Accurate ab intio determination of binding energies for rare-gas dimers by basis set extrapolation

To investigate the electron correlation effect on the binding energies of very weakly bound complexes at highly correlated levels, an extrapolation scheme exploiting the convergent behavior of the binding energy differences between two correlation levels with the correlation-consistent basis set aug-cc-pVXZ was explored. The scheme is based on extrapolating the binding energy differences between the lower and higher correlation levels (such as second-order Møller–Plesset perturbation theory and the single and double coupled-cluster method with perturbative triple correction level), CCSD(T), by X^–3 for relatively small basis set calculations to estimate the corresponding basis set limit, which is then added to the complete basis set(CBS) limit binding energy at the lower correlation level to derive the CBS limit binding energy at the higher level. Test results on rare-gas dimers Rg₂ (Rg is He, Ne, Ar) show that the CCSD(T) CBS limit binding energies estimated by this scheme with aug-cc-pVXZ and aug-cc-pV(X+1)Z basis sets are more accurate than the CBS limit estimated by direct extrapolation of correlation energies by X^–3 with aug-cc-pV(X+1)Z and aug-cc-pV(X+2)Z basis sets in most cases, which signifies the utility of the proposed extrapolation scheme as the level of electron correlation treatment increases. The nonnegligible discrepancy in the well depth near equilibrium between the experimental and the full connected single, double, and triple coupled-cluster method CBS limit estimate obtained by this procedure in the case of Ar₂ suggests that the previous semiempirical potential may be too attractive near equilibrium compared with the actual one.Acknowledgement The major portion of this work was carried out while the author was visiting the Quantum Theory Project (QTP) at the University of Florida. The author is thankful to Rodney Bartlett for hospitality and support during the visit. The author is also thankful to Ajith Perera for assistance in using the ACESII program package. Computational support from the QTP at the University of Florida and the Institute for Basic Science at Ajou University is gratefully acknowledged.     

Characterization of the potential energy surfaces of two small but challenging noncovalent dimers: (P2)2 and (PCCP)2

This work characterizes eight stationary points of the P₂ dimer and six stationary points of the PCCP dimer, including a newly identified minimum on both potential energy surfaces. Full geometry optimizations and corresponding harmonic vibrational frequencies were computed with the second‐order Møller–Plesset (MP2) electronic structure method and six different basis sets: aug‐cc‐pVXZ, aug‐cc‐pV(X+d)Z, and aug‐cc‐pCVXZ where X = T, Q. A new L‐shaped structure with C₂ symmetry is the only minimum for the P₂ dimer at the MP2 level of theory with these basis sets. The previously reported parallel‐slipped structure with C_2h symmetry and a newly identified cross configuration with D₂ symmetry are the only minima for the PCCP dimer. Single point energies were also computed using the canonical MP2 and CCSD(T) methods as well as the explicitly correlated MP2‐F12 and CCSD(T)‐F12 methods and the aug‐cc‐pVXZ (X = D, T, Q, 5) basis sets. The energetics obtained with the explicitly correlated methods were very similar to the canonical results for the larger basis sets. Extrapolations were performed to estimate the complete basis set (CBS) limit MP2 and CCSD(T) binding energies. MP2 and MP2‐F12 significantly overbind the P₂ and PCCP dimers relative to the CCSD(T) and CCSD(T)‐F12 binding energies by as much as 1.5 kcal mol^?1 for the former and 5.0 kcal mol^?1 for the latter at the CBS limit. The dominant attractive component of the interaction energy for each dimer configuration was dispersion according to several symmetry‐adapted perturbation theory analyses. © 2014 Wiley Periodicals, Inc.     

Auxiliary basis sets for density‐fitting second‐order Møller–Plesset perturbation theory: Weighted core‐valence correlation consistent basis sets for the 4d elements Y–Pd

Auxiliary basis sets (ABS) specifically matched to the cc‐pwCVnZ‐PP and aug‐cc‐pwCVnZ‐PP orbital basis sets (OBS) have been developed and optimized for the 4d elements Y‐Pd at the second‐order Møller‐Plesset perturbation theory level. Calculation of the core‐valence electron correlation energies for small to medium sized transition metal complexes demonstrates that the error due to the use of these new sets in density fitting is three to four orders of magnitude smaller than that due to the OBS incompleteness, and hence is considered negligible. Utilizing the ABSs in the resolution‐of‐the‐identity component of explicitly correlated calculations is also investigated, where it is shown that i‐type functions are important to produce well‐controlled errors in both integrals and correlation energy. Benchmarking at the explicitly correlated coupled cluster with single, double, and perturbative triple excitations level indicates impressive convergence with respect to basis set size for the spectroscopic constants of 4d monofluorides; explicitly correlated double‐ζ calculations produce results close to conventional quadruple‐ζ, and triple‐ζ is within chemical accuracy of the complete basis set limit. © 2013 Wiley Periodicals, Inc.     

Phosphorus mononitride: A difficult case for theory

Phosphorus nitride (PN) is the simplest molecule formed solely by phosphorus and nitrogen. It represents an interesting model for materials, where phosphorus is directly attached to nitrogen. Nevertheless, both theoretical and experimental studies often provide an incomplete picture on the structural, electronic, and spectral properties of PN. Theoretical predictions often suffer from insufficient level of theory, incomplete basis set, or from neglecting several effects, for example, zero-point vibrational correction (ZPVC). Therefore, we performed an extensive benchmark study on structural, electronic, and spectral properties of PN at the Hartree-Fock, density functional theory (DFT), or even the coupled-cluster levels. We paid special attention to the basis set effect. We tested three variants of Dunning's aug-cc-pVXZ basis sets with the size from double-ζ to sextuple-ζ, as well as Jensen's aug-pc-n, aug-pcJ-n, and aug-pcSseg-n basis sets, where n = 1-4. Obtained energetics, PN distance, dipole moment, vibrational frequencies, and nuclear magnetic resonance (NMR) parameters were extrapolated to the complete basis set limit (CBS) using three- or two-parameter formulas. The ³¹P NMR shieldings estimated with the aug-cc-pVXZ and aug-cc-pV(X + d)Z basis sets strongly depend on the basis set size providing scattered convergence patterns toward CBS. The Hartree-Fock self-consistent field (HF-SCF) NMR parameters evinced similar behavior as the coupled-cluster data. The only smooth convergence was achieved using the aug-cc-pCVXZ basis sets that include core-valence effects. The KT3 functional underestimated the phosphorus CBS shieldings by about 12 ppm compared to coupled cluster with singles and doubles (CCSD) (T). Nevertheless, KT3 unambiguously surpasses the HF-SCF and CCSD levels that provide ³¹P shieldings that are lower by about 150 ppm and 24 ppm compared to CCSD(T). The convergence of nitrogen shieldings was regular for all basis set hierarchies and all theoretical methods. Relativistic and vibrational effects on selected properties were also discussed.     

Substituent effects in second-row molecules: Basis set performance in calculations of normal valency phosphorus and sulfur compounds

Ab inito molecular orbital calculations of the phosphorus- and sulfur-containing series PH₂X, PH₃X⁺, SHX, and SH₂X⁺ (X = H, CH₃, NH₂, OH, F) have been carried out over a range of Gaussian basis sets and the results (optimized geometrical structures, relative energies, and electron distributions) critically compared. As in first-row molecules there are large discrepancies between substituent interaction energies at different basis set levels, particularly in electron-rich molecules; use of basis sets lower than the supplemented 6-31G basis incurs the risk of obtaining substituent stabilizations with large errors, including the wrong sign. Only a small part of the discrepancies is accounted for by structural differences between the optimized geometries. Supplementation of low level basis sets by d functions frequently leads to exaggerated stabilization energies for π-donor substituents. Poor performance also results from the use of split valence basis sets in which the valence shell electron density is too heavily concentrated in diffuse component of the valence shell functions, again likely to occur in electron-rich molecules. Isodesmic reaction energies are much less sensitive to basis set variation, but d function supplementation is necessary to achieve reliable results, suggesting a marginal valence role for d functions, not merely polarization of the bonding density. Optimized molecular geometries are relatively insensitive to basis set and electron population analysis data, for better-than-minimal bases, are uniform to an unexpected degree.     

Basis set convergence of explicitly correlated double-hybrid density functional theory calculations

The basis set convergence of explicitly correlated double-hybrid density functional theory (DFT) is investigated using the B2GP-PLYP functional. As reference values, we use basis set limit B2GP-PLYP-F12 reaction energies extrapolated from the aug(')-cc-pV(Q+d)Z and aug(')-cc-pV(5+d)Z basis sets. Explicitly correlated double-hybrid DFT calculations converge significantly faster to the basis set limit than conventional calculations done with basis sets saturated up to the same angular momentum (typically, one "gains" one angular momentum in the explicitly correlated calculations). In explicitly correlated F12 calculations the VnZ-F12 basis sets converge faster than the orbital A(')VnZ basis sets. Furthermore, basis set convergence of the MP2-F12 component is apparently faster than that of the underlying Kohn-Sham calculation. Therefore, the most cost-effective approach consists of combining the MP2-F12 correlation energy from a comparatively small basis set such as VDZ-F12 with a DFT energy from a larger basis set such as aug(')-cc-pV(T+d)Z.     

Accurate Theoretical Study of LiS Radical and Its Singly Charged Cation and Anion in their Ground Electronic State

Potential energies of LiS(²Π), LiS^-(¹Σ⁺) and LiS⁺(³Σ^-) are calculated by using the multireference configuration interaction method including Davidson correction and the augmented correlation-consistent basis sets aug-cc-PV(X+d)Z (X=T, Q). Such obtained potential energies are subsequently extrapolated to the complete basis set limit. Both the core-valence correction and the relativistic effect are also considered. The analytical potential energy functions are then obtained by fitting such accurate energies utilizing a least-squares fitting procedure. By using such analytical potential energy functions, we obtain the accurate spectroscopic parameters, complete set of vibrational levels and classical turning points. The present results are compared well with the experimental and other theoretical work.     

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.     

TiCl, TiH, and TiH+ bond energies: a test of a correlation-consistent Ti basis set

Correlation-consistent basis sets are developed for the Ti atom. The polarization functions are optimized for the average of the ³F and ⁵F states. One series of correlation-consistent basis sets is for 3d and 4s correlation, while the second series includes 3s and 3p correlation as well as 3d and 4s correlation. These basis sets are tested using the Ti ³F–⁵F separation and the dissociation energies of TiCl X⁴Φ, TiH X⁴Φ, and TiH⁺ X³Φ. The CCSD(T) complete basis set limit values are determined by extrapolation. The Douglas–Kroll approach is used to compute the scalar relativistic effect. Spin-orbit effects are taken from experiment and/or are computed at the CASSCF level. The Ti³F–⁵F separation is in excellent agreement with experiment, while the TiCl, TiH, and TiH⁺ bond energies are in good agreement with experiment. Extrapolation with the valence basis set is consistent with other atoms, while including 3s and 3p correlation appears to make extrapolation more difficult. Received: 20 January 1999 / Accepted: 26 February 1999 / Published online: 7 June 1999     

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

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

Can extrapolation to the basis set limit be an alternative to the counterpoise correction? A study on the helium dimer

Configuration interaction and coupled cluster calculations are reported for He₂ using various orbital basis sets of the d-aug-AVXZ type, with the results being extrapolated to the one electron basis set limit both with counterpoise and without counterpoise correction. A generalized uniform singlet- and triplet-pair extrapolation scheme has been utilized for such a purpose. Using appropriate corrections to mimic full configuration interaction, the energies were predicted in excellent agreement with the best available estimates. The results also suggest that extrapolation to the complete basis set limit may be a general alternative to the counterpoise correction that yields a more accurate potential energy while being more economical.     

Quantum dynamics of vibrationally activated OH-CO reactant complexes

The MP2 complete basis set (CBS) limit for the binding energy of the two low-lying water octamer isomers of D2d and S4 symmetry is estimated at -72.7+/-0.4 kcal/mol using the family of augmented correlation-consistent orbital basis sets of double through quintuple zeta quality. The largest MP2 calculation with the augmented quintuple zeta (aug-cc-pV5Z) basis set produced binding energies of -73.70 (D2d) and -73.67 kcal/mol (S4). The effects of higher correlation, computed at the CCSD(T) level of theory, are estimated at <0.1 kcal/mol. The newly established MP2/CBS limit for the water octamer is reproduced quite accurately by the newly developed all atom polarizable, flexible interaction potential (TTM2-F). The TTM2-F binding energies of -73.21 (D2d) and -73.24 kcal/mol (S4) for the two isomers are just 0.5 kcal/mol (or 0.7%) larger than the MP2/CBS limit.     

Binding energies of hydrogen-bonded clusters from extrapolation-oriented basis sets

The MP2 and CCSD(T) basis set limit binding energies of various hydrogen-bonded clusters were estimated via basis set extrapolation employing the correlation consistent aug-cc-pVDZ and modified aug-cc-pVDZ set containing extra polarization functions from cc-pVTZ set. By adopting the optimal interval for the difference between the cardinal numbers (X) corresponding to two basis sets in the X ⁻³ type extrapolation scheme the estimated binding energies for (H₂O)_n and (HF)_n (n=3−5) are shown to be close to the reference basis set limit values within the error bounds in many cases, manifesting the significance of these basis sets in studying the structures and binding of large hydrogen-bonded clusters.     

Benchmark calculations with correlated molecular wave functions XII. Core correlation effects on the homonuclear diatomic molecules B2-F2

Using systematic sequences of the newly developed correlation consistent core-valence basis sets from cc-pCVDZ through cc-pCV6Z, the spectroscopic constants of the homonuclear diatomic molecules containing first row atoms, B–F, are calculated both with and without inclusion of 1s correlation. Internally contracted multireference configuration interaction (IC-MRCI) and singles and doubles coupled cluster (CCSD) theory with a perturbational estimate of connected triple excitations, CCSD(T), have been investigated. By exploiting the convergence of the correlation consistent basis sets, complete basis set (CBS) limits have been estimated for total energies, dissociation energies, equilibrium geometries, and harmonic frequencies. Based on the estimated CBS limits the effects of 1s correlation on D _e (kcal/mol), r _e (?), and ω _e (cm⁻¹) are: +1.1, −0.0070, +10 for B₂; +1.5, −0.0040, +13 for C₂; +0.9, −0.0020, +9 for N₂; +0.3, −0.0020, +6 for O₂; and −0.1, −0.0015, +1 for F₂. Received: 20 January 1997 / Accepted: 6 May 1997     

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

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

Complete basis set model chemistry applied to molecules of increasing molecular complexity: Thermochemical properties of organic sulfur derivatives

To estimate the thermochemical properties, bond dissociation energies and atomization energies of sulfur organic derivatives, the complete basis set (CBS) method was employed at the lower computational level (CBS‐4) owing to the large molecular size of a number of the molecules chosen. By comparison with experimental values, calculated values of thermochemical properties are subject to error, which increases in line with the increase in molecular complexity. The main source of error affecting the calculated enthalpy of formation stems from the difference between the energy of the molecule and that of the single atoms: the greater the size of the molecule, the greater the accumulation of error. By acting on the empirical correction to the CBS energy and minimizing the error due to the contribution of the single atoms to the dissociation energy a parameter d_i for each atom i is obtained. Application of these corrections does not greatly affect the heats of formation of the small molecules included in test sets employed for previous comparisons of calculated and experimental values, while there is a great improvement in the case of large molecules, for example, diphenyl disulfide. The mean absolute deviation turns out to be 2.52, which is greater than that obtained in recent reexaminations of model chemistry methods including the G3 and G3(MP3) approaches. The improvement in the results calculated for large molecules, whose heats of formation are calculated with large errors at the CBS‐4 level, in comparison also with the CBS‐4M version, justify our approach. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1405–1418, 2000     

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     

Quality of correlating functions generated from commonly used basis sets

Tests have been performed on the quality of correlating functions generated from commonly used Gaussian basis sets, such as the 4-31G and MIDI-4 sets. The atoms tested were carbon, nitrogen, and oxygen. Self-consistent field and configuration interaction (CI) calculations were performed for the ground and lower excited states of neutral atoms as well as for positive and negative ions, using the original sets. Next, after adding (1) one d, and (2) two d and one f primitive Gaussian-type functions (GTFs) to the original sets, the CI calculations were repeated. In order to investigate the quality of the correlating orbitals generated from the GTF sets, parallel calculations to those for the GTF sets were carried out with an extended set of Slater-type functions. It was found that the excitation energies change in a stepwise manner as the basis sets changed from the original sets to the original set + 1d and the original set +2d1f. The improvements in excitation energies and ionization energies were almost independent of the original sets and were found to be strongly dependent on the augmented correlation functions. © 1996 by John Wiley & Sons, Inc.