Equation-of-motion Coupled-cluster Singles And Doubles Research Articles

State-Averaged Pair Natural Orbitals for Excited States: A Route toward Efficient Equation of Motion Coupled-Cluster.

A reduced-complexity variant of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method is formulated in terms of state-averaged excited-state pair natural orbitals (PNO) designed to describe manifolds of excited states. State-averaged excited-state PNOs for the target manifold are determined by averaging CIS(D) pair densities over the model manifold. The performance of the PNO-EOM-CCSD approach has been tested with the help of a distributed-memory parallel canonical EOM-CCSD implementation within the Massively Parallel Quantum Chemistry program that allows treatment of systems with 50+ atoms using realistic basis sets with 1000+ functions. The use of state-averaged PNOs offers several potential advantages relative to the recently proposed state-specific PNOs: our approach is robust with respect to root flipping and state degeneracies, it is more economical when computing large manifolds of states, and it simplifies evaluation of transition-specific observables such as dipole moments. With the PNO truncation threshold of 10-7, the errors in excitation energies are on average below 0.02 eV for the first six singlet states of 28 organic molecules included in the standard test set of Thiel and co-workers ( J. Chem. Phys. 2008, 128, 134110) with 50-70 state-averaged PNOs per pair.

Understanding Processes Following Resonant Electron Attachment: Minimum-Energy Crossing Points between Anionic and Neutral Potential Energy Surfaces.

The equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method with and without a complex absorbing potential (CAP) is applied for the study of the complex potential energy surfaces (CPES) of temporary anions and their parent neutral molecules. Crossings between the anionic state and the neutral state can be connected to the emission of nearly zero-energy electrons, which is demonstrated by the examples of acrylonitrile and methacrylonitrile. We show that the location of the minimum-energy crossing point (MECP) relative to the equilibrium structures of the neutral molecule and the anion can explain experimentally observed peaks on the threshold line of two-dimensional electron-energy loss spectra. The location and energy of the MECP is also crucial in dissociative electron attachment as we illustrate for chloro-substituted ethylenes. It is demonstrated that both the metastable region of the anionic CPES and the crossing with the neutral PES need to be considered to explain trends in the chloride ion formation cross sections of dichloroethylenes.

Can TDDFT Describe Excited Electronic States of Naphthol Photoacids? A Closer Look with EOM-CCSD

The 1Lb and 1La excited states of naphthols are characterized by using time-dependent density functional theory (TDDFT), configuration interaction with singles (CIS), and equation-of-motion coupled cluster singles and doubles (EOM-CCSD) methods. TDDFT fails dramatically at predicting the energy and ordering of the 1La and 1Lb excited states as observed experimentally, while EOM-CCSD accurately predicts the excited states as characterized by natural transition orbital analysis. The limitations of TDDFT are attributed to the absence of correlation from doubly excited configurations as well as the inconsistent description of excited electronic states of naphthol photoacids revealed by excitation analysis based on the one-electron transition density matrix.

Simple many-body based screening mixing ansatz for improvement of GW /Bethe-Salpeter equation excitation energies of molecular systems

We propose a simple many-body based screening mixing strategy to considerably enhance the performance of the Bethe-Salpeter equation (BSE) approach for prediction of excitation energies of molecular systems. This strategy enables us to closely reproduce results of highly correlated equation of motion coupled cluster singles and doubles (EOM-CCSD) through optimal use of cancellation effects. We start from the Hartree-Fock (HF) reference state and take advantage of local density approximation (LDA) based random phase approximation (RPA) screening, denoted as ${W}_{0}$-RPA@LDA with ${W}_{0}$ as the dynamically screened interaction built upon LDA wave functions and energies. We further use this ${W}_{0}$-RPA@LDA screening as an initial screening guess for calculation of quasiparticle energies in the framework of ${G}_{0}{W}_{0}$@HF. The ${W}_{0}$-RPA@LDA screening is further injected into the BSE. By applying such an approach on a set of 22 molecules for which the traditional $GW$/BSE approaches fail, we observe good agreement with respect to EOM-CCSD references. The reason for the observed good accuracy of this mixing ansatz (scheme A) lies in an optimal damping of HF exchange effect through the ${W}_{0}$-RPA@LDA strong screening, leading to substantial decrease of typically overestimated HF electronic gap, and hence to better excitation energies. Further, we present a second multiscreening ansatz (scheme B), which is similar to scheme A with the exception that now the ${W}_{0}$-RPA@HF screening is used in the BSE in order to further improve the overestimated excitation energies of carbonyl sulfide (COS) and disilane $({\mathrm{Si}}_{2}{\mathrm{H}}_{6})$. The reason for improvement of the excitation energies in scheme B lies in the fact that ${W}_{0}$-RPA@HF screening is less effective (and weaker than ${W}_{0}$-RPA@LDA), which gives rise to stronger electron-hole effects in the BSE.

Pushing the Limits of EOM-CCSD with Projector-Based Embedding for Excitation Energies.

The calculation of accurate excitation energies using ab initio electronic structure methods such as standard equation of motion coupled cluster singles and doubles (EOM-CCSD) has been cost prohibitive for large systems. In this work, we use a simple projector-based embedding scheme to calculate the EOM-CCSD excitation energies of acrolein solvated in water molecules modeled using density functional theory (DFT). We demonstrate the accuracy of this approach gives excitation energies within 0.01 eV of full EOM-CCSD, but with significantly reduced computational cost. This approach is also shown to be relatively invariant to the choice of functional used in the environment and allows for the description of systems with large numbers of basis functions (>1000) to be treated using state-of-the-art wave function methods. The flexibility of embedding to select orbitals to add to the excited-state method provides insights into the origins of the excitations and can reduce artifacts that could arise in traditional linear response time-dependent DFT (LR-TDDFT).

Excited states from modified coupled cluster methods: Are they any better than EOM CCSD?

Simplifications or modifications of coupled cluster methods such as the CCSD (coupled cluster singles and doubles) model often perform better than the original method in providing the total energy, equilibrium geometries, and harmonic vibration frequencies for the ground state. Three such methods that have been recently proposed include 2CC, parameterized CCSD generalizations, and distinguishable cluster singles and doubles (DCSD) approach. In this paper, we lay the theoretical foundation needed to treat excited states via the equation of motion (EOM) approach using these ground state CC methods. As these ground state approximations to CCSD share its property of being exact for two-electron systems, so will their excited state extensions. These methods are tested for two complementary benchmark sets of excited states for a wide range of organic molecules with focus on singlet and triplet excited states of both valence and Rydberg nature. We also test these methods for doubly excited states, taking CH+ as an example to test their performance at equilibrium and stretched bond geometries. Finally, we assess if any of these methods perform consistently better than EOM CCSD.

Static polarizabilities for excited states within the spin-conserving and spin-flipping equation-of-motion coupled-cluster singles and doubles formalism: Theory, implementation, and benchmarks.

We present the theory and implementation for calculating static polarizabilities within the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) framework for electronically excited states and its spin-flip variant. We evaluate the second derivatives of the EOM-CCSD Lagrangian with respect to electric-field perturbations. The relaxation of reference molecular orbitals is not included. In our approach, the wave function amplitudes satisfy the 2n + 1 rule and the amplitude-response Lagrange multipliers satisfy the 2n + 2 rule. The new implementation is validated against finite-field and CCSD response-theory calculations of the excited-state polarizabilities of pyrimidine and s-tetrazine. We use the new method to compute static polarizabilities of different types of electronic states (valence, charge-transfer, singlets, and triplets) in open- and closed-shell systems (uracil, p-nitroaniline, methylene, and p-benzyne). We also present an alternative approach for calculating excited-state static polarizabilities as expectation values by using the EOM-CCSD wave functions and energies in the polarizability expression for an exact state. We find that this computationally less demanding approach may show differences up to ∼30% relative to the excited-state polarizabilities computed using the analytic-derivative formalism.

Boron as an Electron-Pair Donor for B⋅⋅⋅Cl Halogen Bonds.

MP2/aug'-cc-pVTZ calculations were performed to investigate boron as an electron-pair donor in halogen-bonded complexes (CO)2 (HB):ClX and (N2 )2 (HB):ClX, for X=F, Cl, OH, NC, CN, CCH, CH3 , and H. Equilibrium halogen-bonded complexes with boron as the electron-pair donor are found on all of the potential surfaces, except for (CO)2 (HB):ClCH3 and (N2 )2 (HB):ClF. The majority of these complexes are stabilized by traditional halogen bonds, except for (CO)2 (HB):ClF, (CO)2 (HB):ClCl, (N2 )2 (HB):ClCl, and (N2 )2 (HB):ClOH, which are stabilized by chlorine-shared halogen bonds. These complexes have increased binding energies and shorter B-Cl distances. Charge transfer stabilizes all complexes and occurs from the B lone pair to the σ* Cl-A orbital of ClX, in which A is the atom of X directly bonded to Cl. A second reduced charge-transfer interaction occurs in (CO)2 (HB):ClX complexes from the Cl lone pair to the π* C≡O orbitals. Equation-of-motion coupled cluster singles and doubles (EOM-CCSD) spin-spin coupling constants, 1x J(B-Cl), across the halogen bonds are also indicative of the changing nature of this bond. 1x J(B-Cl) values for both series of complexes are positive at long distances, increase as the distance decreases, and then decrease as the halogen bonds change from traditional to chlorine-shared bonds, and begin to approach the values for the covalent bonds in the corresponding ions [(CO)2 (HB)-Cl]+ and [(N2 )2 (HB)-Cl]+ . Changes in 11 B chemical shieldings upon complexation correlate with changes in the charges on B.

Simulation of circularly polarized luminescence spectra using coupled cluster theory.

We report the first computations of circularly polarized luminescence (CPL) rotatory strengths at the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) level of theory. Using a test set of eight chiral ketones, we compare both dipole and rotatory strengths for absorption (electronic circular dichroism) and emission to the results from time-dependent density-functional theory (TD-DFT) and available experimental data for both valence and Rydberg transitions. For two of the compounds, we obtained optimized geometries of the lowest several excited states using both EOM-CCSD and TD-DFT and determined that structures and EOM-CCSD transition properties obtained with each structure were sufficiently similar that TD-DFT optimizations were acceptable for the remaining test cases. Agreement between EOM-CCSD and the Becke three-parameter exchange function and Lee-Yang-Parr correlation functional (B3LYP) corrected using the Coulomb attenuating method (CAM-B3LYP) is typically good for most of the transitions, though agreement with the uncorrected B3LYP functional is significantly worse for all reported properties. The choice of length vs. velocity representation of the electric dipole operator has little impact on the EOM-CCSD transition strengths for nearly all of the states we examined. For a pair of closely related β, γ-enones, (1R)-7-methylenebicyclo[2.2.1]heptan-2-one and (1S)-2-methylenebicyclo[2.2.1]heptan-7-one, we find that EOM-CCSD and CAM-B3LYP agree with the energetic ordering of the two possible excited-state conformations, resulting in good agreement with experimental rotatory strengths in both absorption and emission, whereas B3LYP yields a qualitatively incorrect result for the CPL signal of (1S)-2-methylenebicyclo[2.2.1]heptan-7-one. Finally, we predict that one of the compounds considered here, trans-bicyclo[3.3.0]octane-3,7-dione, is unique in that it exhibits an achiral ground state and a chiral first excited state, leading to a strong CPL signal but a weak circular dichroism signal.

Assessment of low-scaling approximations to the equation of motion coupled-cluster singles and doubles equations.

Methods for fast and reliable computation of electronic excitation energies are in short supply, and little is known about their systematic performance. This work reports a comparison of several low-scaling approximations to the equation of motion coupled cluster singles and doubles (EOM-CCSD) and linear-response coupled cluster singles and doubles (LR-CCSD) equations with other single reference methods for computing the vertical electronic transition energies of 11 small organic molecules. The methods, including second order equation-of-motion many-body perturbation theory (EOM-MBPT2) and its partitioned variant, are compared to several valence and Rydberg singlet states. We find that the EOM-MBPT2 method was rarely more than a tenth of an eV from EOM-CCSD calculated energies, yet demonstrates a performance gain of nearly 30%. The partitioned equation-of-motion approach, P-EOM-MBPT2, which is an order of magnitude faster than EOM-CCSD, outperforms the CIS(D) and CC2 in the description of Rydberg states. CC2, on the other hand, excels at describing valence states where P-EOM-MBPT2 does not. The difference between the CC2 and P-EOM-MBPT2 can ultimately be traced back to how each method approximates EOM-CCSD and LR-CCSD. The results suggest that CC2 and P-EOM-MBPT2 are complementary: CC2 is best suited for the description of valence states while P-EOM-MBPT2 proves to be a superior O(N(5)) method for the description of Rydberg states.

Complex absorbing potentials within EOM-CC family of methods: theory, implementation, and benchmarks.

A production-level implementation of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) for electron attachment and excitation energies augmented by a complex absorbing potential (CAP) is presented. The new method enables the treatment of metastable states within the EOM-CC formalism in a similar manner as bound states. The numeric performance of the method and the sensitivity of resonance positions and lifetimes to the CAP parameters and the choice of one-electron basis set are investigated. A protocol for studying molecular shape resonances based on the use of standard basis sets and a universal criterion for choosing the CAP parameters are presented. Our results for a variety of π(*) shape resonances of small to medium-size molecules demonstrate that CAP-augmented EOM-CCSD is competitive relative to other theoretical approaches for the treatment of resonances and is often able to reproduce experimental results.

A quantum chemical study from a molecular transport perspective: ionization and electron attachment energies for species often used to fabricate single-molecule junctions.

The accurate determination of the lowest electron attachment (EA) and ionization (IP) energies for molecules embedded in molecular junctions is important for correctly estimating, for example, the magnitude of the currents (I) or the biases (V) where an I-V curve exhibits significant non-Ohmic behavior. Benchmark calculations for the lowest electron attachment and ionization energies of several typical molecules utilized to fabricate single-molecule junctions characterized by n-type conduction (4,4'-bipyridine, 1,4-dicyanobenzene and 4,4'-dicyano-1,1'-biphenyl) and p-type conduction (benzenedithiol, biphenyldithiol, hexanemonothiol and hexanedithiol) based on the EOM-CCSD (equation-of-motion coupled-cluster singles and doubles) state-of-the-art method of quantum chemistry are presented. They indicate significant differences from the results obtained within current approaches to molecular transport. The present study emphasizes that, in addition to a reliable quantum chemical method, basis sets much better than the ubiquitous double-zeta set employed for transport calculations are needed. The latter is a particularly critical issue for correctly determining EAs, which is impossible without including sufficient diffuse basis functions. The spatial distribution of the dominant molecular orbitals (MOs) is another important issue, on which the present study draws attention, because it sensitively affects the MO energy shifts Φ due to image charges formed in electrodes. The present results cannot substantiate the common assumption of a point-like MO midway between electrodes, which substantially affects the actual Φ-values.

Vertical Electronic Excitations in Solution with the EOM-CCSD Method Combined with a Polarizable Explicit/Implicit Solvent Model.

The accurate calculation of electronic transition energies and properties of isolated chromophores is not sufficient to provide a realistic simulation of their excited states in solution. In fact, the solvent influences the solute geometry, electronic structure, and response to external fields. Therefore, a proper description of the solvent effect is fundamental. This can be achieved by combining polarizable explicit and implicit representations of the solvent. The former provides a realistic description of solvent molecules around the solute, while the latter introduces the electrostatic effect of the bulk solution and reduces the need of too large a number of explicit solvent molecules. This strategy is particularly appealing when an accurate method such as equation of motion coupled cluster singles and doubles (EOM-CCSD) is employed for the treatment of the chromophore. In this contribution, we present the coupling of EOM-CCSD with a fluctuating charges (FQ) model and polarizable continuum model (PCM) of solvation for vertical excitations in a state-specific framework. The theory, implementation, and prototypical applications of the method are presented. Numerical tests on small solute-water clusters show very good agreement between full EOM-CCSD and EOM-CCSD-FQ calculations, with and without PCM, with differences ≤ 0.1 eV. Additionally, approximated schemes that further reduce the computational cost of the method are introduced and showed to perform well compared to the full method (errors ≤ 0.1 eV).

Benchmarking for Perturbative Triple-Excitations in EE-EOM-CC Methods

Perturbative triples corrections ((T) and (T̃)) to the equation of motion coupled cluster singles and doubles (EOM-CCSD) are rederived and implemented in a pilot parallel code. The vertical excitation energies of molecules in the test set of Sauer et al. [J. Chem. Theor. Comput. 2009, 5, 555] are reported and compared to the iterative EOM-CCSDT-3 method. The average absolute deviations of EOM-CCSD(T) and EOM-CCSD(T̃) from EOM-CCSDT-3 over this wide test set are 0.06 and 0.18 eV, respectively. The poor performance of the latter suggests misbalanced handling of the (T̃) terms. Scaling curves for EOM-CCSD(T) are also presented to demonstrate the performance across multiple compute nodes, thus enabling the routine and accurate study of excited states for ever larger molecular systems.

Electronic structure of the S1 state in methylcobalamin: Insight from CASSCF/MC‐XQDPT2, EOM‐CCSD, and TD‐DFT calculations

The methylcobalamin cofactor (MeCbl), which is one of the biologically active forms of vitamin B12, has been the subject of many spectroscopic and theoretical investigations. Traditionally, the lowest-energy part of the photoabsorption spectrum of MeCbl (the so-called α/β band) has been interpreted as an S0→S1 electronic transition dominated by π→π* excitations associated with the C=C stretching of the corrin ring. However, a more quantitative band-shape analysis of the α/β spectral region, along with circular dichroism (CD), magnetic CD, and resonance Raman data, has revealed the presence of a second electronic transition that involves the Co-C(Me) bond weakening. Conversely, the lowest-energy excitations based on transient absorption spectroscopy measurements have been interpreted as metal-to-ligand charge transfer (MLCT) transitions. To resolve the existing controversy about the interpretation of the S1 state of MeCbl, calculations have been performed using two independent ab initio wavefunction-based methods. These include the modified variant of the second-order multiconfigurational quasi-degenerate perturbation theory (MC-XQDPT2), using complete active space self-consistent field orbitals, and the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) approach using restricted Hartree-Fock orbitals. It is shown that both ab initio methods provide a consistent description of the S1 state as having an MLCT character. In addition, the performance of different types of functionals, including hybrid (B3LYP, MPW1PW91, TPSSh), generalized-gradient-approximation-type (GGA-type) (BP86, BLYP, MPWPW91), meta-GGA (TPSS), and range-separated (CAM-B3LYP, LC-BLYP) approaches, has been examined and the results of the corresponding time-dependent density functional theory calculations have been benchmarked against the MC-XQDPT2 and EOM-CCSD data. The hybrid functionals support the interpretation in which the S1 state represents a π→π* transition localized on corrin, while pure GGA, meta-GGA, and LC-BLYP functionals produce results consistent with the MLCT assignment.

Absorption and Emission Spectra of Solvated Molecules with the EOM-CCSD-PCM Method.

The accurate calculation of transition energies and properties of isolated molecules is not enough for realistic simulations of their absorption and emission spectra in solution. In fact, the solvent influences the solute geometry, electronic structure, and response to external fields, and a proper description of the solvent effect is fundamental. However, the computational cost of including explicit solvent molecules around the solute becomes rather onerous when an accurate method such as the equation of motion coupled cluster singles and doubles (EOM-CCSD) is employed. The polarizable continuum model of solvation (PCM) may provide an efficient alternative to explicit models, since the sampling of solvent configurations is implicit and the solute-solvent mutual polarization is naturally accounted for. In this contribution, the absorption and emission spectra of molecules in solution are modeled through the EOM-CCSD-PCM method. The equilibrium solvation regime is employed for the geometry optimization of the solute molecule in the ground and excited states, while the nonequilibrium solvation regime is employed for vertical transitions. The theory, implementation, and prototypical applications of the method are presented. The numerical tests involve solvents that are particularly challenging for PCM: low-polar and protic polar solvents. Nonetheless, the experimental trends are well reproduced, and the overall agreement with the measured data is remarkable.

Exploring Potential Energy Surfaces of Electronic Excited States in Solution with the EOM-CCSD-PCM Method.

The effect of the solvent on the structure of a molecule in an electronic excited state cannot be neglected. However, the computational cost of including explicit solvent molecules around the solute becomes rather onerous when an accurate method such as the equation of motion coupled cluster singles and doubles (EOM-CCSD) is employed. Solvation continuum models like the polarizable continuum model (PCM) provide an efficient alternative to explicit models, since the solvent conformational average is implicit and the solute-solvent mutual polarization is naturally accounted for. In this work, the coupling of EOM-CCSD and PCM in a state specific approach is presented for the evaluation of energy and analytic energy gradients. Also, various approximations are explored to maintain the computational cost comparable to gas phase EOM-CCSD. Numerical examples are used to test the different schemes.

Projected Coupled Cluster Amplitudes from a Different Basis Set As Initial Guess.

Many model chemistry schemes require a sequence of high level calculations, often at the coupled cluster (CC) level, with increasingly larger basis sets. The CC equations are solved iteratively, and the rate of convergence strongly depends on the quality of the initial guess. Here, we propose a strategy to define a better guess from a previous CC calculation with a different basis set by employing the concept of corresponding orbitals. (1, 2) Only the part of the converged amplitudes from the previous calculation that has a large correspondence to the space spanned by the new basis set is projected and used as a new starting point. The computational time for the projection is negligible and significantly reduces the number of cycles necessary for convergence in comparison to the standard guess based on the first order wave function. Numerical results are presented for ground and excited state calculations with the CC singles and doubles (CCSD) and the equation of motion CCSD (EOM-CCSD) methods with the restricted and unrestricted Hartree-Fock (HF) reference functions. However, this approach is more general and can be applied to any truncation of the cluster expansion and reference function.

Oscillator Strength: How Does TDDFT Compare to EOM-CCSD?

In this work, we compare a large variety of density functionals against the equation of motion coupled cluster singles and doubles (EOM-CCSD) method for the calculation of oscillator strengths. Valence and Rydberg states are considered for a test set composed of 11 small organic molecules. In our previous work, the same systems and methods were tested against experimental results for the excitation energies. The results from this investigation confirm our previous findings, i.e., that there is a large difference between the functionals. For the oscillator strength, the average best agreement with EOM-CCSD is provided by CAM-B3LYP followed by LC-ωPBE and, to a lesser extent, B3P86 and LC-BLYP.

Oscillator Strengths in ONIOM Excited State Calculations.

We compute oscillator strengths with the ONIOM (Our own N-layer Integrated molecular Orbital molecular Mechanics) hybrid method between ground and valence excited states and compare the results with the high level of theory equation of motion coupled cluster singles and doubles (EOM-CCSD). This work follows our previous studies in which we validated the ability of ONIOM to compute accurate transition energies compared to EOM-CCSD. We test various levels of theory and molecular systems, as well as the effect of the link atom bond length. Our results show that oscillator strengths can be accurately computed with ONIOM, provided that a sensible choice of the partitioning and of the low level method is made. Being able to calculate both the transition energy and the oscillator strength, ONIOM represents a promising approach to completely characterize valence excited states of molecules that are too large to be studied with a conventional high-accuracy method.