of-motion coupled-cluster (EOM-CC). In addition to site-specific values, obtained as one-body properties, pair and triple corrections are added to account for nonlocal EOM-CC contributions of the molecular environment of a chromophore. The developed method is applied to carboxylic acids, alkyl cations, a protein ubiquitin (Protein Data Bank ID 1UBQ), and a nano ribbon of white graphene elucidating the effect of environment on ionization potential and electron affinity.

Studying excited-state-specific perturbation theory on the Thiel set.

We explore the performance of a recently introduced N5-scaling excited-state-specific second order perturbation theory (ESMP2) on the singlet excitations of the Thiel benchmarking set. We find that, without regularization, ESMP2 is quite sensitive to π system size, performing well in molecules with small π systems but poorly in those with larger π systems. With regularization, ESMP2 is far less sensitive to π system size and shows a higher overall accuracy on the Thiel set than CC2, equation of motion-coupled cluster with singles and doubles, CC3, and a wide variety of time-dependent density functional approaches. Unsurprisingly, even regularized ESMP2 is less accurate than multi-reference perturbation theory on this test set, which can, in part, be explained by the set's inclusion of some doubly excited states but none of the strong charge transfer states that often pose challenges for state-averaging. Beyond energetics, we find that the ESMP2 doubles normoffers a relatively low-cost way to test for doubly excited character without the need to define an active space.

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.

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.

Reference Energies for Cyclobutadiene: Automerization and Excited States.

Cyclobutadiene is a well-known playground for theoretical chemists and is particularly suitable to test ground- and excited-state methods. Indeed, due to its high spatial symmetry, especially at the D4h square geometry but also in the D2h rectangular arrangement, the ground and excited states of cyclobutadiene exhibit multiconfigurational characters and single-reference methods, such as standard adiabatic time-dependent density-functional theory (TD-DFT) or standard equation-of-motion coupled cluster (EOM-CC), are notoriously known to struggle in such situations. In this work, using a large panel of methods and basis sets, we provide an extensive computational study of the automerization barrier (defined as the difference between the square and rectangular ground-state energies) and the vertical excitation energies at D2h and D4h equilibrium structures. In particular, selected configuration interaction (SCI), multireference perturbation theory (CASSCF, CASPT2, and NEVPT2), and coupled-cluster (CCSD, CC3, CCSDT, CC4, and CCSDTQ) calculations are performed. The spin-flip formalism, which is known to provide a qualitatively correct description of these diradical states, is also tested within TD-DFT (combined with numerous exchange-correlation functionals) and the algebraic diagrammatic construction [ADC(2)-s, ADC(2)-x, and ADC(3)]. A theoretical best estimate is defined for the automerization barrier and for each vertical transition energy.

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.

Linear response properties of solvated systems: a computational study.

We present a computational study of static and dynamic linear polarizabilities in solution. We use different theoretical approaches to describe solvent effects, ranging from quantum mechanics/molecular mechanics (QM/MM) to quantum embedding approaches. In particular, we consider non-polarizable and polarizable QM/MM methods, the latter based on the fluctuating charge (FQ) force field. In addition, we use a quantum embedding method defined in the context of multilevel Hartree-Fock (MLHF), where the system is divided into active and inactive regions, and combine it with a third layer described by means of the FQ model. The multiscale approaches are then used as reference wave functions for equation-of-motion coupled cluster (EOM-CC) response properties, allowing for the account of electron correlation. The developed models are applied to the calculation of linear response properties of two organic moieties-namely, para-nitroaniline and benzonitrile-in non-aqueous solvents-1,4-dioxane, acetonitrile, and tetrahydrofuran. The computed polarizabilities are then discussed in terms of the physico-chemical solute-solvent interactions described by each method (electrostatic, polarization and Pauli repulsion), and finally compared with the available experimental references.

Announcement of the winner of the Longuet-Higgins Early Career Researcher Prize 2020

The recently proposed approach to excited electronic states, in which the deterministic equation-of-motion coupled-cluster (EOMCC) framework is merged with stochastic configuration interaction Quantum Monte Carlo (CIQMC) computations [J.E. Deustua et al., J. Chem. Phys. 150, 111101 (2019)], is combined with the noniterative energy corrections derived from the CC(P;Q) formalism. By examining vertical excitations in CH+ at the equilibrium and stretched geometries and adiabatic excitations in CH and CNC, we demonstrate that the resulting semi-stochastic CC(P;Q) methodology converges target high-level energetics, represented in this study by the EOMCC method with singles, doubles, and triples, in the early stages of CIQMC propagations.

Internal Conversion between Bright (11Bu+) and Dark (21Ag-) States in s-trans-Butadiene and s-trans-Hexatriene.

Internal conversion (IC) between the two lowest singlet excited states, 11Bu+ and 21Ag-, of s-trans-butadiene and s-trans-hexatriene is investigated using a series of single- and multi- reference wave function and density functional theory (DFT) methodologies. Three independent types of the equation-of-motion coupled-cluster (EOMCC) theory capable of providing an accurate and balanced description of one- as well as two-electron transitions, abbreviated as δ-CR-EOMCC(2,3), DIP-EOMCC(4h2p){No}, and DEA-EOMCC(4p2h){Nu} or DEA-EOMCC(3p1h,4p2h){Nu}, consistently predict that the 11Bu+/21Ag- crossing in both molecules occurs along the bond length alternation coordinate. However, the analogous 11Bu+ and 21Ag- potentials obtained with some multireference approaches, such as CASSCF and MRCIS(D), as well as with the linear-response formulation of time-dependent DFT (TDDFT), do not cross. Hence, caution needs to be exercised when studying the low-lying singlet excited states of polyenes with conventional multiconfigurational methods and TDDFT. The multistate many-body perturbation theory methods, such as XMCQDPT2, do correctly reproduce the curve crossing. Among the simplest and least expensive computational methodologies, the DFT approaches that incorporate the contributions of doubly excited configurations, abbreviated as MRSF (mixed reference spin-flip) TDDFT and SSR(4,4), accurately reproduce our best EOMCC results. This is highly promising for nonadiabatic molecular dynamics simulations in larger systems.

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.

LCAO Electronic Structure of Nucleic Acid Bases and Other Heterocycles and Transfer Integrals in B-DNA, Including Structural Variability.

To describe the molecular electronic structure of nucleic acid bases and other heterocycles, we employ the Linear Combination of Atomic Orbitals (LCAO) method, considering the molecular wave function as a linear combination of all valence orbitals, i.e., 2s, 2p, 2p, 2p orbitals for C, N, and O atoms and 1s orbital for H atoms. Regarding the diagonal matrix elements (also known as on-site energies), we introduce a novel parameterization. For the non-diagonal matrix elements referring to neighboring atoms, we employ the Slater–Koster two-center interaction transfer integrals. We use Harrison-type expressions with factors slightly modified relative to the original. We compare our LCAO predictions for the ionization and excitation energies of heterocycles with those obtained from Ionization Potential Equation of Motion Coupled Cluster with Singles and Doubles (IP-EOMCCSD)/aug-cc-pVDZ level of theory and Completely Normalized Equation of Motion Coupled Cluster with Singles, Doubles, and non-iterative Triples (CR-EOMCCSD(T))/aug-cc-pVDZ level of theory, respectively, (vertical values), as well as with available experimental data. Similarly, we calculate the transfer integrals between subsequent base pairs, to be used for a Tight-Binding (TB) wire model description of charge transfer and transport along ideal or deformed B-DNA. Taking into account all valence orbitals, we are in the position to treat deflection from the planar geometry, e.g., DNA structural variability, a task impossible for the plane Hückel approach (i.e., using only 2p orbitals). We show the effects of structural deformations utilizing a 20mer evolved by Molecular Dynamics.

Why the CC Stretch in HCC Is So Anharmonic

The marked anharmonicity of the CC stretching mode in the ethynyl radical (C2H) is investigated in terms of a vibronic coupling model. It is demonstrated that the large difference between the harmonic frequency and the fundamental level (about 5-10 times that for triple-bond stretches in the related species HCCH, HCN, HNC and CN) can be attributed to the well-known vibronic interaction between the X̃2Σ+ and Ã2Π states of CCH. Although the mode has σ symmetry and it is the perturbations of π symmetry that mix the two electronic states, a combination of large intrinsic coupling strength, modest energy gap, and-most importantly-the strong tuning of the gap energy by the CC stretch mode leads to a profound vibronic influence on parts of the anharmonic force field that sample the CC stretch. Finally, calculations of the force field for the X̃2Σ+ state with different flavors of coupled-cluster (CC) theory provide insight and underscore an intrinsic advantage of equation-of-motion CC (EOM-CC) methods.

Capturing Correlation Effects on Photoionization Dynamics.

A highly correlated combination of the equation-of-motion coupled cluster (EOM-CC) Dyson orbital and the multicentric B-spline time-dependent density functional theory (TDDFT)-based approach is proposed and implemented within the single-channel approximation to describe molecular photoionization processes. The twofold objective of the approach is to capture interchannel coupling effects, missing in the B-spline DFT treatment, and to explore the response of Dyson orbitals to strong correlation effects and its influence on the photoionization observables. We validate our scheme by computing partial cross sections, branching ratios, asymmetry parameters, and molecular frame photoelectron angular distributions of simple molecules. Finally, the method has been applied to the study of photoelectron spectra of the Ni(C3H5)2 molecule, where giant correlation effects completely destroy the Koopmans picture.

Equation-of-Motion Coupled-Cluster Protocol for Calculating Magnetic Properties: Theory and Applications to Single-Molecule Magnets.

We present a new computational protocol for computing macroscopic magnetic properties of transition-metal complexes using the equation-of-motion coupled-cluster (EOM-CC) framework. The approach follows a two-step state-interaction scheme: we first compute zero-order states using nonrelativistic EOM-CC and then use these states to evaluate matrix elements of the spin-orbit and Zeeman operators. Diagonalization of the resulting Hamiltonian yields spin-orbit- and field-perturbed eigenstates. Temperature- and field-dependent magnetization and susceptibility are computed by numerical differentiation of the partition function. To compare with powder-sample experiments, these quantities are numerically averaged over field orientations. We applied this protocol to several single-molecule magnets (SMMs) with Fe(II) and Fe(III) in trigonal pyramidal, linear, and trigonal bipyramidal coordination environments. We described the underlying electronic structure by the electron-attachment (EOM-EA) and spin-flip (EOM-SF) variants of EOM-CC. The computed energy barriers for spin inversion, and macroscopic magnetization and susceptibility agree well with experimental data. Trends in magnetic anisotropy and spin-reversal energy barriers are explained in terms of a molecular orbital picture rigorously distilled from spinless transition density matrices between many-body states. The results illustrate excellent performances of EOM-CC in describing magnetic behavior of mononuclear transition-metal SMMs.

Short Iterative Lanczos Integration in Time-Dependent Equation-of-Motion Coupled-Cluster Theory.

A time-dependent (TD) formulation of equation-of-motion coupled-cluster (EOM-CC) theory can provide excited-state information over an arbitrarily wide energy window with a reduced memory footprint relative to conventional, frequency-domain EOM-CC theory. However, the floating-point costs of the time-integration required by TD-EOM-CC are generally far larger than those of the frequency-domain form of the approach. This work considers the potential of the short iterative Lanczos (SIL) integration scheme [J. Chem. Phys. 1986, 85, 5870-5876] to reduce the floating-point costs of TD-EOM-CC simulations. Low-energy and K-edge absorption features for small molecules are evaluated using TD-EOM-CC with single and double excitations, with the time-integrations carried out via SIL and fourth-order Runge-Kutta (RK4) schemes. Spectra derived from SIL- and RK4-driven simulations are nearly indistinguishable, and with an appropriately chosen subspace dimension, the SIL requires far fewer floating-point operations than are required by RK4. For K-edge spectra, SIL is the more efficient scheme by an average factor of 7.2.

Relativistic coupled‐cluster and equation‐of‐motion coupled‐cluster methods

AbstractThe development of relativistic coupled‐cluster (CC) and equation‐of‐motion coupled‐cluster (EOM‐CC) methods is reviewed. An emphasis is placed on recent efforts to improve the computational efficiency of CC and EOM‐CC calculations with non‐perturbative treatments of spin‐orbit coupling (SO‐CC and EOM‐CC) by partially recovering spin symmetry in the formulations. Example calculations of electronic ground state as well as valence‐excited and core‐excited states for molecules containing heavy elements are presented to demonstrate the applicability and usefulness of the SO‐CC and EOM‐CC methods. Future directions for the development of the SO‐CC and EOM‐CC methods are also discussed.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

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.

The role of high-order electron correlation effects in a model system for non-valence correlation-bound anions.

The diffusion Monte Carlo (DMC), auxiliary field quantum Monte Carlo (AFQMC), and equation-of-motion coupled cluster (EOM-CC) methods are used to calculate the electron binding energy (EBE) of the non-valence anion state of a model (H2O)4 cluster. Two geometries are considered, one at which the anion is unbound and the other at which it is bound in the Hartree-Fock (HF) approximation. It is demonstrated that DMC calculations can recover from the use of a HF trial wave function that has collapsed onto a discretized continuum solution, although larger EBEs are obtained when using a trial wave function for the anion that provides a more realistic description of the charge distribution and, hence, of the nodal surface. For the geometry at which the cluster has a non-valence correlation-bound anion, both the inclusion of triples in the EOM-CC method and the inclusion of supplemental diffuse d functions in the basis set are important. DMC calculations with suitable trial wave functions give EBE values in good agreement with our best estimate EOM-CC result. AFQMC using a trial wave function for the anion with a realistic electron density gives a value of the EBE nearly identical to the EOM-CC result when using the same basis set. For the geometry at which the anion is bound in the HF approximation, the inclusion of triple excitations in the EOM-CC calculations is much less important. The best estimate EOM-CC EBE value is in good agreement with the results of DMC calculations with appropriate trial wave functions.

Equation-of-motion coupled-cluster theory for double electron attachment with spin-orbit coupling.

We report implementation of the equation-of-motion coupled-cluster (EOM-CC) method for double electron-attachment (DEA) with spin-orbit coupling (SOC) at the CC singles and doubles (CCSD) level using a closed-shell reference in this work. The DEA operator employed in this work contains two-particle and three-particle one-hole excitations, and SOC is included in post-Hartree-Fock treatment. Time-reversal symmetry and spatial symmetry are exploited to reduce computational cost. The EOM-DEA-CCSD method with SOC allows us to investigate SOC effects of systems with two-unpaired electrons. According to our results on atoms, double ionization potentials (DIPs), excitation energies (EEs), and SO splittings of low-lying states are calculated reliably using the EOM-DEA-CCSD method with SOC. Its accuracy is usually higher than that of EOM-CCSD for EEs or DIPs if the same target can be reached from single excitations by choosing a proper closed-shell reference. However, performance of the EOM-DEA-CCSD method with SOC on molecules is not as good as that for atoms. Bond lengths for the ground and the several lowest excited states of GaH, InH, and TlH are underestimated pronouncedly, although reasonable EEs are obtained, and splittings of the 3Σ- state from the π2 configuration are calculated to be too small with EOM-DEA-CCSD.