Coupled Cluster Singles And Doubles Calculations Research Articles

Exploring Alternate Methods for the Calculation of High-Level Vibrational Corrections of NMR Spin-Spin Coupling Constants.

Traditional nuclear magnetic resonance (NMR) calculations typically treat systems with a Born-Oppenheimer-derived electronic wave function that is solved for a fixed nuclear geometry. One can numerically account for this neglected nuclear motion by averaging over property values for all nuclear geometries with a vibrational wave function and adding this expectation value as a correction to an equilibrium geometry property value. Presented are benchmark coupled-cluster singles and doubles (CCSD) vibrational corrections to spin-spin coupling constants (SSCCs) computed at the level of vibrational second-order perturbation theory (VPT2) using the vibrational averaging driver of the CFOUR program. As CCSD calculations of vibrational corrections are very costly, cheaper electronic structure methods are explored via a newly developed Python vibrational averaging program within the Dalton Project. Namely, results obtained with the second-order polarization propagator approximation (SOPPA) and density functional theory (DFT) with the B3LYP and PBE0 exchange-correlation functionals are compared to the benchmark CCSD//CCSD(T) and experimental values. CCSD//CCSD(T) corrections are also combined with literature CC3 equilibrium geometry values to form the highest-order vibrationally corrected values available (i.e., CC3//CCSD(T) + CCSD//CCSD(T)). CCSD//CCSD(T) statistics showed favorable statistics in comparison to experimental values, albeit at an unfavorably high computational cost. A cheaper CCSD//CCSD(T) + B3LYP method showed quite similar mean absolute deviation (MAD) values as CCSD//CCSD(T), concluding that CCSD//CCSD(T) + B3LYP is optimal in terms of cost and accuracy. With reference to experimental values, a vibrational correction was not worth the cost for all of the other methods tested. Finally, deviation statistics showed that CC3//CCSD(T) + CCSD//CCSD(T) vibrational-corrected equilibrium values deteriorated in comparison to CCSD//CCSD(T) attributed to the use of a smaller basis set or lack of solvation effects for the CC3 equilibrium calculations.

Rank-reduced coupled-cluster. III. Tensor hypercontraction of the doubles amplitudes.

We develop a quartic-scaling implementation of coupled-cluster singles and doubles (CCSD) based on low-rank tensor hypercontraction (THC) factorizations of both the electron repulsion integrals (ERIs) and the doubles amplitudes. This extends our rank-reduced (RR) coupled-cluster method to incorporate higher-order tensor factorizations. The THC factorization of the doubles amplitudes accounts for most of the gain in computational efficiency as it is sufficient, in conjunction with a Cholesky decomposition of the ERIs, to reduce the computational complexity of most contributions to the CCSD amplitude equations. Further THC factorization of the ERIs reduces the complexity of certain terms arising from nested commutators between the doubles excitation operator and the two-electron operator. We implement this new algorithm using graphical processing units and demonstrate that it enables CCSD calculations for molecules with 250 atoms and 2500 basis functions using a single computer node. Furthermore, we show that the new method computes correlation energies with comparable accuracy to the underlying RR-CCSD method.

Geometries and electronic states of iron trimer (Fe3) by CCSD and CCSD(T) calculations

Geometries and energy separations of various low-lying electronic states of an iron trimer (Fe3) are investigated by coupled-cluster singles and doubles (CCSD) and coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] calculations. The ground state is found to be a 13A′ state with Cs symmetry, whereas a nearly isoenergetic state, 13A1 (C2v), is degenerate to the ground state. The ground and five low-lying states with a spin multiplicity of 13 are found below 0.20 eV at the CCSD(T) level. On the other hand, the low-lying states with spin multiplicities of 9, 11, and 15 appear only above 0.20 eV. From detailed natural bond orbital analyses, Fe3 has Fe–Fe bonds composed of σ-bond orbitals only in theβ-spin part with higher s-character in low-lying states with a spin multiplicity of 13. The polarization coefficients indicate that the σFe–Fe bonds are nearly complete covalent bonds with little polarization.

Particle-particle ladder based basis-set corrections applied to atoms and molecules using coupled-cluster theory.

We investigate the basis-set convergence of electronic correlation energies calculated using coupled cluster theory and a recently proposed finite basis-set correction technique. The correction is applied to atomic and molecular systems and is based on a diagrammatically decomposed coupled cluster singles and doubles (CCSD) correlation energy. Only the second-order energy and the particle-particle ladder term are corrected for their basis-set incompleteness error. We present absolute correlation energies and results for a large benchmark set. Our findings indicate that basis set reductions by two cardinal numbers are possible for atomization energies, ionization potentials, and electron affinities without compromising accuracy when compared to conventional CCSD calculations. In the case of reaction energies, we find that reductions by one cardinal number are possible compared to conventional CCSD calculations. The employed technique can readily be applied to other many-electron theories without the need for three- or four-electron integrals.

Orbital spaces in the divide-expand-consolidate coupled cluster method.

The theoretical foundation for solving coupled cluster singles and doubles (CCSD) amplitude equations to a desired precision in terms of independent fragment calculations using restricted local orbital spaces is reinvestigated with focus on the individual error sources. Four different error sources are identified theoretically and numerically and it is demonstrated that, for practical purposes, local orbital spaces for CCSD calculations can be identified from calculations at the MP2 level. The development establishes a solid theoretical foundation for local CCSD calculations for the independent fragments, and thus for divide-expand-consolidate coupled cluster calculations for large molecular systems with rigorous error control. Based on this theoretical foundation, we have developed an algorithm for determining the orbital spaces needed for obtaining the single fragment energies to a requested precision and numerically demonstrated the robustness and precision of this algorithm.

Electronic Couplings for Resonance Energy Transfer from CCSD Calculations: From Isolated to Solvated Systems.

Quantum mechanical (QM) calculations of electronic couplings provide great insights for the study of resonance energy transfer (RET). However, most of these calculations rely on approximate QM methods due to the computational limitations imposed by the size of typical donor-acceptor systems. In this work, we present a novel implementation that allows computing electronic couplings at the coupled cluster singles and doubles (CCSD) level of theory. Solvent effects are also taken into account through the polarizable continuum model (PCM). As a test case, we use a dimer of indole, a common model system for tryptophan, which is routinely used as an intrinsic fluorophore in Förster resonance energy transfer studies. We consider two bright π → π* states, one of which has charge transfer character. Lastly, the results are compared with those obtained by applying TD-DFT in combination with one of the most popular density functionals, B3LYP.

Systematic comparison of DFT and CCSD dipole moments, polarizabilities and hyperpolarizabilities

A comparative study of dipole moments, polarizabilities and hyperpolarizabilities of 16 different molecules is performed employing two completely different theoretical approaches namely, density functional theory (DFT) and coupled cluster singles and doubles (CCSD). Both methods include electron correlation. The CCSD method is more accurate but highly expensive. DFT with auxiliary density allows non-iterative solutions which is computational advantage and useful for large molecules. Dipole moments and polarizability calculations from DFT are in very good agreement with CCSD calculations. However, negative hyperpolarizability values from DFT differ significantly from their CCSD counterparts, whereas positive hyperpolarizabilities show reasonable agreement between these methodologies.

Efficient Sparse Matrix Algorithm to Speed Up the Calculation of the Ladder Term in Coupled Cluster Programs.

A new algorithm is presented for the calculation of the ladder-type term of the coupled cluster singles and doubles (CCSD) equations using two-electron integrals in atomic orbital (AO) basis. The method is based on an orbital grouping scheme, which results in an optimal partitioning of the AO integral matrix into sparse and dense blocks allowing efficient matrix multiplication. Carefully chosen numerical tests have been performed to analyze the performance of all aspects of the new algorithm. It is shown that the suggested scheme allows an efficient utilization of modern highly parallel architectures and devices in CCSD calculations. Details of the implementation in the development version of CFOUR quantum chemical program package are also presented.

Geometries and adiabatic excitation energies of the low-lying valence states of CNC, C2N, N3 and NCO studied with the electron-attached and ionized equation-of-motion coupled-cluster methodologies

The full and active-space variants of the electron-attached (EA) equation-of-motion (EOM) coupled-cluster (CC) method with up to three-particle–two-hole (3p–2h) excitations in the electron-attaching operator Rμ(N+1) that use the CC singles and doubles (CCSD) approach to obtain the ground state of the reference N-electron closed-shell system, abbreviated as EA-EOMCCSD(3p–2h), and their ionized (IP) counterparts with up to three-hole–two-particle (3h–2p) excitations in the ionizing operator Rμ(N−1), abbreviated as IP-EOMCCSD(3h–2p), are used to optimize the geometries of the ground and low-lying excited states of four open-shell molecules, CNC, C2N, NCO and N3, and determine the corresponding adiabatic excitation energies. The full and active-space EA-EOMCCSD(3p–2h) results for the CNC and C2N molecules, obtained with the correlation-consistent basis sets as large as cc-pVTZ and cc-pVQZ, respectively, are compared with one another, with the corresponding EA-EOMCCSD(2p–1h) calculations, with the previously generated small basis set EA-EOMCC and symmetry-adapted-cluster configuration-interaction (SAC-CI-SDT-R/PS) data, and, wherever possible, with experiment. The analogous comparison of the full and active-space IP-EOMCCSD(3h–2p) results with the IP-EOMCCSD(2h–1p), SAC-CI-SDT-R/PS and experimental data is performed for the NCO and N3 molecules. It is shown that the active-space EA-EOMCCSD(3p–2h) and IP-EOMCCSD(3h–2p) approaches using small numbers of active orbitals, which have computational costs that are of the order of the CCSD calculations, provide excitation energies and optimized geometries that are in excellent agreement with the results of the significantly more expensive parent EA-EOMCCSD(3p–2h) and IP-EOMCCSD(3h–2p) calculations, independent of the basis set. It is also demonstrated that the basic EA-EOMCCSD(2p–1h) and IP-EOMCCSD(2h–1p) methods, while generally inadequate for a reliable description of the excitation energies, describe the geometries in a reasonable manner, including excited states dominated by two-electron transitions. Although the full and active-space EA-EOMCCSD(3p–2h) calculations for the most challenging CNC and C2N molecules improve the EA-EOMCCSD(2p–1h) excitation energies, some differences with the available experimental data remain in spite of the use of larger correlation-consistent basis sets and complete basis set extrapolations.

Localized orbitals in the coupled cluster singles and doubles model

In the vast majority of many-body perturbation theory and coupled-cluster method (MBPT/CCM) calculations, the self-consistent field (SCF) wavefunction is employed as the zeroth-order wave-function, and its associated canonical orbitals as the molecular orbital basis. The recent development of the coupled-cluster singles and doubles (CCSD) formalism provides an opportunity to explore alternative reference function and orbital choices, on a consistent basis. Here we examine, specifically, sets of orbitals that are more “localized” than the SCF orbitals. In particular, the bonding orbitals are selected to resemble traditional chemical bonds. CCSD calculations in the localized basis on methane employing a better than double-zeta basis set are performed and compared with conventional SCF basis results.

Calculations of Molecular Properties in Hybrid Coupled-Cluster and Molecular Mechanics Approach

We report benchmark calculations obtained with our new coupled-cluster singles and doubles (CCSD) code for calculating the first- and second-order molecular properties. This code can be easily incorporated into combined [Valiev, M.; Kowalski, K. J. Chem. Phys. 2006, 125, 211101] classical molecular mechanics (MM) and ab initio coupled-cluster (CC) calculations using NWChem, enabling us to study molecular properties in a realistic environment. To test this methodology, we discuss the results of calculations of dipole moments and static polarizabilities for the Cl2O system in the CCl4 solution using the CCSD (CC with singles and doubles) linear response approach. We also discuss the application of the asymptotic extrapolation scheme (AES) [Kowalski, K.; Valiev, M. J. Phys. Chem. A 2006, 110, 13106] in reducing the numerical cost of CCSD calculations.

The Equilibrium Structure of Ferrocene

The molecular structures of ferrocene in the eclipsed (equilibrium) and staggered (saddle-point) conformations have been determined by full geometry optimizations at the levels of second-order Møller-Plesset (MP2) theory, coupled-cluster singles-and-doubles (CCSD) theory, and CCSD theory with a perturbative triples correction [CCSD(T)] in a TZV2P+f basis set. Existing experimental results are reviewed. The agreement between the CCSD(T) results and experiment is in all cases excellent; the calculated structure parameters and the barrier to internal rotation of the ligand rings differ from the most accurate experimental values by less than two estimated standard deviations. The CCSD(T) calculations for single-configuration-dominated transition metal complexes such as ferrocene thus appear to have an accuracy comparable to that observed for molecules containing only first- and second-row atoms, and to be of a quality similar to that obtained experimentally. A comparison with previous DFT results indicates that the B3LYP model gives overall the best DFT results, with a deviation of around 2 pm for the metal-carbon distance and smaller errors for the cyclopentadienyl rings.

The effect of local approximations in coupled-cluster wave functions on dipole moments and static dipole polarisabilities

The influence of local approximations in electron correlation treatments on electric dipole moments and static dipole polarisabilities is examined for a test set of 16 molecules for the case of coupled-cluster singles and doubles (CCSD) theory. Utilising standard local approximations as originally proposed by Pulay and used in our previous work, the average errors relative to the corresponding conventional CCSD calculations amount to 0.48% for mean dipole polarisabilities and to 1.61% for dipole moments. The accuracy of the computed mean dipole polarisabilities can be reduced to 0.32% by extending the domains of the strong pairs, while for the dipole moments the domain extension reduces the error to 0.91%. It is found that orbital relaxation effects are much more important in the local than in the conventional case. Weak pairs contribute substantially to the computed dipole polarisabilities, but the effect of distant pairs is small and can be neglected. This means that linear scaling local electron correlation methods can be faithfully used to compute these molecular properties.

Theoretical study of the potential stability of the peroxo nitrate radical

The existence of the peroxo nitrate radical (ONOO) has been discussed for some time and its formation has been used to explain aspects of the NO3 scavenging in the Earth’s atmosphere. In this study we report our thorough investigation of the stability of this species by means of highly correlated ab initio calculations. Single-reference coupled-cluster singles and doubles (CCSD) as well as multireference configuration interaction (MRCI) calculations were performed to optimize equilibrium structures and obtain harmonic force fields. The force fields were used to calculate the harmonic frequencies as well as isotopic shifts. The CCSD calculations result in shallow minima for both the |2A″〉 ground as well as the |2A′〉 excited state. However, the calculated isotopic shifts of the ground state show that the experimentally observed shift of 50 cm−1 cannot be due to ONOO. In contrast, no minima were found by the MRCI calculations. The analysis of the wave functions indicates that the potential wells obtained by CCSD are artifacts which are due to the single-reference nature of the CCSD method. Our conclusion is that ONOO is not a bound structure and cannot be observed experimentally. Our calculations also show that thermal decomposition of NO3 into NO and O2 is not possible under atmospheric conditions and thus this channel cannot be responsible for the unknown NO3 scavenging process discussed in the literature.

Calculation of X‐ray scattering intensities by means of the coupled cluster singles and doubles model

AbstractTotal X‐ray scattering intensity σee(q) is very sensitive to electron correlation effects. In this study σee(q) of N2, CO, and N2O have been computed by the coupled cluster singles and doubles (CCSD) method and compared with configuration interaction singles and doubles (CISD) calculations as well as experimental observations. σee(q) curves by CCSD calculations are rather close to those by CISD, but although small, there still exist some discrepancies between calculated and observed values. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1315–1320, 2001

Coupled-cluster methods with internal and semi-internal triply excited clusters: Vibrational spectrum of the HF molecule

The recently proposed, single-reference, coupled-cluster (CC) methods with singly, doubly, and triply excited clusters, in which triexcited clusters T3 are restricted to internal and semi-internal components defined through active orbitals, designated as the CCSD{t′} and CCSDt approaches [P. Piecuch, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 110, 6103 (1999)] have been used to obtain the potential energy function and the vibrational spectrum of the HF molecule, as described by the large, aug-cc-pvtz, basis set. A comparison has been made with the vibrational term values obtained at the very high, full CCSDT (CC singles, doubles, and triples), level and with the experimental (RKR) data. In spite of using the restricted Hartree–Fock reference, the calculated CCSD{t′} and CCSDt vibrational term values have been found to be in much better agreement with the full CCSDT and RKR data than the results of conventional CCSD (CC singles and doubles) calculations, which produce errors of an order of 2000 cm−1 for high-lying states near dissociation. It has been demonstrated that the CCSDt approach is capable of providing the vibrational spectrum of the full CCSDT quality.

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.

Property evaluation and orbital relaxation in coupled cluster methods

Molecular electronic properties such as dipole moments, polarizabilities and hyperpolarizabilities and quadrupole moments and polarizabilities, and spin properties such as hyperfine splitting constants and nuclear magnetic coupling constants are predicted by ab initio coupled cluster (CC) methods for a variety of molecules. We compare the results of property evaluation using orbitals that have been allowed to relax in the presence of the perturbation and results obtained using nonrelaxed orbitals. It is demonstrated numerically, and proven formally, that the coupled cluster singles and doubles (CCSD) model using nonrelaxed orbitals is able to include most of the relaxation effects for the evaluation of first- and second-order properties. Thus there is little reason to perform coupled (perturbed) Hartree–Fock calculations as a precursor to correlated CCSD calculations of such properties.