Coupled Cluster Singles And Doubles Model Research Articles

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.

Assessment of the accuracy of coupled cluster perturbation theory for open-shell systems. I. Triples expansions.

The accuracy at which total energies of open-shell atoms and organic radicals may be calculated is assessed for selected coupled cluster perturbative triples expansions, all of which augment the coupled cluster singles and doubles (CCSD) energy by a non-iterative correction for the effect of triple excitations. Namely, the second- through sixth-order models of the recently proposed CCSD(T-n) triples series [J. J. Eriksen et al., J. Chem. Phys. 140, 064108 (2014)] are compared to the acclaimed CCSD(T) model for both unrestricted as well as restricted open-shell Hartree-Fock (UHF/ROHF) reference determinants. By comparing UHF- and ROHF-based statistical results for a test set of 18 modest-sized open-shell species with comparable RHF-based results, no behavioral differences are observed for the higher-order models of the CCSD(T-n) series in their correlated descriptions of closed- and open-shell species. In particular, we find that the convergence rate throughout the series towards the coupled cluster singles, doubles, and triples (CCSDT) solution is identical for the two cases. For the CCSD(T) model, on the other hand, not only its numerical consistency, but also its established, yet fortuitous cancellation of errors breaks down in the transition from closed- to open-shell systems. The higher-order CCSD(T-n) models (orders n > 3) thus offer a consistent and significant improvement in accuracy relative to CCSDT over the CCSD(T) model, equally for RHF, UHF, and ROHF reference determinants, albeit at an increased computational cost.

Assessing the distinguishable cluster approximation based on the triple bond-breaking in the nitrogen molecule.

Obtaining the correct potential energy curves for the dissociation of multiple bonds is a challenging problem for ab initio methods which are affected by the choice of a spin-restricted reference function. Coupled cluster (CC) methods such as CCSD (coupled cluster singles and doubles model) and CCSD(T) (CCSD + perturbative triples) correctly predict the geometry and properties at equilibrium but the process of bond dissociation, particularly when more than one bond is simultaneously broken, is much more complicated. New modifications of CC theory suggest that the deleterious role of the reference function can be diminished, provided a particular subset of terms is retained in the CC equations. The Distinguishable Cluster (DC) approach of Kats and Manby [J. Chem. Phys. 139, 021102 (2013)], seemingly overcomes the deficiencies for some bond-dissociation problems and might be of use in quasi-degenerate situations in general. DC along with other approximate coupled cluster methods such as ACCD (approximate coupled cluster doubles), ACP-D45, ACP-D14, 2CC, and pCCSD(α, β) (all defined in text) falls under a category of methods that are basically obtained by the deletion of some quadratic terms in the double excitation amplitude equation for CCD/CCSD (coupled cluster doubles model/coupled cluster singles and doubles model). Here these approximate methods, particularly those based on the DC approach, are studied in detail for the nitrogen molecule bond-breaking. The N2 problem is further addressed with conventional single reference methods but based on spatial symmetry-broken restricted Hartree-Fock (HF) solutions to assess the use of these references for correlated calculations in the situation where CC methods using fully symmetry adapted SCF solutions fail. The distinguishable cluster method is generalized: 1) to different orbitals for different spins (unrestricted HF based DCD and DCSD), 2) by adding triples correction perturbatively (DCSD(T)) and iteratively (DCSDT-n), and 3) via an excited state approximation through the equation of motion (EOM) approach (EOM-DCD, EOM-DCSD). The EOM-CC method is used to identify lower-energy CC solutions to overcome singularities in the CC potential energy curves. It is also shown that UHF based CC and DC methods behave very similarly in bond-breaking of N2, and that using spatially broken but spin preserving SCF references makes the CCSD solutions better than those for DCSD.

Linear-Scaling Coupled Cluster with Perturbative Triple Excitations: The Divide-Expand-Consolidate CCSD(T) Model.

We propose a reformulation of the traditional (T) triples correction to the coupled cluster singles and doubles (CCSD) energy in terms of local Hartree-Fock (HF) orbitals such that its structural form aligns with our recently developed linear-scaling divide-expand-consolidate (DEC) coupled cluster family of local correlation methods. In a DEC-CCSD(T) calculation, a basis of local occupied and virtual HF orbitals is used to partition the correlated calculation on the full system into a number of independent atomic fragment and pair fragment calculations, each performed within a truncated set of the complete orbital space. In return, this leads to a massively parallel algorithm for the evaluation of the DEC-CCSD(T) correlation energy, which formally scales linearly with the size of the full system and has a tunable precision with respect to a conventional CCSD(T) calculation via a single energy-based input threshold. The theoretical developments are supported by proof of concept DEC-CCSD(T) calculations on a series of medium-sized molecular systems.

On the convergence of perturbative coupled cluster triples expansions: error cancellations in the CCSD(T) model and the importance of amplitude relaxation.

Recently, we proposed a novel Lagrangian-based perturbation series-the CCSD(T-n) series-which systematically corrects the coupled cluster singles and doubles (CCSD) energy in orders of the Møller-Plesset fluctuation potential for effects due to triple excitations. In the present study, we report numerical results for the CCSD(T-n) series up through fourth order which show the predicted convergence trend throughout the series towards the energy of its target, the coupled cluster singles, doubles, and triples (CCSDT) model. Since effects due to the relaxation of the CCSD singles and doubles amplitudes enter the CCSD(T-n) series at fourth order (the CCSD(T-4) model), we are able to separate these effects from the total energy correction and thereby emphasize their crucial importance. Furthermore, we illustrate how the ΛCCSD[T]/(T) and CCSD[T]/(T) models, which in slightly different manners augment the CCSD energy by the [T] and (T) corrections rationalized from many-body perturbation theory, may be viewed as approximations to the second-order CCSD(T-2) model. From numerical comparisons with the CCSD(T-n) models, we show that the extraordinary performance of the ΛCCSD[T]/(T) and CCSD[T]/(T) models relies on fortuitous, yet rather consistent, cancellations of errors. As a side product of our investigations, we are led to reconsider the asymmetric ΛCCSD[T] model due to both its rigorous theoretical foundation and its performance, which is shown to be similar to that of the CCSD(T) model for systems at equilibrium geometry and superior to it for distorted systems. In both the calculations at equilibrium and distorted geometries, however, the ΛCCSD[T] and CCSD(T) models are shown to be outperformed by the fourth-order CCSD(T-4) model.

Accurate atomization energies from combining coupled-cluster computations with interference-corrected explicitly correlated second-order perturbation theory

Interference-corrected explicitly correlated second-order perturbation theory (INT-MP2-F12) is applied to accelerate the convergence to the complete-basis-set limit of coupled-cluster computations. Adding energy terms obtained from INT-MP2-F12 theory to the energies obtained from coupled-cluster singles-and-doubles (CCSD) computations yields a mean absolute deviation (MAD) from explicitly correlated CCSD results below 1 kJ/mol for a test set of 106 molecules. A composite scheme for the computation of atomization energies is assessed. This scheme is denoted as CCSD(T)+F12+INT and consists of the CCSD model with perturbative triples (CCSD(T)) supplemented with INT-MP2-F12 corrections, using a quadruple-zeta quality basis set (cc-pVQZ-F12). The composite scheme achieves chemical accuracy with respect to experimentally derived or computed reference values. Using Boys localized molecular orbitals, the MAD of the CCSD(T)+F12+INT/cc-pVQZ-F12 atomization energies from the reference values is below 1 kJ/mol for the G2/97 test set.

The ‘tailored’ CCSD(T) description of the automerization of cyclobutadiene

An alternative route to extend the CCSD(T) approach to multireference problems is presented. The well-known defect of the CCSD(T) model in describing the non-dynamic electron correlation effects is remedied by ‘tailoring’ the underlying coupled-cluster singles and doubles (CCSD) approach and applying the perturbative triples correction to it. The TCCSD(T) approach suggested in the paper has the same computational demands as the CCSD(T) method, though being mostly free from its drawbacks pertinent to multireference (quasidegenerate) situations. To test the approach we calculate the potential energy surface for the automerization of cyclobutadiene where the transition state exhibits a strong multireference character.

Highly accurate CCSD(R12) and CCSD(F12) optical response properties using standard triple-ζ basis sets

Coupled-cluster response theory for frequency-dependent optical properties within the coupled-cluster singles-and-doubles model (CCSD) has been derived and implemented for ansatz 2 of the explicitly correlated CCSD(R12) and CCSD(F12) methods as part of the program package DALTON. The basis set convergence of static dipole moments, polarizabilities, and parallel averages of first and second hyperpolarizabilities has been investigated for Ne, BH, N(2), CO, and BF. The frequency-dependent results are presented for the electronic second-harmonic generation of N(2). With triple-zeta basis sets, the CCSD(F12) correlation contributions using ansatz 2 are close to the basis set limits for dipole moments and second hyperpolarizabilities; the CCSD(R12) results are better than the CCSD results obtained with at least quintuple-zeta basis sets for polarizabilities and first hyperpolarizabilities. The exponent of Slater-type correlation factor for CCSD(F12) ground state energy may not be optimal and has to be re-examined for response properties. We also suggest that the remaining one-electron basis set errors arising within the coupled-cluster singles should be reduced by allowing excitations into the auxiliary orbital space.

Variational formulation of perturbative explicitly-correlated coupled-cluster methods

We present a variational formulation of the recently-proposed CCSD(2)(R12) method [Valeev, Phys. Chem. Chem. Phys., 2008, 10, 106]. The centerpiece of this approach is the CCSD(2)(R12) Lagrangian obtained via Löwdin partitioning of the coupled-cluster singles and doubles (CCSD) Hamiltonian. Extremization of the Lagrangian yields the second-order basis set incompleteness correction for the CCSD energy. We also developed a simpler Hylleraas-type functional that only depends on one set of geminal amplitudes by applying screening approximations. This functional is used to develop a diagonal orbital-invariant version of the method in which the geminal amplitudes are fixed at the values determined by the first-order cusp conditions. Extension of the variational method to include perturbatively the effect of connected triples produces the method that approximates the complete basis-set limit of the standard CCSD plus perturbative triples [CCSD(T)] method. For a set of 20 small closed-shell molecules, the method recovered at least 94.5/97.3% of the CBS CCSD(T) correlation energy with the aug-cc-pVDZ/aug-cc-pVTZ orbital basis set. For 12 isogyric reactions involving these molecules, combining the aug-cc-pVTZ correlation energies with the aug-cc-pVQZ Hartree-Fock energies produces the electronic reaction energies with a mean absolute deviation of 1.4 kJ mol(-1) from the experimental values. The method has the same number of optimized parameters as the corresponding CCSD(T) model, does not require any modification of the coupled-cluster computer program, and only needs a small triple-zeta basis to match the precision of the considerably more expensive standard quintuple-zeta CCSD(T) computation.

Density-Functional and Coupled-Cluster Singles-and-Doubles Calculations of the Nuclear Shielding and Indirect Nuclear Spin-Spin Coupling Constants of o-Benzyne.

Density-functional theory (DFT) and coupled-cluster singles-and-doubles (CCSD) theory are applied to compute the nuclear magnetic resonance (NMR) shielding and indirect nuclear spin-spin coupling constants of o-benzyne, whose biradical nature makes it difficult to study both experimentally and theoretically. Because of near-equilibrium triplet instabilities that follow from its biradical character, the calculated DFT NMR properties of o-benzyne are unusually sensitive to details of the exchange-correlation functional. However, this sensitivity is greatly reduced if these properties are calculated at the equilibrium of the chosen functional. A strong correlation is demonstrated between the quality of the calculated indirect spin-spin coupling constants and the quality of the calculated lowest triplet excitation energy in o-benzyne. Orbital-unrelaxed coupled-cluster theory should be less affected by such instabilities, and the CCSD NMR properties were only calculated at the experimental equilibrium geometry. For the shielding constants, the results in best agreement with experimental results are obtained with CCSD theory and with the Keal-Tozer KT1 and KT2 functionals. For the triply bonded carbon atoms, these models yield an isotropic shielding of 1.3, -3.3, and -1.2 ppm, respectively, compared with the experimentally observed shielding of 3.7 ppm for incarcerated o-benzyne. For the indirect spin-spin coupling constants, the CCSD model and the Perdew-Burke-Ernzerhof functional both yield reliable results; for the most interesting spin-spin coupling constant, (1)J (C⋮C), we obtain 210 and 209 Hz with these two models, respectively, somewhat above the recently reported experimental value of 177.9 ± 0.7 Hz for o-benzyne inside a molecular container, suggesting large incarceration effects.

Full configuration interaction benchmarking of coupled-cluster models for the lowest singlet energy surfaces of N2

The potential-energy curves for the X 1Σg+, a 1Πg, a′ 1Σu−, w 1Δu, c3 1Πu, and b 1Πu states of N2 have been investigated in full configuration interaction (FCI) and coupled-cluster response calculations. The equilibrium bond lengths, adiabatic excitation energies, and harmonic frequencies have been obtained with the coupled-cluster singles model (CCS), an approximate coupled-cluster singles and doubles model (CC2), the coupled-cluster singles and doubles model (CCSD), and an approximate coupled-cluster singles, doubles, and triples model (CC3), and subsequently compared to FCI results. The weak and strong features of the coupled-cluster models are discussed and illustrated. Overall, improvements towards FCI are obtained in the hierarchy CCS–CC2–CCSD–CC3. CC3 is always consistently better than CCSD, and for all the considered spectroscopic constants CC3 provides excellent results. Examples where the CC3 model fails are also given. The noniterative triples model, CCSDR(3), is compared to the iterative triples models CC3 and FCI. CCSDR(3) recovers the major part of the CC3 correlation contribution and is thus a cheap alternative to the CC3 model.

The prediction of molecular equilibrium structures by the standard electronic wave functions

A systematic investigation has been carried out of the accuracy of molecular equilibrium structures of 19 small closed-shell molecules containing first-row atoms as predicted by the following standard electronic ab initio models: Hartree–Fock (HF) theory, Mo/ller–Plesset theory to second, third, and fourth orders (MP2, MP3, and MP4), coupled-cluster singles and doubles (CCSD) theory; CCSD theory with perturbational triples corrections [CCSD(T)], and the configuration-interaction singles and doubles (CISD) model. For all models, calculations were carried out using the correlation-consistent polarized valence double-zeta (cc-pVDZ) basis, the correlation-consistent polarized valence triple-zeta (cc-pVTZ) basis, and the correlation-consistent polarized valence quadruple-zeta (cc-pVQZ) basis. Improvements in the basis sets shorten the bond distances at all levels. Going from cc-pVDZ to cc-pVTZ, bond distances are on the average reduced by 0.8 pm at the Hartree–Fock level and by 1.6 pm at the correlated levels. From cc-pVTZ to cc-pVQZ, the contractions are about ten times smaller and the cc-pVTZ basis set appears to yield bond distances close to the basis-set limit for all models. The models HF, MP2, and CCSD(T) give improved accuracy at increased computational cost. The accuracy of the Mo/ller–Plesset series oscillates, with MP3 being considerably less accurate than MP2 and MP4. The MP2 geometries are remarkably accurate, being only very slightly improved upon at the MP4 level for the cc-pVQZ basis. The CCSD equilibrium structures are only moderately accurate, being intermediate between MP2 and MP3. The accuracy of the CCSD(T) model, in contrast, is high and comparable to that observed in most experimental studies and it has been used to challenge the experimentally determined equilibrium structure of HNO. The CISD wave function provides structures of low quality.

Response functions in the CC3 iterative triple excitation model

The derivation of response functions for coupled cluster models is discussed in a context where approximations can be introduced in the coupled cluster equations. The linear response function is derived for the approximate coupled cluster singles, doubles, and triples model CC3. The linear response functions for the approximate triples models, CCSDT-1a and CCSDT-1b, are obtained as simplifications to the CC3 linear response function. The consequences of these simplifications are discussed for the evaluation of molecular properties, in particular, for excitation energies. Excitation energies obtained from the linear response eigenvalue equation are analyzed in orders of the fluctuation potential. Double replacement dominated excitations are correct through second order in all the triples models mentioned, whereas they are only correct to first order in the coupled cluster singles and doubles model (CCSD). Single replacement dominated excitation energies are correct through third order in CC3, while in CCSDT-1a, CCSDT-1b, and CCSD they are only correct through second order. Calculations of excitation energies are reported for CH+, N2, and C2H4 to illustrate the accuracy that can be obtained in the various triples models. The CH+ results are compared to full configuration interaction results, the C2H4 results are compared with complete active space second order perturbation theory (CASPT2) and experiment, and the N2 results are compared to experiment. Double replacement dominated excitations are improved significantly relative to CCSD in all the triples models mentioned, and is of the same quality in CC3 and CCSDT-1a. The single replacement dominated excitation are close to full configuration interaction results for the CC3 model and significantly improved relative to CCSD. The CCSDT-1 results for the single replacement dominated excitations are not improved compared to CCSD.

Analytic evaluation of energy gradients at the coupled-cluster singles and doubles level using quasi-restricted Hartree–Fock open-shell reference functions

Analytic formulas for the evaluation of energy gradients at the coupled-cluster singles and doubles (CCSD) level for a special class of open-shell quasi-restricted Hartree–Fock (QRHF) reference functions are presented and their computational implementation is discussed. The reference functions considered here are those which are formed from a closed-shell determinant for the molecule in a different ionization state by either adding or removing a single electron. Due to the approximate invariance of the CCSD model with respect to the choice of orbitals, results obtained with QRHF-CCSD closely approximate those from CCSD calculations using unrestricted Hartree–Fock (UHF) or restricted open-shell Hartree–Fock (ROHF) reference functions in cases where the latter references are well behaved. However, in cases where spin contamination (UHF) or symmetry breaking (UHF and ROHF) effects are important, QRHF-CCSD offers an attractive alternative since the resulting wave function is a projected eigenfunction of the spin operator Ŝ2 and transforms as a pure irreducible representation of the molecular point group.

A full coupled-cluster singles and doubles model: The inclusion of disconnected triples

The coupled-cluster singles and doubles model (CCSD) is derived algebraically, presenting the full set of equations for a general reference function explicitly in spin–orbital form. The computational implementation of the CCSD model, which involves cubic and quartic terms, is discussed and results are reported and compared with full CI calculations for H2O and BeH2. We demonstrate that the CCSD exponential ansatz sums higher-order correlation effects efficiently even for BeH2, near its transition state geometry where quasidegeneracy efforts are quite large, recovering 98% of the full CI correlation energy. For H2O, CCSD plus the fourth-order triple excitation correction agrees with the full CI energy to 0.5 kcal/mol. Comparisons with low-order models provide estimates of the effect of the higher-order terms T1T2, T21T2, T31, and T41 on the correlation energy.