Truncation of the correlation consistent basis sets: extension to third-row (Ga-Kr) molecules

The systematic reduction of the commonly used correlation consistent basis sets [cc-pVnZ where n=D(2), T(3), Q(4), and 5] as a means to reduce computational cost has been extended to hydrogen-containing third-row (Ga-Kr) molecules of the G2 test suite. Coupled cluster with singles, doubles, and quasiperturbative triple excitations [CCSD(T)] calculations were performed using both the full correlation consistent basis sets and a series of truncated basis sets in order to assess the impact of basis set reduction upon the structures and energies of the species. The impact that truncation of the basis sets for hydrogen has upon extrapolation of energies to the complete basis set limit also has been examined, and the cost savings that can be achieved are discussed. Overall, basis set reduction can be accomplished which preserves the systematic convergence behavior of the full correlation consistent basis sets.     

Correlation energy extrapolation by intrinsic scaling. IV. Accurate binding energies of the homonuclear diatomic molecules carbon, nitrogen, oxygen, and fluorine

The method of extrapolation by intrinsic scaling, recently introduced to obtain correlation energies, is generalized to multiconfigurational reference functions and used to calculate the binding energies of the diatomic molecules C2, N2, O2, and F2. First, accurate approximations to the full configuration interaction energies of the individual molecules and their constituent atoms are determined, employing Dunning's correlation consistent double-, triple- and quadruple zeta basis sets. Then, these energies are extrapolated to their full basis set limits. Chemical accuracy is attained for the binding energies of all molecules.     

Convergence of third order correlation energy in atoms and molecules

We have investigated the convergence of third order correlation energy within the hierarchies of correlation consistent basis sets for helium, neon, and water, and for three stationary points of hydrogen peroxide. This analysis confirms that singlet pair energies converge much slower than triplet pair energies. In addition, singlet pair energies with (aug)-cc-pVDZ and (aug)-cc-pVTZ basis sets do not follow a converging trend and energies with three basis sets larger than aug-cc-pVTZ are generally required for reliable extrapolations of third order correlation energies, making so the explicitly correlated R12 calculations preferable.     

Basis set limits of the second order Moller-Plesset correlation energies of water, methane, acetylene, ethylene, and benzene

We report second order Moller-Plesset (MP2) and MP2-F12 total energies on He, Ne, Ar, H(2)O, CH(4), C(2)H(2), C(2)H(4), and C(6)H(6), using the correlation consistent basis sets, aug-cc-pVXZ (X=D-7). Basis set extrapolation techniques are applied to the MP2 and MP2-F12/B methods. The performance of the methods is tested in the calculations of the atoms, He, Ne, and Ar. It is indicated that the two-point extrapolation of MP2-F12/B with the basis sets (X=5,6) is the most reliable. Similar accuracy is obtained using two-point extrapolated conventional MP2 with the basis sets (X=6,7). For the molecules investigated the valence MP2 correlation energy is estimated within 1 mE(h).     

On the effectiveness of CCSD(T) complete basis set extrapolations for atomization energies

The leading cause of error in standard coupled cluster theory calculations of thermodynamic properties such as atomization energies and heats of formation originates with the truncation of the one-particle basis set expansion. Unfortunately, the use of finite basis sets is currently a computational necessity. Even with basis sets of quadruple zeta quality, errors can easily exceed 8 kcal/mol in small molecules, rendering the results of little practical use. Attempts to address this serious problem have led to a wide variety of proposals for simple complete basis set extrapolation formulas that exploit the regularity in the correlation consistent sequence of basis sets. This study explores the effectiveness of six formulas for reproducing the complete basis set limit. The W4 approach was also examined, although in lesser detail. Reference atomization energies were obtained from standard coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) calculations involving basis sets of 6ζ or better quality for a collection of 141 molecules. In addition, a subset of 51 atomization energies was treated with explicitly correlated CCSD(T)-F12b calculations and very large basis sets. Of the formulas considered, all proved reliable at reducing the one-particle expansion error. Even the least effective formulas cut the error in the raw values by more than half, a feat requiring a much larger basis set without the aid of extrapolation. The most effective formulas cut the mean absolute deviation by a further factor of two. Careful examination of the complete body of statistics failed to reveal a single choice that out performed the others for all basis set combinations and all classes of molecules.     

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.     

Relativistic effects determined using the Douglas-Kroll contracted basis sets and correlation consistent basis sets with small-core relativistic pseudopotentials

The coupled cluster approximation with single, double, and quasiperturbative triple excitations [CCSD(T)] was used in combination with the Douglas-Kroll contracted correlation consistent basis sets [cc-pVnZ-DK, where n = D(2), T(3), Q(4), and 5] and small-core relativistic pseudopotentials (PP) with correlation consistent polarized valence basis sets (cc-pVnZ-PP and aug-cc-pVnZ-PP) to investigate the impact of scalar relativistic corrections on energetic and structural properties of small molecules containing third-row (Ga-Kr) atoms. These molecules were taken from the Gaussian-2 extended test set for third-row atoms. Atomization energies, ionization energies, electron affinities, and proton affinities for molecules in the test set were determined and compared with nonrelativistic results which were obtained in a recent study in which the standard and augmented correlation consistent basis sets were used in combination with CCSD(T). Several schemes were used to extrapolate the energies to the complete basis set limit.     

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.     

Estimating the Hartree—Fock limit from finite basis set calculations

It is demonstrated that the polarization-consistent basis sets, which are optimized for density functional methods, are also suitable for Hartree–Fock calculations, and can be used for estimating the Hartree–Fock basis set limit to within a few micro-hartree accuracy. Various two- and three-point extrapolation schemes are tested and exponential functions are found to be superior compared to functions depending on the inverse power of the highest angular momentum function in the basis set. Total energies can be improved by roughly an order of magnitude, but atomization energies are only marginally improved by extrapolation.     

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.     

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.     

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.     

Accuracy of basis-set extrapolation schemes for DFT-RPA correlation energies in molecular calculations

We construct a reference benchmark set for atomic and molecular random phase approximation (RPA) correlation energies in a density functional theory framework at the complete basis-set limit. This set is used to evaluate the accuracy of some popular extrapolation schemes for RPA all-electron molecular calculations. The results indicate that for absolute energies, accurate results, clearly outperforming raw data, are achievable with two-point extrapolation schemes based on quintuple- and sextuple-zeta basis sets. Moreover, we show that results in good agreement with the benchmark can also be obtained by using a semiempirical extrapolation procedure based on quadruple- and quintuple-zeta basis sets. Finally, we analyze the performance of different extrapolation schemes for atomization energies.     

Compatibility of correlation‐consistent basis sets with a hybrid Hartree–Fock/density functional method

The present study examines the feasibility of combining the correlation‐consistent basis sets developed by Dunning and coworkers with the hybrid Hartree–Fock/density functional method B3LYP. Furthermore, extrapolation to the complete basis set (CBS) limit minimizes errors due to the presence of an incomplete basis set and can act as a rigorous test of the limitations of the B3LYP method. Equilibrium geometries, energies, and harmonic vibrational frequencies were determined for a series of well‐studied, yet computationally challenging, small inorganics and their respective ions. The results were then extrapolated to the CBS limit, where applicable, and compared to experiment. It was found that a union between the hybrid Hartree–Fock/density functional B3LYP method and Dunning's augmented correlation‐consistent basis sets gave results that were comparable to molecular orbital methods that explicitly account for electron correlation. Furthermore, the minimum basis set necessary to attain reasonable results for the systems studied was aug‐cc‐pVTZ. Upgrading to the aug‐cc‐pVQZ level and subsequent extrapolation to the CBS limit further improved the overall agreement with the experiment. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 207–216, 1999     

The correlation-consistent composite approach: application to the G3/99 test set

The correlation-consistent composite approach (ccCA), an ab initio composite technique for computing atomic and molecular energies, recently has been shown to successfully reproduce experimental data for a number of systems. The ccCA is applied to the G3/99 test set, which includes 223 enthalpies of formation, 88 adiabatic ionization potentials, 58 adiabatic electron affinities, and 8 adiabatic proton affinities. Improvements on the original ccCA formalism include replacing the small basis set quadratic configuration interaction computation with a coupled cluster computation, employing a correction for scalar relativistic effects, utilizing the tight-d forms of the second-row correlation-consistent basis sets, and revisiting the basis set chosen for geometry optimization. With two types of complete basis set extrapolation of MP2 energies, ccCA results in an almost zero mean deviation for the G3/99 set (with a best value of -0.10 kcal mol(-1)), and a 0.96 kcal mol(-1) mean absolute deviation, which is equivalent to the accuracy of the G3X model chemistry. There are no optimized or empirical parameters included in the computation of ccCA energies. Except for a few systems to be discussed, ccCA performs as well as or better than Gn methods for most systems containing first-row atoms, while for systems containing second-row atoms, ccCA is an improvement over Gn model chemistries.     

Informatics‐Based Energy Fitting Scheme for Correlation Energy at Complete Basis Set Limit

Energy fitting schemes based on informatics techniques using hierarchical basis sets with small cardinal numbers were numerically investigated to estimate correlation energies at the complete basis set limits. Numerical validations confirmed that the conventional two‐point extrapolation models can be unified into a simple formula with optimal parameters obtained by the same test sets. The extrapolation model was extended to two‐point fitting models by a relaxation of the relationship between the extrapolation coefficients or a change of the fitting formula. Furthermore, n‐scheme fitting models were developed by the combinations of results calculated at several theory levels and basis sets to compensate for the deficiencies in the fitting model at one level of theory. Systematic assessments on the Gaussian‐3X and Gaussian‐2 sets revealed that the fitting models drastically reduced errors with equal or smaller computational effort. © 2016 Wiley Periodicals, Inc.     

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.     

Focal-point conformational analysis of ethanol, propanol, and isopropanol.

Conformational analysis of three small alcohols--ethanol, propanol, and isopropanol--was carried out by systematically improving the basis set and the level of electron correlation. Correlation energy contributions to conformational energies are strongly basis-set-dependent but accurate energy contributions can be obtained by extrapolation to the basis-set limit. At the basis-set limit, second- and third-order electron correlation effects play a significant role for rotations around the CC-OH, HC-CO, and CC-CO bonds. Specifically, second- and third-order correlation effects strongly stabilize structures in which the hydroxylic hydrogen eclipses with the adjacent carbon; a lesser stabilization is present in structures where the CC-OH moiety is in the gauche form. Fourth-order correlation effects to the CC-OH rotation are small due to a partial cancellation of the singles, doubles, and quadruples contribution by the triples contribution. Electron correlation significantly lowers barriers for methyl-group rotations in ethanol and isopropanol, and in these cases the fourth-order correlation effects are noticeable. The relatively large overall importance of third-order correlation energy contributions raises a concern that the inability to accurately estimate this slowly converging contribution may become a limiting factor when highly accurate conformational energies in larger molecules are sought.     

Polarization consistent basis sets. VII. The elements K, Ca, Ga, Ge, As, Se, Br, and Kr

Polarization consistent basis sets, optimized for density functional calculations, are proposed for the elements K, Ca, Ga, Ge, As, Se, Br, and Kr. The basis set composition in terms of number of primitive functions and the contraction is defined based on energetic analyses of atoms and molecules along the lines used in previous work and on the performance for molecular systems. The performance for atomization energies and dipole moments is compared to other widely used basis sets, and it is shown that the new basis sets allow a systematic reduction of basis set errors and in general perform better than existing ones.