A comparative assessment of the perturbative and renormalized coupled cluster theories with a noniterative treatment of triple excitations for thermochemical kinetics, including a study of basis set and core correlation effects

The CCSD, CCSD(T), and CR-CC(2,3) coupled cluster methods, combined with five triple-zeta basis sets, namely, MG3S, aug-cc-pVTZ, aug-cc-pV(T+d)Z, aug-cc-pCVTZ, and aug-cc-pCV(T+d)Z, are tested against the DBH24 database of diverse reaction barrier heights. The calculations confirm that the inclusion of connected triple excitations is essential to achieving high accuracy for thermochemical kinetics. They show that various noniterative ways of incorporating connected triple excitations in coupled cluster theory, including the CCSD(T) approach, the full CR-CC(2,3) method, and approximate variants of CR-CC(2,3) similar to the triples corrections of the CCSD(2) approaches, are all about equally accurate for describing the effects of connected triply excited clusters in studies of activation barriers. The effect of freezing core electrons on the results of the CCSD, CCSD(T), and CR-CC(2,3) calculations for barrier heights is also examined. It is demonstrated that to include core correlation most reliably, a basis set including functions that correlate the core and that can treat core-valence correlation is required. On the other hand, the frozen-core approximation using valence-optimized basis sets that lead to relatively small computational costs of CCSD(T) and CR-CC(2,3) calculations can achieve almost as high accuracy as the analogous fully correlated calculations.     

Benchmark database of accurate (MP2 and CCSD(T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs

MP2 and CCSD(T) complete basis set (CBS) limit interaction energies and geometries for more than 100 DNA base pairs, amino acid pairs and model complexes are for the first time presented together. Extrapolation to the CBS limit is done by using two-point extrapolation methods and different basis sets (aug-cc-pVDZ - aug-cc-pVTZ, aug-cc-pVTZ - aug-cc-pVQZ, cc-pVTZ - cc-pVQZ) are utilized. The CCSD(T) correction term, determined as a difference between CCSD(T) and MP2 interaction energies, is evaluated with smaller basis sets (6-31G** and cc-pVDZ). Two sets of complex geometries were used, optimized or experimental ones. The JSCH-2005 benchmark set, which is now available to the chemical community, can be used for testing lower-level computational methods. For the first screening the smaller training set (S22) containing 22 model complexes can be recommended. In this case larger basis sets were used for extrapolation to the CBS limit and also CCSD(T) and counterpoise-corrected MP2 optimized geometries were sometimes adopted.     

Hartree-Fock complete basis set limit properties for transition metal diatomics

Numerical Hartree-Fock (HF) energies accurate to at least 1 microhartree are reported for 27 diatomic transition-metal-containing species. The convergence of HF energies toward this numerical limit upon increasing the basis set size has been investigated, where standard nonrelativistic all-electron correlation consistent basis sets and augmented basis sets, developed by Balabanov and Peterson [J. Chem. Phys. 123, 064107 (2005)], were employed. Several schemes which enable the complete basis set (CBS) limit to be determined have been investigated, and the resulting energies have been compared to the numerical Hartree-Fock energies. When comparing basis set extrapolation schemes, those in the form of exponential functions perform well for our test set, with mean absolute deviations from numerical HF energies of 234 and 153 microE(h), when the CBS limit has been determined using a two-point fit as proposed by Halkier et al. [Chem. Phys. Lett. 302, 437 (1999)] on calculations of triple- and quadruple-zeta basis set qualities and calculations of quadruple- and quintuple-zeta basis set qualities, respectively. Overall, extrapolation schemes in the form of a power series are not recommended for the extrapolation of transition metal HF energies. The impact of basis set superposition error has also been examined.     

Breaking bonds of open-shell species with the restricted open-shell size extensive left eigenstate completely renormalized coupled-cluster method

The recently developed restricted open-shell, size extensive, left eigenstate, completely renormalized (CR), coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as ROCCL, is compared with the unrestricted CCSD(T) [UCCSD(T)] and multireference second-order perturbation theory (MRMP2) methods to assess the accuracy of the calculated potential energy surfaces (PESs) of eight single bond-breaking reactions of open-shell species that consist of C, H, Si, and Cl; these types of reactions are interesting because they account for part of the gas-phase chemistry in the silicon carbide chemical vapor deposition. The full configuration interaction (FCI) and multireference configuration interaction with Davidson quadruples correction [MRCI(Q)] methods are used as benchmark methods to evaluate the accuracy of the ROCCL, UCCSD(T), and MRMP2 PESs. The ROCCL PESs are found to be in reasonable agreement with the corresponding FCI or MRCI(Q) PESs in the entire region R = 1-3Re for all of the studied bond-breaking reactions. The ROCCL PESs have smaller nonparallelity error (NPE) than the UCCSD(T) ones and are comparable to those obtained with MRMP2. Both the ROCCL and UCCSD(T) PESs have significantly smaller reaction energy errors (REE) than the MRMP2 ones. Finally, an efficient strategy is proposed to estimate the ROCCL/cc-pVTZ PESs using an additivity approximation for basis set effects and correlation corrections.     

Application of renormalized coupled-cluster methods to potential function of water

The goal of this paper is to examine the performance of the conventional and renormalized single-reference coupled-cluster (CC) methods in calculations of the potential energy surface of the water molecule. A comparison with the results of the internally contracted multi-reference configuration interaction calculations including the quasi-degenerate Davidson correction (MRCI(Q)) and the spectroscopically accurate potential energy surface of water resulting from the use of the energy switching (ES) approach indicates that the relatively inexpensive completely renormalized (CR) CC methods with singles (S), doubles (D), and a non-iterative treatment of triples (T) or triples and quadruples (TQ), such as CR-CCSD(T), CR-CCSD(TQ), and the recently developed rigorously size extensive extension of CR-CCSD(T), termed CR-CC(2,3), provide substantial improvements in the results of conventional CCSD(T) and CCSD(TQ) calculations at larger internuclear separations. It is shown that the CR-CC(2,3) results corrected for the effect of quadruply excited clusters through the CR-CC(2,3)+Q approach can compete with the highly accurate MRCI(Q) data. The excellent agreement between the CR-CC(2,3)+Q and MRCI(Q) results suggests ways of improving the global potential energy surface of water resulting from the use of the ES approach in the regions of intermediate bond stretches and intermediate energies connecting the region of the global minimum with the asymptotic regions. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed Hamiltonian

Completely renormalized (CR) coupled-cluster (CC) approaches, such as CR-CCSD(T), in which one corrects the standard CC singles and doubles (CCSD) energy for the effects of triply (T) and other higher-than-doubly excited clusters [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], are reformulated in terms of the left eigenstates Phimid R:L of the similarity-transformed Hamiltonian of CC theory. The resulting CR-CCSD(T)(L) or CR-CC(2,3) and other CR-CC(L) methods are derived from the new biorthogonal form of the method of moments of CC equations (MMCC) in which, in analogy to the original MMCC theory, one focuses on the noniterative corrections to standard CC energies that recover the exact, full configuration-interaction energies. One of the advantages of the biorthogonal MMCC theory, which will be further analyzed and extended to excited states in a separate paper, is a rigorous size extensivity of the basic ground-state CR-CC(L) approximations that result from it, which was slightly violated by the original CR-CCSD(T) and CR-CCSD(TQ) approaches. This includes the CR-CCSD(T)(L) or CR-CC(2,3) method discussed in this paper, in which one corrects the CCSD energy by the relatively inexpensive noniterative correction due to triples. Test calculations for bond breaking in HF, F(2), and H(2)O indicate that the noniterative CR-CCSD(T)(L) or CR-CC(2,3) approximation is very competitive with the standard CCSD(T) theory for nondegenerate closed-shell states, while being practically as accurate as the full CC approach with singles, doubles, and triples in the bond-breaking region. Calculations of the activation enthalpy for the thermal isomerizations of cyclopropane involving the trimethylene biradical as a transition state show that the noniterative CR-CCSD(T)(L) approximation is capable of providing activation enthalpies which perfectly agree with experiment.     

Coordination of methanol clusters to benzene: a computational study

Benzene-methanol cluster structures were investigated with theoretical chemistry methods to describe the microsolvation of benzene and the benzene-methanol azeotrope. Benzene-methanol (MeOH) clusters containing up to six methanol molecules have been calculated by ab initio [MP2/6-311++G(d,p)//MP2/6-31+G(d,p) + BSSE correction] method. The BSSE was found quite large with this basis set, hence, different extrapolation schemes in combination with the aug-cc-pVxZ basis sets have been used to estimate the complete basis set limit of the MP2 interaction energy [ΔE(MP2/CBS)]. For smaller clusters, n ≤ 3, DFT procedures (DFTB+, MPWB1K, M06-2X) have also been applied. Geometries obtained for these clusters by M06-2X and MP2 calculations are quite similar. Based on the MP2/CBS results, the most stable C(6)H(6)(MeOH)(3) cluster is characterized by a hydrogen bonded MeOH trimer chain interacting with benzene via π···H-O and O···H-C(benzene) hydrogen bonds. Larger benzene-MeOH clusters with n ≥ 4 consist of cyclic (MeOH)(n) subclusters interacting with benzene by dispersive forces, to be denoted by C(6)H(6) + (MeOH)(n). Interaction energies and cooperativity effects are discussed in comparison with methanol clusters. Besides MP2/CBS calculations, for selected larger clusters the M06-2X/6-311++G(d,p)//M06-2X/6-31+G(d,p) procedure including the BSSE correction was also used. Interaction energies obtained thereby are usually close to the MP2/CBS limit. To model the benzene-MeOH azeotrope, several structures for (C(6)H(6))(2)(MeOH)(3) clusters have been calculated. The most stable structures contain a tilted T-shaped benzene dimer interacting by π···H-O and O···H-C (benzene) hydrogen bonds with a (MeOH)(3) chain. A slightly less negative interaction energy results for a parallel displaced benzene sandwich dimer with a (MeOH)(3) chain atop of one of the benzene molecules.     

Conventional strain energies of 1,2-dihydroazete, 2,3-dihydroazete, 1,2-dihydrophosphete, and 2,3-dihydrophosphete

The conventional strain energies of 1,2-dihydroazete, 2,3-dihydroazete, 1,2-dihydrophosphete, and 2,3-dihydrophosphete are determined within the isodesmic, homodesmotic, and hyperhomodesmotic models. Optimum equilibrium geometries, harmonic vibrational frequencies, and corresponding electronic energies and zero-point vibrational energies are computed for all pertinent molecular systems using SCF theory, second-order perturbation theory, and density functional theory and employing the correlation consistent basis sets cc-pVDZ, cc-pVTZ, and cc-pVQZ. Single-point fourth-order perturbation theory, CCSD, and CCSD(T) calculations employing the cc-pVTZ and the cc-pVQZ basis sets are computed using the MP2/cc-pVTZ and MP2/cc-pVQZ optimized geometries, respectfully, to ascertain the contribution of higher order correlation. Three DFT functionals, B3LYP, wB97XD, and M06-2X, are employed to determine whether they can yield results similar to those obtained at the CCSD(T) level.     

Accurate ab initio binding energies of alkaline earth metal clusters

The effects of basis set superposition error (BSSE) and core-correlation on the electronic binding energies of alkaline earth metal clusters Y(n) (Y = Be, Mg, Ca; n = 2-4) at the Moller-Plesset second-order perturbation theory (MP2) and the single and double coupled cluster method with perturbative triples correction (CCSD(T)) levels are examined using the correlation consistent basis sets cc-pVXZ and cc-pCVXZ (X = D, T, Q, 5). It is found that, while BSSE has a negligible effect for valence-electron-only-correlated calculations for most basis sets, its magnitude becomes more pronounced for all-electron-correlated calculations, including core electrons. By utilizing the negligible effect of BSSE on the binding energies for valence-electron-only-correlated calculations, in combination with the negligible core-correlation effect at the CCSD(T) level, accurate binding energies of these clusters up to pentamers (octamers in the case of the Be clusters) are estimated via the basis set extrapolation of ab initio CCSD(T) correlation energies of the monomer and cluster with only the cc-pVDZ and cc-pVTZ sets, using the basis set and correlation-dependent extrapolation formula recently devised. A comparison between the CCSD(T) and density functional theory (DFT) binding energies is made to identify the most appropriate DFT method for the study of these clusters.     

On the validity of complete basis set extrapolation formula optimized for the equilibrium distance applied to the potential energy surface for the correlation energy of the helium dimer

Full configuration interaction calculations are performed for He₂ using various orbital basis sets of the aug‐cc‐pVXZ type, with the correlation energies being extrapolated to the complete basis set (CBS) limit. A two‐point CBS extrapolation formula has been utilized for such a purpose. It is shown that the extrapolation formula with the offset parameter k(R) optimized for the equilibrium distance is not uniformly applicable to He He distances in the very short region of the potential energy curve. The offset parameter k(R) in the repulsive region of the potential energy curve can be largely different with the one in the long‐range distances especially in the cases of basis‐sets with large cardinality number. It is also noticed that the accuracy of this extrapolation scheme may not be improved with the increasing of the cardinality number.     

Basis set convergence of molecular correlation energy differences within the random phase approximation

The basis set convergence of energy differences obtained from the random phase approximation (RPA) to the correlation energy is investigated for a wide range of molecular interactions. For dispersion bound systems the basis set incompleteness error is most pronounced, as shown for the S22 benchmark [P. Jurecka et al., Phys. Chem. Chem. Phys. 8, 1985 (2006)]. The use of very large basis sets (> quintuple-zeta) or extrapolation to the complete basis set (CBS) limit is necessary to obtain a reliable estimate of the binding energy for these systems. Counterpoise corrected results converge to the same CBS limit, but counterpoise correction without extrapolation is insufficient. Core-valence correlations do not play a significant role. For medium- and short-range correlation, quadruple-zeta results are essentially converged, as demonstrated for relative alkane conformer energies, reaction energies dominated by intramolecular dispersion, isomerization energies, and reaction energies of small organic molecules. Except for weakly bound systems, diffuse augmentation almost universally slows down basis set convergence. For most RPA applications, quadruple-zeta valence basis sets offer a good balance between accuracy and efficiency.     

Accurate theoretical near-equilibrium potential energy and dipole moment surfaces of HgClO and HgBrO

The complexes HgBrO and HgClO have been previously determined by ab initio methods to be strongly bound and were suggested to be important intermediates during mercury depletions events observed in the polar troposphere. In the present work accurate near-equilibrium potential energy surfaces (PESs) of these species are reported. The PESs are determined using accurate coupled cluster methods and a series of correlation consistent basis sets with subsequent extrapolation to the complete basis set limit. Additive corrections for both core-valence correlation energy and relativistic effects are also included. The anharmonic ro-vibrational spectra of HgBrO and HgClO have been calculated in variational calculations. Strong infrared band strengths are predicted for all fundamentals in these species. The spin-orbit splitting dominates over the vibronic coupling effect in both HgClO and HgBrO. The Renner-Teller vibronic energy levels corresponding to the bending mode of these molecules are calculated via perturbation theory.     

Intermolecular bond lengths: extrapolation to the basis set limit on uncorrected and BSSE‐corrected potential energy hypersurfaces

Geometry optimizations were carried out for the (HF)₂, (H₂O)₂, and HF–H₂O intermolecular complexes using the MP2/aug‐cc‐pVXZ {X=2, 3, 4, and 5} theoretical models on both the uncorrected and counterpoise (CP) corrected potential energy hypersurfaces (PES). Our results and the available literature data clearly show that extrapolation of intermolecular distances to the complete basis set (CBS) limit is satisfactory on PESs corrected for BSSE. On the other hand, one should avoid such extrapolations using data obtained from uncorrected PESs. Also, fixing intramolecular parameters at their experimental values could cause difficulties during the extrapolation. As the available literature data and our results clearly show, the MP2/aug‐cc‐pVXZ {X=2, 3, 4} data series of intermolecular distances obtained from the CP‐corrected surfaces can be safely used for the purpose of CBS extrapolations. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 196–207, 2001     

Towards a complete basis set limit of Hartree–Fock method: correlation-consistent versus polarized-consistent basis sets

In this paper the convergence pattern of correlation-consistent (cc-pVxZ) and polarized-consistent (PC-n) hierarchies relative to the complete basis set limit have been considered in a small set of diatomic molecules. Using the sequence of these basis sets it was demonstrated that potential energy surfaces derived from basis-set-dependent solution of the Hartree–Fock equations achieves the exact numerical derived potential energy surfaces (PESs) in an ordered manner. So it was possible to compute the spectroscopic parameters in the complete basis set limit with considerable accuracy using the most extended members of both hierarchies. On the other hand, for the first time the detailed convergence patterns of total energies in three separate inter-nuclear distances have been considered in these molecules and it was demonstrated that the total energies arrive at microhartree accuracy at a considerable rate. Possible performance of extrapolation schemes is discussed and it was demonstrated that reliable extrapolation procedures indeed exist. A successful test of the proposed extrapolation method, using the three most extended members of polarized-consistent basis sets, has been accomplished on selected polyatomic molecules.     

Infinite Basis set extrapolation for Double Hybrid Density Functional Theory 1: effect of applying various extrapolation functions

We applied the Infinite Basis (IB) set extrapolation and Double Hybrid Density Functional Theory (DHDF) to calculate the databases of atomization energies, ionization energies, electron affinities, reaction barrier heights, proton affinities, alkyl bond dissociation energies, and noncovalent interactions. The Complete Basis Set (CBS) limit is estimated by extrapolating the hybrid density functional theory and PT2 energies using extrapolation functions including exponential, inverse power, modified exponential, and the combination of the these functions. We found that the combination of B2KPLYP/cc-pV[D|T]Z (which is the extrapolation based on the energies calculated in cc-pVDZ and cc-pVTZ) gives results in quadruple-ζ quality. However, if we want to reach the ～2 kcal/mol chemical accuracy limit, the cc-pV[T|Q]Z is required. Similar results with various extrapolation functions obtained, because the IB parameters were determined by minimizing the averaged mean unsigned error of the calculated databases. We generalized the IB set extrapolation to include more than two basis sets, but we found that extrapolation with two basis sets is satisfactory to give reasonable results. The largest error occurred in the databases of the electron affinities and the weak interactions between the noble gas and the nonpolar molecules. We expect that performing the DHDF-IB scheme with the basis sets augmented by diffuse basis functions will further improve the results.     

Coupled cluster study of the energetic properties of S2x (x=0,+1,1)

Ab initio electronic structure calculations are reported for S2, and its ions S2+ and S2-. Geometric parameters are calculated using the singles and doubles coupled cluster method, including a perturbational correction for connected triple excitation, together with systematic sequences of correlation consistent basis sets extrapolated to the complete basis set (CBS) limit. Energetic and structural properties of S2 and the S2 cation and anion are reported. The heat of formation of S2 (3Sigmag-) at 0 K in the gas phase is predicted to be 29.8 kcal/mol from the average of CBS two extrapolation procedures, less than the experimental heat of formation of S2 of 30.66+/-0.07 kcal/mol. The 0 K adiabatic ionization potential and electron affinity are predicted to be 9.37 and 1.68 eV, respectively.     

A comparison of quantum chemical models for calculating NMR shielding parameters in peptides: mixed basis set and ONIOM methods combined with a complete basis set extrapolation

This article compares several quantum mechanical approaches to the computation of chemical shielding tensors in peptide fragments. First, we describe the effects of basis set quality up to the complete basis set (CBS) limit and level of theory (HF, MP2, and DFT) for four different atoms in trans N-methylacetamide. For both isotropic shielding and shielding anisotropy, the MP2 results in the CBS limit show the best agreement with experiment. The HF values show quite a different tendency to MP2, and even in the CBS limit they are far from experiment for not only the isotropic shielding of carbonyl carbon but also most shielding anisotropies. In most cases, the DFT values differ systematically from MP2, and small basis-set (double- or triple-zeta) results are often fortuitously in better agreement with the experiment than the CBS ones. Second, we compare the mixed basis set and ONIOM methods, combined with CBS extrapolation, for chemical shielding calculations at a DFT level using various model peptides. From the results, it is shown that the mixed basis set method provides better results than ONIOM, compared to CBS calculations using the nonpartitioned full systems. The information studied here will be useful in guiding the selection of proper quantum chemical models, which are in a tradeoff between accuracy and cost, for shielding studies of peptides and proteins.     

Potential-energy surface for the electronic ground state of NH3 up to 20,000 cm-1 above equilibrium

Ab initio coupled cluster calculations with single and double substitutions and a perturbative treatment of connected triple excitations [CCSD(T)] with the augmented correlation-consistent polarized valence triple-zeta aug-cc-pVTZ basis at 51 816 geometries provide a six-dimensional potential-energy surface for the electronic ground state of NH3. At 3814 selected geometries, CBS+ energies are obtained by extrapolating the CCSD(T) results for the aug-cc-pVXZ(X=T,Q,5) basis sets to the complete basis set (CBS) limit and adding corrections for core-valence correlation and relativistic effects. CBS** ab initio energies are generated at 51,816 geometries by an empirical extrapolation of the CCSD(T)/aug-cc-pVTZ results to the CBS+ limit. They cover the energy region up to 20,000 cm-1 above equilibrium. Parametrized analytical functions are fitted through the ab initio points. For these analytical surfaces, vibrational term values and transition moments are calculated by means of a variational program employing a kinetic-energy operator expressed in the Eckart-Sayvetz frame. Comparisons against experiment are used to assess the quality of the generated potential-energy surfaces. A "spectroscopic" potential-energy surface of NH3 is determined by a slight empirical adjustment of the ab initio potential to the experimental vibrational term values. Variational calculations on this refined surface yield rms deviations from experiment of 0.8 cm-1 for 24 inversion splittings and 0.4 (3.0) cm-1 for 34 (51) vibrational term values up to 6100 (10,300) cm-1.     

Estimation of isotropic nuclear magnetic shieldings in the CCSD(T) and MP2 complete basis set limit using affordable correlation calculations

A linear correlation between isotropic nuclear magnetic shielding constants for seven model molecules (CH₂O, H₂O, HF, F₂, HCN, SiH₄ and H₂S) calculated with 37 methods (34 density functionals, RHF, MP2 and CCSD(T)), with affordable pcS‐2 basis set and corresponding complete basis set results, estimated from calculations with the family of polarization‐consistent pcS‐n basis sets is reported. This dependence was also supported by inspection of profiles of deviation between CBS estimated nuclear shieldings and shieldings obtained with the significantly smaller basis sets pcS‐2 and aug‐cc‐pVTZ‐J for the selected set of 37 calculation methods. It was possible to formulate a practical approach of estimating the values of isotropic nuclear magnetic shielding constants at the CCSD(T)/CBS and MP2/CBS levels from affordable CCSD(T)/pcS‐2, MP2/pcS‐2 and DFT/CBS calculations with pcS‐n basis sets. The proposed method leads to a fairly accurate estimation of nuclear magnetic shieldings and considerable saving of computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.