Revisiting the extrapolation of correlation energies to complete basis set limit

The extrapolation scheme of correlation energy is revisited to evaluate the complete basis set limit from double‐zeta (DZ) and triple‐zeta levels of calculations. The DZ level results are adjusted to the standard asymptotic behavior with respect to the cardinal number, observed at the higher levels of basis sets. Two types of adjusting schemes with effective scaling factors, which recover errors in extrapolations with the DZ level basis set, are examined. The first scheme scales the cardinal number for the DZ level energy, while the second scheme scales the prefactor of the extrapolation function. Systematic assessments on the Gaussian‐3X and Gaussian‐2 test sets reveal that these calibration schemes successfully and drastically reduce errors without additional computational efforts. © 2015 Wiley Periodicals, Inc.     

The convergence of complete active space self-consistent-field energies to the complete basis set limit

Examination of the convergence of full valence complete active space self-consistent-field energies with expansion of the one-electron basis set reveals a pattern very similar to the convergence of single determinant Hartree-Fock energies. Calculations on 26 molecular examples with the sequence of ntuple-zeta augmented polarized (nZaP) basis sets (n=2, 3, 4, 5, and 6) are used to evaluate complete basis set extrapolation schemes. The most effective extrapolation reduces the rms one-electron basis set truncation errors from 3.03, 0.58, and 0.12 mhartree to 0.23, 0.05, and 0.014 mhartree for the 3ZaP, 4ZaP, and 5ZaP basis sets, respectively.     

Dynamical properties of liquid water from ab initio molecular dynamics performed in the complete basis set limit

Dynamical properties of liquid water were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (AIMD) simulations within the Kohn-Sham (KS) density functional theory employing the Becke-Lee-Yang-Parr exchange-correlation functional for the electronic structure. The KS orbitals were expanded in a discrete variable representation basis set, wherein the complete basis set limit can be easily reached and which, therefore, provides complete convergence of ionic forces. In order to minimize possible nonergodic behavior of the simulated water system in a constant energy (NVE) ensemble, a long equilibration run (30 ps) preceded a 60 ps long production run. The temperature drift during the entire 60 ps trajectory was found to be minimal. The diffusion coefficient [0.055 A2/ps] obtained from the present work for 32 D2O molecules is a factor of 4 smaller than the most up to date experimental value, but significantly larger than those of other recent AIMD studies. Adjusting the experimental result so as to match the finite-sized system used in the present study brings the comparison between theory and experiment to within a factor of 3. More importantly, the system is not observed to become "glassy" as has been reported in previous AIMD studies. The computed infrared spectrum is in good agreement with experimental data, especially in the low frequency regime where the translational and librational motions of water are manifested. The long simulation length also made it possible to perform detailed studies of hydrogen bond dynamics. The relaxation dynamics of hydrogen bonds observed in the present AIMD simulation is slower than those of popular force fields, such as the TIP4P potential, but comparable to that of the TIP5P potential.     

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.     

On the accuracy of density-functional theory exchange-correlation functionals for H bonds in small water clusters: benchmarks approaching the complete basis set limit

The ability of several density-functional theory (DFT) exchange-correlation functionals to describe hydrogen bonds in small water clusters (dimer to pentamer) in their global minimum energy structures is evaluated with reference to second order Moller-Plesset perturbation theory (MP2). Errors from basis set incompleteness have been minimized in both the MP2 reference data and the DFT calculations, thus enabling a consistent systematic evaluation of the true performance of the tested functionals. Among all the functionals considered, the hybrid X3LYP and PBE0 functionals offer the best performance and among the nonhybrid generalized gradient approximation functionals, mPWLYP and PBE1W perform best. The popular BLYP and B3LYP functionals consistently underbind and PBE and PW91 display rather variable performance with cluster size.     

Structure of liquid water at ambient temperature from ab initio molecular dynamics performed in the complete basis set limit

Structural properties of liquid water at ambient temperature were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (CPAIMD) simulations combined with the Kohn-Sham (KS) density functional theory and the BLYP exchange-correlation functional for the electronic structure. Unlike other recent work on the same subject, where plane-wave (PW) or hybrid Gaussian/plane-wave basis sets were employed, in the present paper, a discrete variable representation (DVR) basis set is used to expand the KS orbitals, so that with the real-space grid adapted in the present work, the properties of liquid water could be obtained very near the complete basis set limit. Structural properties of liquid water were extracted from a 30 ps CPAIMD-BLYP/DVR trajectory at 300 K. The radial distribution functions (RDFs), spatial distribution functions, and hydrogen bond geometry obtained from the CPAIMD-BLYP/DVR simulation are generally in good agreement with the most up to date experimental measurements. Compared to recent ab initio MD simulations based on PW basis sets, less significant overstructuring was found in the RDFs and the distributions of hydrogen bond angles, suggesting that previous plane-wave and Gaussian basis set calculations have exaggerated the tendency toward overstructuring.     

Vibrational energies of PH3 calculated variationally at the complete basis set limit

The potential energy surface for the electronic ground state of PH(3) was calculated at the CCSD(T) level using aug-cc-pV(Q+d)Z and aug-cc-pVQZ basis sets for P and H, respectively, with scalar relativistic corrections included. A parametrized function was fitted through these ab initio points, and one parameter of this function was empirically adjusted. This analytical PES was employed in variational calculations of vibrational energies with the newly developed program TROVE. The convergence of the calculated vibrational energies with increasing vibrational basis set size was improved by means of an extrapolation scheme analogous to the complete basis set limit schemes used in ab initio electronic structure calculations. The resulting theoretical energy values are in excellent agreement with the available experimentally derived values.     

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.     

The convergence of complete active space self-consistent-field configuration interaction including all single and double excitation energies to the complete basis set limit

Examination of the convergence of full valence complete active space self-consistent-field configuration interaction including all single and double excitation (CASSCF-CISD) energies with expansion of the one-electron basis set reveals a pattern very similar to the convergence of single determinant energies. Calculations on the lowest four singlet states and the lowest four triplet states of N(2) with the sequence of n-tuple-zeta augmented polarized (nZaP) basis sets (n=2, 3, 4, 5, and 6) are used to establish the complete basis set limits. Full configuration-interaction (CI) and core electron contributions must be included for very accurate potential energy surfaces. However, a simple extrapolation scheme that has no adjustable parameters and requires nothing more demanding than CAS(10e(-),8orb)-CISD/3ZaP calculations gives the R(e), omega(e), omega(e)X(e), T(e), and D(e) for these eight states with rms errors of 0.0006 Angstrom, 4.43 cm(-1), 0.35 cm(-1), 0.063 eV, and 0.018 eV, respectively.     

Extrapolation to the limit of a complete basis set for electronic structure calculations on the N2 molecule

Results obtained from nonrelativistic electronic structure calculations using finite Gaussian basis sets are extrapolated to the limit of a complete basis set, employing the results of explicitly correlated coupled-cluster calculations including singles and doubles substitutions (CCSD). For N₂, the basis-set limits for the electronic binding energy, equilibrium bond length and harmonic vibrational wave number are established for the CCSD model including a perturbative correction for triples substitutions and for the internally contracted multireference configuration interaction method. The resulting numbers are in good agreement with experimental values. Received: 2 December 1997 / Accepted: 3 February 1998 / Published online: 17 June 1998     

Extrapolating to the one-electron basis set limit in polarizability calculations

We report calculations of polarizabilities using total energies extrapolated to the complete basis set limit. A dual-level scheme has been employed, with the complete basis set limit of the correlation energy determined by the recently reported uniform singlet- and triplet-pair extrapolation method. The finite field approach has been employed, with tensors and averaged polarizabilities for the ground electronic states of H 2, N 2, CO, and H 2O reported and compared with available experimental data in the literature. Exploratory results are also presented for C 6H 4NO 2NH 2.     

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.     

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.     

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.     

A theoretical study of the photoelectron spectra of dichloroketene with accurate computation of ionization energies via complete basis set limit extrapolation

We calculated the equilibrium geometries and harmonic vibrational frequencies of the ground state and five cationic states of dichloroketene using (TD-)B3LYP, PBE0, and M06/M06-2X approaches. The photoelectron spectra of dichloroketene were simulated by computing Franck-Condon factors. The ionization energies were computed using the CCSD(T) approach with extrapolation to the complete basis set (CBS) limit. We propose two new CBS energy formulas (E = E_CBS + Aexp(-x) + B/(x−1) ⁿ, n = 2 or 3) and compare the performance of different CBS approaches. A new ionic state of dichloroketene belonging to the C_s point group is reported. This state is identified as the first excited state of Cl₂CCO⁺ having a double-well potential-energy curve along the CCO bending mode with a barrier height of 1.335 eV. The simulated photoelectron spectra are in agreement with the experiment. The vertical ionization energies calculated via spectral simulation are more accurate compared with those obtained at the ground-state structure. Among the CBS formulas used, the proposed ansatz with n = 2 performs best, with a mean absolute error of 0.021 and 0.012 eV for the adiabatic and vertical ionization energies, respectively.     

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.     

Assessment of a composite method based on selected density functional theory methods and complete basis set extrapolation formulas

Extrapolation formulas based on power and exponential formulas, as well as alternatives from a Taylor series, were tested and used with density functional theory (DFT) for the calculation of enthalpies of formation. The following four functionals were analyzed: B3LYP, BMK, M06-2X, and B2PLYP. Preliminary tests pointed to B2PLYP and B3LYP as the best and worst functionals, respectively. Taylor series expressions were as accurate as the power formulas and presented better performance than the exponential equation. The power formula (Equation (2)) and one of the simplest Taylor expressions (Equation (13)) were selected for the calculations with B3LYP and B2PLYP, with further empirical adjustments based on the higher level correction (HLC) and scaling of the experimental atomization energies used to calculate enthalpies of formation. HLC improved the B3LYP mean absolute error (MAE) from approximately 4.3 to 3.5 kcal mol⁻¹ using both extrapolation alternatives. For B2PLY, the MAEs were improved from 2.7 to 2.6 kcal mol⁻¹. Regarding the G3/05 test set, a significant improvement in the MAEs around 2.5 and 1.5 kcal mol⁻¹ were achieved using B3LYP and B2PLYP, respectively. The accuracy obtained from these empirical corrections was equivalent to other composite methods. The MAEs from B3LYP and B2PLYP may be suggested as ranges for the possible accuracy to be achieved by some DFT methods. The empirical corrections suggested in this work are improvements that may be considered to provide acceptable accuracy for enthalpies of formation and possibly other properties.     

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.