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.     

Three-electron excitation in open-shell coupled-cluster theory

The three-electron excitation operator T₃ is included approximately in the coupled-cluster method with single and double excitations for open shells (CCSD). The resulting CCSD + T scheme incorporates all energy diagrams up to and including third order in the perturbation. All ten valence-shell energy differences of O and its ions were calculated. They show significant effects of T₃ and are within 0.1 eV of the basis set limit.     

Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals: The CCSD(T)R12 model

To approach the complete basis set limit of the "gold-standard" coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] method, we extend the recently proposed perturbative explicitly correlated coupled-cluster singles and doubles method, CCSD(2)(R12) [E. F. Valeev, Phys. Chem. Chem. Phys. 8, 106 (2008)], to account for the effect of connected three-electron correlations. The natural choice of the zeroth-order Hamiltonian produces a perturbation expansion with rigorously separable second-order energy corrections due to the explicitly correlated geminals and conventional triple and higher excitations. The resulting CCSD(T)(R12) energy is defined as a sum of the standard CCSD(T) energy and an amplitude-dependent geminal correction. The method is technically very simple: Its implementation requires no modification of the standard CCSD(T) program and the formal cost of the geminal correction is small. We investigate the performance of the open-shell version of the CCSD(T)(R12) method as a possible replacement of the standard complete-basis-set CCSD(T) energies in the high accuracy extrapolated ab initio thermochemistry model of Stanton et al. [J. Chem. Phys. 121, 11599 (2004)]. Correlation contributions to the heat of formation computed with the new method in an aug-cc-pCVXZ basis set have mean absolute basis set errors of 2.8 and 1.0 kJmol when X is T and Q, respectively. The corresponding errors of the standard CCSD(T) method are 9.1, 4.0, and 2.1 kJmol when X=T, Q, and 5. Simple two-point basis set extrapolations of standard CCSD(T) energies perform better than the explicitly correlated method for absolute correlation energies and atomization energies, but no such advantage found when computing heats of formation. A simple Schwenke-type two-point extrapolation of the CCSD(T)(R12)aug-cc-pCVXZ energies with X=T,Q yields the most accurate heats of formation found in this work, in error on average by 0.5 kJmol and at most by 1.7 kJmol.     

Conformations of allyl amine: theory vs experiment

The relative stabilities of the five conformers of allyl amine, a medium-size aliphatic molecule, were estimated by applying ab initio quantum mechanical methods at several levels of theory. The second-order M?ller-Plesset perturbation method (MP2), quadratic configuration interaction including single and double excitations (QCISD), coupled-cluster with single and double excitations (CCSD) and CCSD plus perturbative triple excitations [CCSD(T)] were applied. The Dunning correlation consistent basis sets (through aug-cc-pVQZ and cc-pV5Z) were employed. The MP2 energies relative to the energy of the cis-trans conformer reported here appear to approach the basis set limit. The predicted allyl amine conformer energies approaching the Hartree-Fock basis set limit are 158 cm-1 (cis-gauche), -5 cm-1 (gauche-trans), and -146 cm-1 (gauche-gauche). The same three relative energies near the MP2 basis set limit are 135, 103, and 50 cm-1, respectively. The analogous energies deduced from experiment are 173 +/- 12, 92 +/- 8, and 122 +/- 5 cm-1. The theoretical results obtained in the present study suggest that satisfactory predictions of the conformer energetics of allyl amine may be achieved only by theoretical methods that incorporate consideration of correlation effects in conjunction with large basis sets. Evaluation of the zero-point vibrational energy corrections is critical, due to the very small classical energy differences between the five conformers of allyl amine. Agreement between theory and experiment for the gauche-gauche conformational energy remains problematical.     

Explicitly correlated coupled-cluster theory using cusp conditions. II. Treatment of connected triple excitations

The coupled-cluster singles and doubles method augmented with single Slater-type correlation factors (CCSD-F12) determined by the cusp conditions (also denoted as SP ansatz) yields results close to the basis set limit with only small overhead compared to conventional CCSD. Quantitative calculations on many-electron systems, however, require to include the effect of connected triple excitations at least. In this contribution, the recently proposed [A. Ko?hn, J. Chem. Phys. 130, 131101 (2009)] extended SP ansatz and its application to the noniterative triples correction CCSD(T) is reviewed. The approach allows to include explicit correlation into connected triple excitations without introducing additional unknown parameters. The explicit expressions are presented and analyzed, and possible simplifications to arrive at a computationally efficient scheme are suggested. Numerical tests based on an implementation obtained by an automated approach are presented. Using a partial wave expansion for the neon atom, we can show that the proposed ansatz indeed leads to the expected (L(max)+1)(-7) convergence of the noniterative triples correction, where L(max) is the maximum angular momentum in the orbital expansion. Further results are reported for a test set of 29 molecules, employing Peterson's F12-optimized basis sets. We find that the customary approach of using the conventional noniterative triples correction on top of a CCSD-F12 calculation leads to significant basis set errors. This, however, is not always directly visible for total CCSD(T) energies due to fortuitous error compensation. The new approach offers a thoroughly explicitly correlated CCSD(T)-F12 method with improved basis set convergence of the triples contributions to both total and relative energies.     

Accurate Interaction Energies of CO2 with the 20 Naturally Occurring Amino Acids

We have performed a series of highly accurate calculations between CO₂ and the 20 naturally occurring amino acids for the investigation of the attractive noncovalent interactions. Different nucleophilic groups present in the amino acid structures were considered (α-NH₂, COOH, side groups), and the stronger binding sites were identified. A database of accurate reference interactions energies was compiled as computed by explicitly-correlated coupled-cluster singles-and-doubles, together with perturbative triples extrapolated to the complete-basis-set limit. The CCSD(F12)(T)/CBS reference values were used for comparing a variety of popular density functionals with different basis sets. Our results show that most density functionals with the triple-zeta basis set def2-TZVPP align with the CCSD(F12)(T)/CBS reference values, but errors range from 0.1 kcal/mol up to 1.0 kcal/mol.     

Explicitly correlated second-order perturbation theory calculations on molecules containing heavy main-group elements

Slater-type geminals (STGs) have been used as explicitly correlated two-electron basis functions for calculations on the hydrides of N–As and Sb (as well as on the hydrides of O–Se and F–Br with similar, not reported results) in various one-electron basis sets of Gaussian atomic orbitals. The performance of the explicitly correlated theory has been assessed with respect to the exponent of the STG, for example, by using different exponents for individual pair correlation functions and pair energies. It is shown that a correlation factor with an exponent of ${\gamma = 1.4 a_{0}^{-1}}$ can give reliable results within 1% from the basis-set limit for all investigated molecules in an aug-cc-pVQZ basis set for the valence shells, using fixed amplitudes for the STGs in a diagonal orbital-invariant formulation of the theory. The use of relativistic effective core potentials (RECPs) in explicitly correlated second-order perturbation theory has been investigated.     

Explicitly correlated coupled-cluster theory using cusp conditions. I. Perturbation analysis of coupled-cluster singles and doubles (CCSD-F12)

Geminal functions based on Slater-type correlation factors and fixed expansion coefficients, determined by cusp conditions, have in recent years been forwarded as an efficient and numerically stable method for introducing explicit electron correlation into coupled-cluster theory. In this work, we analyze the equations of explicitly correlated coupled-cluster singles and doubles (CCSD-F12) theory and introduce an ordering scheme based on perturbation theory which can be used to characterize and understand the various approximations found in the literature. Numerical results for a test set of 29 molecules support our analysis and give additional insight. In particular, our results help rationalize the success of the CCSD(F12) approximation which is based on a very systematic cancellation of the neglected, otherwise individually large third-order geminal-geminal coupling terms. Further approximations to CCSD(F12) can be introduced without sacrificing the accuracy if the entire set of third-order coupling terms between the conventional doubles cluster amplitudes and the geminal doubles amplitudes is retained, leading to the recently proposed CCSD[F12] and CCSD(F12(?)) models, which have negligible overhead compared to conventional CCSD calculations. Particularly, the importance of the ring-term type contribution is pointed out which may be used to improve on other existing approximations such as CCSD-F12b. For small basis sets, it might be advantageous to keep certain higher-order terms leading to CCSD-F12(?), which, for the case of the SP ansatz, merely involves a noniterative correction to CCSD(F12(?)).     

Slater-type geminals in explicitly-correlated perturbation theory: application to n-alkanols and analysis of errors and basis-set requirements

In the recent years, Slater-type geminals (STGs) have been used with great success to expand the first-order wave function in an explicitly-correlated perturbation theory. The present work reports on this theory's implementation in the framework of the Turbomole suite of programs. A formalism is presented for evaluating all of the necessary molecular two-electron integrals by means of the Obara-Saika recurrence relations, which can be applied when the STG is expressed as a linear combination of a small number (n) of Gaussians (STG-nG geminal basis). In the Turbomole implementation of the theory, density fitting is employed and a complementary auxiliary basis set (CABS) is used for the resolution-of-the-identity (RI) approximation of explicitly-correlated theory. By virtue of this RI approximation, the calculation of molecular three- and four-electron integrals is avoided. An approximation is invoked to avoid the two-electron integrals over the commutator between the operators of kinetic energy and the STG. This approximation consists of computing commutators between matrices in place of operators. Integrals over commutators between operators would have occurred if the theory had been formulated and implemented as proposed originally. The new implementation in Turbomole was tested by performing a series of calculations on rotational conformers of the alkanols n-propanol through n-pentanol. Basis-set requirements concerning the orbital basis, the auxiliary basis set for density fitting and the CABS were investigated. Furthermore, various (constrained) optimizations of the amplitudes of the explicitly-correlated double excitations were studied. These amplitudes can be optimized in orbital-variant and orbital-invariant manners, or they can be kept fixed at the values governed by the rational generator approach, that is, by the electron cusp conditions. Electron-correlation effects beyond the level of second-order perturbation theory were accounted for by conventional coupled-cluster calculations with single, double and perturbative triple excitations [CCSD(T)]. The explicitly-correlated perturbation theory results were combined with CCSD(T) results and compared with literature data obtained by basis-set extrapolation.     

Coupled-cluster methods with perturbative inclusion of explicitly correlated terms: a preliminary investigation

We propose to account for the large basis-set error of a conventional coupled-cluster energy and wave function by a simple perturbative correction. The perturbation expansion is constructed by L?wdin partitioning of the similarity-transformed Hamiltonian in a space that includes explicitly correlated basis functions. To test this idea, we investigate the second-order explicitly correlated correction to the coupled-cluster singles and doubles (CCSD) energy, denoted here as the CCSD(2)(R12) method. The proposed perturbation expansion presents a systematic and easy-to-interpret picture of the "interference" between the basis-set and correlation hierarchies in the many-body electronic-structure theory. The leading-order term in the energy correction is the amplitude-independent R12 correction from the standard second-order M?ller-Plesset R12 method. The cluster amplitudes appear in the higher-order terms and their effect is to decrease the basis-set correction, in accordance with the usual experience. In addition to the use of the standard R12 technology which simplifies all matrix elements to at most two-electron integrals, we propose several optional approximations to select only the most important terms in the energy correction. For a limited test set, the valence CCSD energies computed with the approximate method, termed , are on average precise to (1.9, 1.4, 0.5 and 0.1%) when computed with Dunning's aug-cc-pVXZ basis sets [X = (D, T, Q, 5)] accompanied by a single Slater-type correlation factor. This precision is a roughly an order of magnitude improvement over the standard CCSD method, whose respective average basis-set errors are (28.2, 10.6, 4.4 and 2.1%). Performance of the method is almost identical to that of the more complex iterative counterpart, CCSD(R12). The proposed approach to explicitly correlated coupled-cluster methods is technically appealing since no modification of the coupled-cluster equations is necessary and the standard M?ller-Plesset R12 machinery can be reused.     

A theoretical study on CH<Subscript>2</Subscript>N<Subscript>2</Subscript> isomers: structure and energetics

Five CH₂N₂ isomers, namely cyanamide, carbodiimide, diazomethane, isocyanamide and nitrilimine, have been investigated at a high level of accuracy. The singles and doubles coupled-cluster method including a perturbational correction for connected triple excitations, CCSD(T), in conjunction with correlation-consistent basis sets ranging in size from triple to quintuple zeta have been employed. Extrapolation to the complete basis set limit has been used with treatments of core-valence correlation effects in order to accurately predict structures, relative energies as well as N–H and C–H bond dissociation energies. The latter required to also investigate the HNNC radical with the same methodology used for CH₂N₂ isomers, while HCNN and HNCN data are available in the literature by the same authors (Puzzarini and Gambi in J Chem Phys 122:064316, 2005). For all the species studied, harmonic vibrational frequencies have also been evaluated at the CCSD(T) level in order to obtain zero-point corrections to total energies.     

New accurate benchmark energies for large water clusters: DFT is better than expected

In this work, we use MP2 and coupled‐cluster with single, double, and perturbative triple excitations [CCSD(T)] as well as their corresponding explicitly correlated (F12) counterparts to compute the interaction energies of water icosamers. The incremental scheme is used to compute benchmark energies at the CCSD(T)/CBS(45) and CCSD(T)(F12*)/cc‐pVQZ‐F12 level of theory. The four structures, dodecahedron, edge sharing, face sharing, and fused cubes, are part of the WATER27 test set and therefore, highly accurate interaction energies are required. All methods applied in this work lead to new benchmark energies for these four systems. To obtain these values, we carefully analyze the convergence of the interaction energies with respect to the basis set. Furthermore, we investigate the influence of the basis set superposition error and the core‐valence correlation. The interaction energies are: dodecahedron ?198.6 kcal/mol, edge sharing ?209.7 kcal/mol, face sharing ?208.0 kcal/mol, and fused cubes ?208.0 kcal/mol. For water clusters, we recommend to use the PW6B95 density functional of Truhlar in combination with Grimme's dispersion correction (D3), as the mean absolute error is 0.9 and the root mean‐squared deviation is only 1.4 kcal/mol. © 2014 Wiley Periodicals, Inc.     

Theoretical investigation for spectroscopic constants of ground-state alkaline-earth dimers with high accuracy

In this work, we provide highly accurate theoretical estimates for spectroscopic constants of the ground-state alkaline-earth dimers (Ca₂, Sr₂, and Ba₂). Electron correlation energies are calculated with coupled-cluster method at the single, double, and noniterative triple excitations [CCSD(T)] level, and the effects of full triples as well as quadruple excitations are also taken into account at the CCSDT and the CCSDT(Q) level. Our results demonstrate that high-order electron correlation is important to achieve results with high accuracy. We also find that results for Ca₂ with counterpoise corrections, which are designed to eliminate the basis set superposition error, deviate further away from those at the complete basis set limit than the uncorrected ones. The calculated binding energies and equilibrium bond lengths for Ca₂ and Sr₂ are in excellent agreement with recent experimental data. On the other hand, our results for Ba₂ are quite different from previous theoretical data, and there is no available experimental equilibrium bond length and binding energy for calibration. Based on the performance of the adopted approach for Ca₂ and Sr₂, our results should be more reliable and could be helpful for future investigations.     

Accurate pair interaction energies for helium from supermolecular Gaussian geminal calculations

Nonrelativistic clamped-nuclei pair interaction energy for ground-state helium atoms has been computed for 12 interatomic separations ranging from 3.0 to 9.0 bohr. The calculations applied the supermolecular approach. The major part of the interaction energy was obtained using the Gaussian geminal implementation of the coupled-cluster theory with double excitations (CCD). Relatively small contributions from single, triple, and quadruple excitations were subsequently included employing the conventional orbital coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)] and the full configuration interaction (FCI) method. For three distances, the single-excitation contribution was taken from literature Gaussian-geminal calculations at the CCSD level. The orbital CCSD(T) and FCI calculations used very large basis sets, up to doubly augmented septuple- and sextuple-zeta size, respectively, and were followed by extrapolations to the complete basis set limits. The accuracy of the total interaction energies has been estimated to be about 3 mK or 0.03% at the minimum of the potential well. For the attractive part of the well, the relative errors remain consistently smaller than 0.03%. In the repulsive part, the accuracy is even better, except, of course, for the region where the potential goes through zero. For interatomic separations smaller than 4.0 bohr, the relative errors do not exceed 0.01%. Such uncertainties are significantly smaller than the expected values of the relativistic and diagonal Born-Oppenheimer contributions to the potential.     

Canonical transcorrelated theory with projected Slater-type geminals

An effective Hamiltonian perturbed with explicit interelectronic correlation is derived from similarity transformation of Hamiltonian using a unitary operator with Slater-type geminals. The Slater-type geminal is projected onto the excitation (and deexcitation) component as in the F12 theory. Simplification is made by truncating higher-body operators, resulting in a correlated Hamiltonian which is Hermitian and has exactly the same complexity as the original Hamiltonian in the second quantized form. It can thus be easily combined with arbitrary correlation models proposed to date. The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite. Our theory is naturally extensible to multireference calculations on the basis of the generalized normal ordering. The construction of the effective Hamiltonian is non-iterative. The numerical assessments demonstrate that the present scheme improves the basis set convergence of the post-mean-field calculations at a similar rate to the explicitly correlated methods proposed by others that couple geminals and conventional excitations.     

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.     

Three-dimensional Potential Energy Surface and Bound States of the Ar2-Ne Complex

The first three-dimensional interaction potential energy surface (PES) of the Ar₂-Ne complex is developed using the single and double excitation coupled cluster theory with noniterative treatment of triple excitations CCSD(T). The aug-cc-pVQZ basis sets are employed for all atoms, including an additional (3s3p2d2f1g) set of midpoint bond functions. The calculated single point energies are fitted to an analytic two-dimensional potential model at each of seven fixed r_Ar2 values. The seven model potentials are then used to construct the three-dimensional PES by interpolating along (r—r_e) using a sixth-order polynomial. The PES is used in the following rovibrational energy levels calculations. The comparisons of theoretical transition frequencies and spectroscopic constants with the experimental results are given.     

pCCSD: parameterized coupled-cluster theory with single and double excitations

The primary characteristics of single reference coupled-cluster (CC) theory are size-extensivity and size-consistency, invariance under orbital rotations of the occupied or virtual space, the exactness of CC theory for N electron systems when the cluster operator is truncated to N-tuple excitations, and the relative insensitivity of CC theory to the choice of the reference determinant. In this work, we propose a continuous class of methods which display the desirable features of the coupled-cluster approach with single and double excitations (CCSD). These methods are closely related to the CCSD method itself and are inspired by the coupled electron pair approximation (CEPA). It is demonstrated that one can systematically improve upon CCSD and obtain geometries, harmonic vibrational frequencies, and total energies from a parameterized version of CCSD or pCCSD(α,β) by selecting a specific member from this continuous family of approaches. In particular, one finds that one such approach, the pCCSD(-1,1) method, is a significant improvement over CCSD for the calculation of equilibrium structures and harmonic frequencies. Moreover, this method behaves surprisingly well in the calculation of potential energy surfaces for single bond dissociation. It appears that this methodology has significant promise for chemical applications and may be particularly useful in applications to larger molecules within the framework of a high accuracy local correlation approach.     

Structure and stability of the organo-noble gas molecules XNgCCX and XNgCCNgX (Ng = Kr, Ar; X = F, Cl)

Density functional theory (B3LYP) and ab initio theory [second-order M?ller-Plesset perturbation theory (MP2) and coupled-cluster theory including single, double, and quasiperturbative triple excitations (CCSD(T))] have been used in combination with the standard and augmented correlation consistent basis sets (cc-pVnZ and aug-cc-pVnZ, where n = D, T, and Q) to investigate potential new noble gas compounds. Two classes of molecules were studied: XNgCCNgX and XNgCCX, where Ng = Kr and Ar and X = F and Cl. These molecules were characterized by finding the ground-state structures and calculating the relative energies, charge distributions, and vibrational frequencies. In addition, transition-state structures were also determined and decomposition pathways were identified through intrinsic reaction coordinate calculations.     

What are the most efficient basis set strategies for correlated wave function calculations of reaction energies and barrier heights?

As electronic structure methods are being used to obtain quantitatively accurate reaction energies and barrier heights for increasingly larger systems, the choice of an efficient basis set is becoming more critical. The optimum strategy for achieving basis set convergence can depend on the way that electron correlation is treated and can take advantage of flexibility in the order in which basis functions are added. Here we study several approaches for estimating accurate reaction energies and barrier heights from post-Hartree-Fock electronic structure calculations. First and second, we evaluate methods of estimating the basis set limit of second order Mo?ller-Plesset perturbation theory and of coupled cluster theory with single and double excitations and a quasiperturbative treatment of connected triple excitations by using explicitly correlated basis functions (in the F12a implementation) along with valence, polarization, and diffuse one-electron basis functions. Third, we test the scheme of adding a higher-order correction to MP2 results (sometimes called MP2∕CBS + ΔCCSD(T)). Finally, we evaluate the basis set requirements of these methods in light of comparisons to Weizmann-3.2, Weizmann-4, and CCSDT(2)(Q)∕CBS+CV+R results.