Equation Of Motion Coupled Cluster Method 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].

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.

Electronic Structure of Liquid Alkanes: A Representative Case of Liquid Hexanes and Cyclohexane Studied Using Polarization-Dependent Two-Photon Absorption Spectroscopy.

Two-photon absorption (2PA) spectra of liquid cyclohexane and hexanes are reported for the energy range 6.4-8.5 eV (177-145 nm), providing detailed information about their electronic structures in bulk liquid. Using a broadband pump-probe fashion, we measured the continuous 2PA spectra by simultaneous absorption of a 266 nm (4.6 eV) pump photon and one UV-vis probe photon from the white-light continuum (1.8-3.9 eV). Theoretical one-photon absorption (1PA) and 2PA cross sections of isolated gas phase molecules are computed by the equation of motion coupled-cluster method with single and double substitutions (EOM-CCSD) to substantiate the assignment of the experimental spectra, and the natural transition orbital (NTO) analysis provides visualization of the participating orbitals in a transition. Our analysis suggests that upon solvation transitions at the lowest excitation energy involving promotion of electron to the 3s Rydberg orbitals are blue-shifted (∼0.55 eV for cyclohexane and ∼0.18 eV for hexanes) to a greater extent as compared to those involving other Rydberg orbitals, which is similar to the behavior observed for water and alcohols. All other transitions experience negligible (cyclohexane) or minor red-shift by ∼0.15-0.2 eV (hexane) upon solvation. In both alkanes, the spectra are entirely dominated by Rydberg transitions: the most intense bands in 1PA and 2PA spectra are due to the excitation of electrons to the Rydberg "p" and "d" type orbitals, respectively, although one transition terminating in the 3s Rydberg has significant 2PA strength. This work demonstrates that the gas phase electronic transition properties in alkanes are not significantly altered upon solvation. In addition, electronic structure calculations using an isolated-molecule framework appear to provide a reasonable starting point for a semiquantitative picture for spectral assignment and also to analyze the solvatochromic shifts for liquid phase absorption spectra.

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.

Accelerating convergence of equation-of-motion coupled-cluster computations using the semi-stochastic CC(P;Q) formalism

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.

A Multilayer Approach to the Equation of Motion Coupled-Cluster Method for the Electron Affinity.

We have presented a multilayer implementation of the equation of motion coupled-cluster method for the electron affinity, based on local and pair natural orbitals. The method gives consistent accuracy for both localized and delocalized anionic states and results in many-fold speed-up as compared to the canonical and DLPNO-based implementation of the EA-EOM-CCSD method. We have also developed an explicit fragment-based approach which can lead to even higher speed-up with little loss in accuracy. The multilayer method can be used to treat the environmental effect of both bonded and nonbonded nature on the electron attachment process in large molecules.

Predictive Simulations of Ionization Energies of Solvated Halide Ions with Relativistic Embedded Equation of Motion Coupled Cluster Theory

A subsystem approach for obtaining electron binding energies in the valence region is presented and applied to the case of halide ions (X^{-},X=F-At) in water. This approach is based on electronic structure calculations combining the relativistic equation-of-motion coupled cluster method for electron detachment and density functional theory via the frozen density embedding approach, using structures from classical molecular dynamics with polarizable force fields for discrete systems (in our study, droplets containing the anion and 50 water molecules). Our results indicate that one can accurately capture both the large solvent effect observed for the halides and the splitting of their ionization signals due to the increasingly large spin-orbit coupling of the p_{3/2}-p_{1/2} manifold across the series, at an affordable computational cost. Furthermore, owing to the quantum mechanical treatment of both solute and solvent electron binding energies of semiquantitative quality are also obtained for (bulk) water as by-products of the calculations for the halogens (in droplets).

Semiconductor color-center structure and excitation spectra: Equation-of-motion coupled-cluster description of vacancy and transition-metal defect photoluminescence

Valence excitation spectra are computed for all deep-center silicon-vacancy defect types in 3C, 4H, and 6H silicon carbide (SiC) and comparisons are made with literature photoluminescence measurements. Nuclear geometries surrounding the defect centers are optimized within a Gaussian basis-set framework using many-body perturbation theory or density functional theory (DFT) methods, with computational expenses minimized by a QM/MM technique called SIMOMM. Vertical excitation energies are subsequently obtained by applying excitation-energy, electron-attached, and ionized equation-of-motion coupled-cluster (EOMCC) methods, where appropriate, as well as time-dependent (TD) DFT, to small models including only a few atoms adjacent to the defect center. We consider the relative quality of various EOMCC and TD-DFT methods for (i) energy-ordering potential ground states differing incrementally in charge and multiplicity, (ii) accurately reproducing experimentally measured photoluminescence peaks, and (iii) energy-ordering defects of different types occurring within a given polytype. The extensibility of this approach to transition-metal defects is also tested by applying it to silicon-substitutional chromium defects in SiC and comparing with measurements. It is demonstrated that, when used in conjunction with SIMOMM-optimized geometries, EOMCC-based methods can provide a reliable prediction of the ground-state charge and multiplicity, while also giving a quantitative description of the photoluminescence spectra, accurate to within 0.1 eV of measurement in all cases considered.

Modeling Photoelectron Spectra of CuO, Cu2O, and CuO2 Anions with Equation-of-Motion Coupled-Cluster Methods: An Adventure in Fock Space.

The experimental photoelectron spectra of di- and triatomic copper oxide anions have been reported previously. We present an analysis of the experimental spectra of the CuO-, Cu2O-, and CuO2- anions using equation-of-motion coupled-cluster (EOM-CC) methods. The open-shell electronic structure of each molecule demands a unique combination of EOM-CC methods to achieve an accurate and balanced representation of the multiconfigurational anionic- and neutral-state manifolds. Analysis of the Dyson orbitals associated with photodetachment from CuO- reveals the strong non-Koopmans character of the CuO states. For the lowest detachment energy, a good agreement between theoretical and experimental values is obtained with CCSD(T) (coupled-cluster with single and double excitations and perturbative account of triple excitations). The (T) correction is particularly important for Cu2O-. Use of a relativistic pseudopotential and matching basis set improves the quality of results in most cases. EOM-DIP-CCSD analysis of the low-lying states of CuO2- reveals multiple singlet and triplet anionic states near the triplet ground state, adding an extra layer of complexity to the interpretation of the experimental CuO2- photoelectron spectrum.

Benchmark calculations on the nuclear quadrupole-coupling parameters for open-shell molecules using non-relativistic and scalar-relativistic coupled-cluster methods.

Quantum-chemical computations of nuclear quadrupole-coupling parameters for 24 open-shell states of small molecules based on non-relativistic and spin-free exact two-component (SFX2C) relativistic equation-of-motion coupled-cluster (EOM-CC) as well as spin-orbital-based restricted open-shell Hartree-Fock coupled-cluster (ROHF-CC) methods are reported. Relativistic effects, the performance of the EOM-CC and ROHF-CC methods for treating electron correlation, as well as basis-set convergence have been carefully analyzed. Consideration of relativistic effects is necessary for accurate calculations on systems containing third-row (K-Kr) and heavier elements, as expected, and the SFX2C approach is shown to be a useful cost-effective option here. Further, it is demonstrated that the EOM-CC methods constitute flexible and accurate alternatives to the ROHF-CC methods in the calculations of nuclear quadrupole-coupling parameters for open-shell states.

EOMCC over excited state Hartree-Fock solutions (ESHF-EOMCC): An efficient approach for the entire ground state potential energy curves of higher-order bonds

The equation-of-motion coupled cluster (EOMCC) method based on the excited state Hartree-Fock (ESHF) solutions is shown to be appropriate for computing the entire ground state potential energy curves of strongly correlated higher-order bonds. The new approach is best illustrated for the homolytic dissociation of higher-order bonds in molecules. The required multireference character of the true ground state wavefunction is introduced through the linear excitation operator of the EOMCC method. Even at the singles and doubles level of cluster excitation truncation, the nonparallelity error of the ground state potential energy curve from the ESHF based EOMCC method is small.

A relativistic equation-of-motion coupled-cluster investigation of the trends of single and double ionization potentials in the He and Be isoelectronic systems

We employ a four-component spinor relativistic equation-of-motion coupled-cluster (EOMCC) method within the single- and double- excitation approximation to calculate the single ionization potentials (IPs) and double ionization potentials (DIPs) of the He and Be isoelectronic sequences up to Ne. The obtained results are compared with the available results from the National Institute of Standards and Technology (NIST) database to test the performance of the EOMCC method. We also present intermediate results at different levels of approximations in the EOMCC framework to gain insight of the effect of electron correlation. Furthermore, we investigate the dependence of the IPs and DIPs of these ions on the ionic charge and observe that these follow parabolic trends. Similarities between the trends of IPs and DIPs in both the classes of considered systems are categorically demonstrated.

Modeling Excited States in TiO2 Nanoparticles: On the Accuracy of a TD-DFT Based Description.

We have investigated the suitability of Time-Dependent Density Functional Theory (TD-DFT) to describe vertical low-energy excitations in naked and hydrated titanium dioxide nanoparticles. Specifically, we compared TD-DFT results obtained using different exchange-correlation (XC) potentials with those calculated using Equation-of-Motion Coupled Cluster (EOM-CC) quantum chemistry methods. We demonstrate that TD-DFT calculations with commonly used XC potentials (e.g., B3LYP) and EOM-CC methods give qualitatively similar results for most TiO2 nanoparticles investigated. More importantly, however, we also show that, for a significant subset of structures, TD-DFT gives qualitatively different results depending upon the XC potential used and that only TD-CAM-B3LYP and TD-BHLYP calculations yield results that are consistent with those obtained using EOM-CC theory. Moreover, we demonstrate that the discrepancies for such structures originate from a particular combination of defects that give rise to charge-transfer excitations, which are poorly described by XC potentials that do not contain sufficient Hartree–Fock like exchange. Finally, we consider that such defects are readily healed in the presence of ubiquitously present water and that, as a result, the description of vertical low-energy excitations for hydrated TiO2 nanoparticles is nonproblematic.

Depletion of atmospheric ozone by nitrogen dioxide: a bifurcated reaction pathway

Theoretical investigation on the reaction of NO2 with O3 in the context of ozone depletion is reported in this study. The progress of this reaction in single or concrete steps is an open and highly challenging question for both experimentalists and theoreticians. The reaction is studied through DFT methods, M06-2X, BH and HLYP and MPW1K with 6-311++G(2df,2p) basis set and also at UCCSD and CASPT2 methods with 6-31G(d,p) basis set. The single-point energies of the reactive species are calculated at CASSCF, UCCSD(T), UQCISD(T) and MRCI methods with 6-311++G(2df,2p) basis set. The reaction proceeds along an intermediate through a rate-limiting first transition state. This intermediate is followed by a second transition state, and the intrinsic reaction coordinate (IRC) calculations from this transition state show that the reaction path follows a product channel and another transition state. Two sequential transition states are calculated and the potential energy surface bifurcation is observed through a valley-ridge inflection point. The IRC calculations from third transition state lead to second product channel in which NO and O2 are formed. The energy barrier calculated for the formation of NO3 and O2 in the first product channel is 33.96 kcal/mol, and an energy barrier of 34.07 kcal/mol is calculated for the formation of NO + 2O2 product channel. The rate constant calculated at 298 K for the two product channels is 2.31 × 10−7 cm3 molecule−1 s−1. These results show that the formation of the two product channels is equally probable. The photodissociation wavelength of the reactive species involved in the reaction system is calculated using equation of motion coupled cluster (EOMCC) method. The calculated photodissociation wavelength shows that a significant amount of NO2 photofragment enters into the stratosphere and potentially involves in the depletion of ozone. The photodissociation wavelength of the reactive species calculated using EOMCC method is in good agreement with the available experimental data.

Communication: Existence of the doubly excited state that mediates the photoionization of azulene

We use the high-level electronic structure computations based on the equation-of-motion coupled-cluster (EOMCC) theory to show that the previously postulated [V. Blanchet et al., J. Chem. Phys. 128, 164318 (2008)] doubly excited state of azulene, located below the ionization threshold and mediating the 1 + 2' multi-photon ionization that leads to a Rydberg fingerprint, exists. This supports the crucial role of doubly excited states in the Rydberg fingerprint spectroscopy, while demonstrating the usefulness of EOMCC methods in capturing such states.

Conical Intersection and Potential Energy Surface Features of a Model Retinal Chromophore: Comparison of EOM-CC and Multireference Methods.

This work investigates the performance of equation-of-motion coupled-cluster (EOM-CC) methods for describing the changes in the potential energy surfaces of the penta-2,4-dieniminium cation, a reduced model of the retinal chromophore of visual pigments, due to dynamical electron correlation effects. The ground-state wave function of this model includes charge-transfer and diradical configurations whose weights vary along different displacements and are rapidly changing at the conical intersection between the ground and the first excited states, making the shape of the potential energy surface sensitive to a balanced description of nondynamical and dynamical correlation. Recently, variational (MRCISD) and perturbative (MRPT2) approaches for including dynamical correlation in CASSCF-based calculations were tested along three representative ground state paths. Here, we use the same three paths to compare the performance of single-reference EOM-CC methods against MRCISD and MRCISD+Q. We find that the spin-flip variant of EOM-CCSD with perturbative inclusion of triple excitations (dT or fT) produces potential energy profiles of the two lowest electronic states in quantitative agreement with MRCISD+Q (our highest-quality reference method). The nonparallelity errors and differences in vertical energy differences of the two surfaces along these scans are less than 1.4 kcal/mol (EOM-SF-CCSD(dT) versus MRCISD+Q). For comparison, the largest error of MRCISD versus MRCISD+Q is 1.7 kcal/mol. Our results show that the EOM-CC methods provide an alternative to multireference approaches and may be used to study photochemical systems like the one used in this work.

Equation of motion coupled cluster method for electron attached states with spin–orbit coupling

We report implementation of the equation of motion coupled-cluster method for electron attachment (EOMEA-CC) based on a previously developed CC approach with spin–orbit coupling included in post-Hartree–Fock treatment at the CC singles and doubles level (CCSD). Time-reversal symmetry is exploited by dealing with one partner of a Kramers pair explicitly and determining the other one through time reversal. This EOMEA-CCSD approach has been applied to calculate properties of some open-shell atoms and molecules containing heavy elements and is found to be able to describe SOC effects with reasonable accuracy.

Symmetry-adapted-cluster configuration-interaction and equation-of-motion coupled-cluster studies of electronically excited states of copper tetrachloride and copper tetrabromide dianions

The valence excitation spectra of the copper tetrachloride and copper tetrabromide open-shell dianions, CuCl42- and CuBr42-, respectively, are investigated by a variety of symmetry-adapted-cluster configuration-interaction (SAC-CI) and equation-of-motion coupled-cluster (EOMCC) methods. The valence excited states of the CuCl42- and CuBr42- species that correspond to transitions from doubly occupied molecular orbitals (MOs) to a singly occupied MO (SOMO), for which experimental spectra are available, are examined with the ionized (IP) variants of the SAC-CI and EOMCC methods. The higher-energy excited states of CuCl42- and CuBr42- that correspond to transitions from SOMO to unoccupied MOs, which have not been characterized experimentally, are determined using the electron-attached (EA) SAC-CI and EOMCC approaches. An emphasis is placed on the scalar relativistic SAC-CI and EOMCC calculations based on the spin-free part of the second-order Douglass–Kroll–Hess Hamiltonian (DKH2) and on a comparison of the results of the IP and EA SAC-CI and EOMCC calculations with up to 2-hole-1-particle (2h-1p) and 2-particle-1-hole (2p-1h) excitations, referred to as the IP-SAC-CI SD-R and IP-EOMCCSD(2h-1p) methods in the IP case and EA-SAC-CI SD-R and EA-EOMCCSD(2p-1h) approaches in the EA case, with those obtained with the higher-level IP-EOMCC and EA-EOMCC theories with up to 3-hole-2-particle (3h-2p) and 3-particle-2-hole (3p-2h) excitations treated via active orbitals, abbreviated as IP-EOMCCSD(3h-2p) and EA-EOMCCSD(3p-2h), respectively, as well as with the available experimental data. It is demonstrated that all of the employed DKH2-based IP-SAC-CI and IP-EOMCC methods offer a reliable description of the valence excited states of the CuCl42- and CuBr42- complexes that correspond to transitions from doubly occupied MOs to SOMO, accurately reproducing the observed UV–vis absorption spectra in both peak positions and intensities, which enables a rigorous assignment of the observed strong bands and weaker shoulder transitions. It is also shown that the scalar relativistic effects have a non-negligible effect on the excitation energies, on the order of 0.1–0.2eV, and a substantial effect on the calculated ground-state geometries, particularly in the case of the CuBr42- complex and the Cu–Br bond length, although the effect of relativity on the oscillator strengths is generally very small.

What made possible the accurate calculations of NMR spin–spin coupling constants?

The 1990s were an exciting time at the Quantum Theory Project of the University of Florida, during which several new developments in coupled cluster (CC) theory took place. One in particular is the development of the equation of motion coupled cluster (EOM-CC) method. It was realized that the EOM-CC excited states could be used to compute second-order properties in a sum over state formulation and the benchmark molecular polarizability calculations have since shown that EOM-CC performs remarkably well. The NMR spin–spin coupling is also a second-order property, and thus calculable by the new EOM-CC methodology. It was a common observation of reviews published at the time that a need existed for accurate nuclear magnetic resonance (NMR) indirect spin–spin coupling constants for medium-size molecules. This article is a first-hand account of the development of the EOM-CC methods for NMR spin–spin coupling constants and their successful applications to problems of interest.

Effects of Donor/Acceptor Strengths on the Multiphoton Absorption:  An EOM-CCSD Correction Vector Study

We have developed a correction method (CV) to calculate the single- and multiphoton absorption (MPA) spectra of organic pi-conjugated systems within the equation of motion coupled-cluster method with single and double excitations (EOM-CCSD). The effects of donor/acceptor strengths on the multiphoton absorption in a series of symmetrically substituted stilbene derivatives have been reinvestigated at both the ab initio and the semiempirical intermediate neglect of differential overlap (INDO) Hamiltonian levels. Both ab initio and INDO calculations show that the electron-donating or electron-withdrawing substituents lead to enhancements of two- and three-photon absorption cross sections, more pronounced for two-photon absorption than for three-photon absorption. The ab initio calculations usually produce larger excitation energies than the semiempirical, which lead to lower MPA cross sections.