Performance of the spin-flip and multireference methods for bond breaking in hydrocarbons: a benchmark study

Benchmark results for spin-flip (SF) coupled-cluster and multireference (MR) methods for bond-breaking in hydrocarbons are presented. The nonparallelity errors (NPEs), which are defined as an absolute value of the difference between the maximum and minimum values of the errors in the potential energy along bond-breaking curves, are analyzed for (i) the entire range of nuclear distortions from equilibrium to the dissociation limit and (ii) in the intermediate range (2.5-4.5 A), which is the most relevant for kinetics modeling. For methane, the spin-flip and MR results are compared against full configuration interaction (FCI). For the entire potential energy curves, the NPEs for the SF model with single and double substitutions (SF-CCSD) are slightly less than 3 kcal/mol. Inclusion of triple excitations reduces the NPEs to 0.32 kcal/mol. The corresponding NPEs for the MR-CI are less than 1 kcal/mol, while those of multireference perturbation theory are slightly larger (1.2 kcal/mol). The NPEs in the intermediate range are smaller for all of the methods. The largest errors of 0.35 kcal/mol are observed, surprisingly, for a spin-flip approach that includes triple excitations, while MR-CI, CASPT2, and SF-CCSD curves are very close to each other and are within 0.1-0.2 kcal/mol of FCI. For a larger basis set, the difference between MR-CI and CASPT2 is about 0.2 kcal/mol, while SF-CCSD is within 0.4 kcal/mol of MR-CI. For the C-C bond breaking in ethane, the results of the SF-CCSD are within 1 kcal/mol of MR-CI for the entire curve and within 0.4 kcal/mol in the intermediate region. The corresponding NPEs for CASPT2 are 1.8 and 0.4 kcal/mol, respectively. Including the effect of triples by energy-additivity schemes is found to be insignificant for the intermediate region. For the entire range of nuclear separations, sufficiently large basis sets are required to avoid artifacts at small internuclear separations.     

Interactions in diatomic dimers involving closed-shell metals

Interaction energies of dimers containing alkaline earth (Be, Mg, and Ca) metals have been investigated using symmetry-adapted perturbation theory (SAPT) and supermolecular (SM) methods. Also, to enable broader comparisons, some calculations have been performed on the Zn dimer and on the He-Mg dimer. Although all of the investigated metallic atoms have closed electronic shells, the quasidegeneracy of the ground states of these atoms with the lowest-lying excited states leads to convergence problems in theories based on a single-determinant reference state. The main goal of the present work was to establish how the quality of the interaction energies computed using various electronic-structure methods changes across the range of atoms. We show that although the convergence problems become somewhat less severe with the increase of the atomic number, single-determinant-based methods do not provide reliable interaction energies for any of the investigated metallic dimers even at the level of the coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)]. However, interaction energies accurate to within a few percent can be obtained if CCSD(T) calculations in large basis sets are extrapolated to the complete basis set limit and followed by full configuration interaction (FCI) calculations with a frozen-core (FC) approximation. Since the systems considered contain only two valence electrons, FCI/FC calculations have been feasible for all of them except for Zn2, providing the best theoretical estimates of the binding energies to date. We found that a large part of the error of the SAPT results originates from limiting some exchange components to terms proportional to the squares of the intermonomer orbital overlap integrals. When the neglected terms were approximately accounted for, the accuracy improved significantly and became comparable to that of CCSD(T), allowing us to obtain for the first time a physical interpretation of the interaction energies in metallic dimers.     

Ab initio calculations of dispersion coefficients for nucleic acid base pairs

The results of ab initio calculations of two- and three-body dispersion coefficients for the four most important nucleic acid bases are reported. The isotropic as well as anisotropic coefficients were found by using the time-dependent Hartree-Fock approach and the aug-cc-pVDZ basis set. Single and double excitation coupled-cluster theory with noniterative treatment of triple excitations [CCSD(T)] was used to find the values of static polarizabilities which were subsequently used to estimate the values of the CCSD(T) dispersion coefficients. A comparison of these estimated CCSD(T) dispersion coefficients with coefficients found by using empirical approaches based on atomic contributions revealed that the latter are not reliable.     

An explicitly correlated local coupled cluster method for calculations of large molecules close to the basis set limit

A new explicitly correlated local coupled-cluster method with single and double excitations and a perturbative treatment of triple excitations [DF-LCCSD(T0)-F12x (x = a,b)] is presented. By means of truncating the virtual orbital space to pair-specific local domains (domain approximation) and a simplified treatment of close, weak and distant pairs using LMP2-F12 (pair approximation) the scaling of the computational cost with molecular size is strongly reduced. The basis set incompleteness errors as well as the errors due to the domain approximation are largely eliminated by the explicitly correlated terms. All integrals are computed using efficient density fitting (DF) approximations. The accuracy of the method is investigated for 52 reactions involving medium size molecules. A comparison of DF-LCCSD(T0)-F12x reaction energies with canonical CCSD(T)-F12x calculations shows that the errors introduced by the domain approximation are indeed very small. Care must be taken to keep the errors due to the additional pair approximation equally small, and appropriate distance criteria are recommended. Using these parameters, the root mean square (RMS) deviations of DF-LCCSD(T0)-F12a calculations with triple-ζ basis sets from estimated CCSD(T) complete basis set (CBS) limits and experimental data amount to only 1.5 kJ mol(-1) and 2.9 kJ mol(-1), respectively. For comparison, the RMS deviation of the CCSD(T)/CBS values from the experimental values amounts to 3.0 kJ mol(-1). The potential of the method is demonstrated for five reactions of biochemical or pharmacological interest which include molecules with up to 61 atoms. These calculations show that molecules of this size can now be treated routinely and yield results that are close to the CCSD(T) complete basis set limits.     

Theoretical investigation of rotationally inelastic collisions of the methyl radical with helium

Rotationally inelastic collisions of the CH(3) molecule in its ground X(2)A(2)' electronic state have been investigated. We have determined a potential energy surface (PES) for the interaction of rigid CH(3), frozen at its equilibrium geometry, with a helium atom, using a coupled-cluster method that includes all single and double excitations, as well as perturbative contributions of connected triple excitations [RCCSD(T)]. The anisotropy of the PES is dominated by repulsion of the helium by the hydrogen atoms. The dissociation energy D(e) was computed to equal 27.0 cm(-1). At the global minimum, the helium atom lies in the CH(3) plane between two C-H bonds at an atom-molecule separation R = 6.52 bohr. Cross sections for collision-induced rotational transitions have been determined through quantum scattering calculations for both nuclear spin modifications. Rotationally inelastic collisions can cause a change in the rotational angular momentum n and its body-frame projection k. Because of the anisotropy of the PES due to the hydrogen atoms, there is a strong propensity for Δk = ±3 transitions. Thermal rate constants for state-specific total collisional removal have also been determined.     

Theoretical reference values for the AE6 and BH6 test sets from explicitly correlated coupled-cluster theory

We propose a new computational protocol to obtain highly accurate theoretical reference data. This protocol employs the explicitly correlated coupled-cluster method with iterative single and double excitations as well as perturbative triple excitations, CCSD(T)(F12), using quadruple-z\zeta basis sets. Higher excitations are accounted for by conventional CCSDT(Q) calculations using double-z\zeta basis sets, while core/core-valence correlation effects are estimated by conventional CCSD(T) calculations using quadruple-z\zeta basis sets. Finally, scalar-relativistic effects are accounted for by conventional CCSD(T) calculations using triple-z\zeta basis sets. In the present article, this protocol is applied to the popular test sets AE6 and BH6. An error analysis shows that the new reference values obtained by our computational protocol have an uncertainty of less than 1 kcal/mol (chemical accuracy). Furthermore, concerning the atomization energies, a cancellation of the basis set incompleteness error in the CCSD(T)(F12) perturbative triples contribution with the corresponding error in the contribution from higher excitations is observed. This error cancellation is diminished by the CCSD(T*)(F12) method. Thus, we recommend the use of the CCSD(T*)(F12) method only for small- and medium-sized basis sets, while the CCSD(T)(F12) approach is preferred for high-accuracy calculations in large basis sets.     

A new ab initio interaction energy surface and high-resolution spectra of the H2-CO van der Waals complex

A new four-dimensional intermolecular potential-energy surface for the H(2)-CO complex is presented. The ab initio points have been computed on a five-dimensional grid including the dependence on the H-H separation (the C-O separation was fixed). The surface has then been obtained by averaging over the intramolecular vibration of H(2). The coupled-cluster supermolecular method with single, double, and noniterative triple excitations has been used to calculate the interaction energy. The correlation part of the interaction energy has been obtained from extrapolations based on calculations in a series of basis sets. An analytical fit of the ab initio potential-energy surface has the global minimum of -93.049 cm(-1) at the intermolecular separation of 7.92 bohr for the linear geometry with the C atom pointing toward the H(2) molecule. For the other linear geometry, with the O atom pointing toward H(2), the local minimum of -72.741 cm(-1) has been found for the intermolecular separation of 7.17 bohr. The potential has been used to calculate the rovibrational energy levels of the para-H(2)-CO complex. The results agree very well with those observed by McKellar [A. R. W. McKellar J. Chem. Phys. 108, 1811 (1998)]: the discrepancies are smaller than 0.1 cm(-1). The calculated dissociation energy is equal to 19.527 cm(-1) and significantly smaller than the value of 22 cm(-1) estimated from the experiment. Predictions of rovibrational energy levels for ortho-H(2)-CO have also been done and can serve as a guidance to assign recorded experimental spectra. The interaction second virial coefficient has been calculated and compared with the experimental data.     

Accurate computational thermochemistry from explicitly correlated coupled-cluster theory

Explicitly correlated coupled-cluster theory has developed into a valuable computational tool for the calculation of electronic energies close to the limit of a complete basis set of atomic orbitals. In particular at the level of coupled-cluster theory with single and double excitations (CCSD), the space of double excitations is quickly extended towards a complete basis when Slater-type geminals are added to the wave function expansion. The purpose of the present article is to demonstrate the accuracy and efficiency that can be obtained in computational thermochemistry by a CCSD model that uses such Slater-type geminals. This model is denoted as CCSD(F12), where the acronym F12 highlights the fact that the Slater-type geminals are functions f(r ₁₂) of the interelectronic distances r ₁₂ in the system. The performance of explicitly correlated CCSD(F12) coupled-cluster theory is demonstrated by computing the atomization energies of 73 molecules (containing H, C, N, O, and F) with an estimated root-mean-square deviation from the values compiled in the Active Thermochemical Tables of σ = 0.10 kJ/mol per valence electron. To reach this accuracy, not only the frozen-core CCSD basis-set limit but also high-order excitations (connected triple and quadruple excitations), core–valence correlation effects, anharmonic vibrational zero-point energies, and scalar and spin–orbit relativistic effects must be taken into account.     

How accurate is the density functional theory combined with symmetry-adapted perturbation theory approach for CH-pi and pi-pi interactions? A comparison to supermolecular calculations for the acetylene-benzene dimer

Five different orientations of the acetylene-benzene dimer including the T-shaped global minimum structure are used to assess the accuracy of the density functional theory combined with symmetry adapted perturbation theory (DFT-SAPT) approach in its density-fitting implementation (DF-DFT-SAPT) for the study of CH-pi and pi-pi interactions. The results are compared with the outcome of counterpoise corrected supermolecular calculations employing second-order M?ller-Plesset (MP2), spin-component scaled MP2 (SCS-MP2) and single and double excitation coupled cluster theory including perturbative triple excitations (CCSD(T)). For all considered orientations MP2 predicts much deeper potential energy curves with considerably shifted minima compared to CCSD(T) and DFT-SAPT. In spite of being an improvement over the results of MP2, SCS-MP2 tends to underestimate the well depth while DFT-SAPT, employing an asymptotically corrected hybrid exchange-correlation potential in conjunction with the adiabatic local density approximation for the exchange-correlation kernel, is found to be in excellent agreement with CCSD(T). Furthermore, DFT-SAPT provides a detailed understanding of the importance of the electrostatic, induction and dispersion contributions to the total interaction energy and their repulsive exchange corrections.     

Three-body contribution to the helium interaction potential

Two nonadditive three-body analytic potentials for helium were obtained: one based on three-body symmetry-adapted perturbation theory (SAPT) and the other one on supermolecular coupled-cluster theory with single, double, and noniterative triple excitations [CCSD(T)]. Large basis sets were used, up to the quintuple-zeta doubly augmented size. The fitting functions contain an exponentially decaying component describing the short-range interactions and damped inverse powers expansions for the third- and fourth-order dispersion contributions. The SAPT and CCSD(T) potentials are very close to each other. The largest uncertainty of the potentials comes from the truncation of the level of theory and can be estimated to be about 10 mK or 10% at trimer's minimum configuration. The relative uncertainties for other configurations are also expected to be about 10% except for regions where the nonadditive contribution crosses zero. Such uncertainties are of the same order of magnitude as the current uncertainties of the two-body part of the potential.     

Spectroscopy of K+ x Rg and transport coefficients of K+ in Rg (Rg=He-Rn)

Ab initio calculations employing the coupled-cluster method, with single and double substitutions and accounting for triple excitations noniteratively [CCSD(T)], are used to obtain accurate potential energy curves for the K(+)He, K(+)Ne, K(+)Ar, K(+)Kr, K(+)Xe, and K(+)Rn cationic complexes. From these potentials, rovibrational energy levels and spectroscopic parameters are calculated. In addition, mobilities and diffusion coefficients for K(+) cations moving through the six rare gases are calculated, under conditions that match previous experimental determinations. A detailed statistical comparison of the present and previous potentials is made with available experimental data, and critical conclusions are drawn as to the reliability of each set of data. It is concluded that the present ab initio potentials match the accuracy of the best model potentials and the most reliable experimental data.     

Interaction of dihydrogen with small and light molecules

Second-order M?ller-Plesset (MP2) calculations (using the approximate resolution of the identity, RI-MP2), explicitly correlated MP2 (MP2-R12) calculations, and coupled-cluster calculations including all single and double excitations with a perturbative estimate of triple excitations [CCSD(T)] are performed to study the interaction of molecular hydrogen with the small molecules HF, H2O, NH3, and LiOH. Different adsorption positions are studied. In the cases of H2O and NH3, the most favorable configuration places H2 in an end-on fashion on the O or N atom, respectively. In the cases of HF and LiOH, the H2 molecule takes a side-on position on the H atom of HF or the Li atom. With respect to MP2 calculations in a triple-zeta basis, both the enlargement of the basis set and the extension of the correlation treatment (CCSD(T) vs MP2) increase the interaction energy. The basis set limit CCSD(T) estimates of the interaction energy of H2 with the HF, H2O, NH3, and LiOH molecules amount to 4.40, 2.67, 3.02, and 10.74 kJ mol-1, respectively. The interaction energy for the simultaneous interaction of H2 with two LiOH molecules does not significantly exceed the value obtained for the interaction with a single LiOH molecule. Furthermore, the interaction energies (by MP2) of H2 with glycine, the glycine dimer, and imidazolium chloride amount to 2.78, 5.00, and 6.30 kJ mol-1, respectively.     

The p-difluorobenzene-argon S1 excited state intermolecular potential energy surface

The first excited state (S1) intermolecular potential energy surface for the p-difluorobenzene-Ar van der Waals complex is evaluated using the coupled-cluster method and the augmented correlation consistent polarized valence double-zeta basis set extended with a set of 3s3p2d1f1g midbond functions. In order to calculate the S1 interaction energies we use the ground state surface evaluated with the same basis set and the coupled-cluster singles and doubles [CCSD] including connected triple excitations [CCSD(T)] model, and interaction and excitation energies evaluated at the CCSD level. The surface minima are characterized by the Ar atom located above and below the p-difluorobenzene center of mass at a distance of 3.4736 A. The corresponding interaction energy is -435.233 cm-1. The surface is used in the evaluation of the intermolecular level structure of the complex.     

Quantitative prediction of gas-phase 19F nuclear magnetic shielding constants

Benchmark calculations of (19)F nuclear magnetic shielding constants are presented for a set of 28 molecules. Near-quantitative accuracy (ca. 2 ppm deviation from experiment) is achieved if (1) electron correlation is adequately treated by employing the coupled-cluster singles and doubles (CCSD) model augmented by a perturbative correction for triple excitations [CCSD(T)], (2) large (uncontracted) basis sets are used, (3) gauge-including atomic orbitals are used to ensure gauge-origin independence, (4) calculations are performed at accurate equilibrium geometries [obtained from CCSD(T)/cc-pVTZ calculations correlating all electrons], and (5) vibrational averaging and temperature corrections via second-order vibrational perturbation theory (VPT2) are included. For the CCSD(T)/13s9p4d3f calculations corrected for vibrational effects, mean and standard deviation from experiment are -1.9 and 1.6 ppm, respectively. Less elaborate theoretical treatments result in larger errors. Consideration of relative shifts can reduce the mean deviation (through an appropriately chosen reference compound), but does not change the standard deviation. Density-functional theory calculations of absolute and relative (19)F nuclear magnetic shielding constants are found to be, at best, as accurate as the corresponding Hartree-Fock self-consistent-field calculations and are not improved by consideration of vibrational effects. Molecular systems containing fluorine-oxygen, fluorine-nitrogen, and fluorine-fluorine bonds are found to be more challenging than the other investigated molecules for the considered theoretical methods.     

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.     

Making More Extensive Use of the Coupled-cluster Wave Function: from the Standard Energy Expression to the Energy Expectation Value

The standard coupled-cluster (CC) scheme with single and double excitations in the cluster operator (CCSD) includes only up to quadruple excitations in the equations. The CCSD exponential expansion generates, however, all possible excitations out of the reference function through products of the cluster operators. Clearly, in all standard approximate CC approaches only a part of the CC wave function is used in the equations. If the standard CCSD wave function is inserted into the energy expectation value expression then the complete CCSD wave function contributes to the energy. Such an energy expectation value expression can be presented as a sum of the standard CCSD energy formula plus correction terms. The correction terms provide an information about the quality of the total CC function. Contributions associated with the presence of higher than double excitations in the bra CCSD wave function supplement the CCSD energy obtained within the standard scheme. These contributions can be generated in a sequential way by considering intermediate excitation levels for the bra CCSD wave function in the expectation value expression before reaching the highest excitation level. In this way the importance of particular components differing in the standard and expectation value CCSD energies can be investigated. Some of the contributions can be recognized as close to or identical with the so-called renormalized noniterative corrections to the CC methods. We try to see to what an extent the nonstandard energy expressions, like the energy expectation value or the asymmetric energy formula, can be used to extend the applicability of the CCSD method illustrating our considerations with some numerical examples.Dedicated to Professor Jean-Paul Malrieu to honor his contribution to quantum chemistry and physics