On the counterpoise correction for the basis set superposition error in large systems

The appropriateness of the use of the counterpoise correction for the basis set superposition error in SCF calculations of the interaction energies for pairs of aliphatic amino acids is analyzed in this paper. Our results show that for this type of molecule where the magnitude of the basis set superposition error can become quite big, the use of the counterpoise method provides interaction energies in good agreement with near Hartree-Fock values. The inaccuracies associated with the counterpoise method are much less important compared with the basis set superposition error itself. It is shown that the use of a well-balanced minimal basis set together with the counterpoise method is a good compromise (quality versus computational cost) for calculating interaction energies in systems involving molecules of biological interest.     

On the validity of the counterpoise correction for the basis set superposition error including the fragment relaxation terms

It is demonstrated that relatively large geometrical changes make Emsley et al.'s assumption (J Am Chem Soc (1978) 100:3303) on the counterpoise correction for the basis set superposition error including the fragment relaxation terms unacceptable. Received: 23 September 1997 / Accepted: 31 October 1997     

Efficiency of numerical basis sets for predicting the binding energies of hydrogen bonded complexes: evidence of small basis set superposition error compared to Gaussian basis sets

Binding energies of selected hydrogen bonded complexes have been calculated within the framework of density functional theory (DFT) method to discuss the efficiency of numerical basis sets implemented in the DFT code DMol3 in comparison with Gaussian basis sets. The corrections of basis set superposition error (BSSE) are evaluated by means of counterpoise method. Two kinds of different numerical basis sets in size are examined; the size of the one is comparable to Gaussian double zeta plus polarization function basis set (DNP), and that of the other is comparable to triple zeta plus double polarization functions basis set (TNDP). We have confirmed that the magnitudes of BSSE in these numerical basis sets are comparative to or smaller than those in Gaussian basis sets whose sizes are much larger than the corresponding numerical basis sets; the BSSE corrections in DNP are less than those in the Gaussian 6-311+G(3df,2pd) basis set, and those in TNDP are comparable to those in the substantially large scale Gaussian basis set aug-cc-pVTZ. The differences in counterpoise corrected binding energies between calculated using DNP and calculated using aug-cc-pVTZ are less than 9 kJ/mol for all of the complexes studied in the present work. The present results have shown that the cost effectiveness in the numerical basis sets in DMol3 is superior to that in Gaussian basis sets in terms of accuracy per computational cost.     

Perturbation expansion theory corrected from basis set superposition error II. Charge transfer,pair correlationand dispersion terms

The second-order perturbation theory based on the locally projected molecular orbitals is developed. A few test calculations with cc-pVDZ and aug-cc-pVDZ basis sets are carried out for the dimers, (H₂O)₂ and (HF)₂. The charge transfer terms remove the deficiency of the locally projected self-consistent field method for molecular interaction (LP SCF MO MI), and the potential energy curves calculated with aug-cc-pVDZ are very close to the corresponding curves of the counterpoise-corrected SCF energy. Only after adding the spin-exchanged dispersion type to the dispersion and intra-molecular pair correlation terms, the calculated potential energy curves become close to those of the counterpoise-corrected second-order Møller–Plesset (MP2). Pragmatic approaches for reducing the influence of the basis set superposition error are proposed.     

The effect of basis set superposition error on the convergence of interaction energies

 For the intermolecular interaction energies of ion-water clusters [OH⁻(H₂O) _n (n=1,2), F⁻(H₂O), Cl⁻(H₂O), H₃O⁺(H₂O) _n (n=1,2), and NH₄ ⁺(H₂O) _n (n=1,2)] calculated with correlation-consistent basis sets at MP2, MP4, QCISD(T), and CCSD(T) levels, the basis set superposition error is nearly zero in the complete basis set (CBS) limit. That is, the counterpoise-uncorrected intermolecular interaction energies are nearly equal to the counterpoise-corrected intermolecular interaction energies in the CBS limit. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are more reliable than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies evaluated using the MP2/aug-cc-pVDZ level is reliable. Received: 14 March 2001 / Accepted: 25 April 2001 / Published online: 9 August 2001     

Basis set superposition effects on properties of interacting systems. Dipole moments and polarizabilities

Basis set superposition effects which are not removed by the counterpoise correction are shown to modify the electric properties of interacting subsystems and influence indirectly the calculated interaction energies. The role of these higher-order basis set superposition effects is illustrated by the calculation of the water molecule dipole moment and polarizability in the basis set of the water dimer.     

Comparison of basis set superposition error corrected perturbation theories for calculating intermolecular interaction energies

Recently, two different but conceptually similar basis set superposition error (BSSE) free second‐order perturbation theoretical schemes were developed by us that are being based on the chemical Hamiltonian approach (CHA). Using these CHA‐MP2 and CHA‐PT2 methods, a comparison is made between the a posteriori and a priori BSSE correction schemes at the correlated level. Sample calculations are presented for four hydrogen bonded complexes (HFH₃N, HFH₂O, H₂SHF, and H₂OHCl) in nine different basis sets (from 6–31G to TZV**++). The results show that the BSSE content is very significant in the interaction energy if electron correlation is accounted for, so removing the BSSE is very important. The differences of the two perturbational theories discussed are connected solely with the different one electron orbital sets used for building up the unperturbed single determinant wave function. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 274–283, 1999     

A comparison of the behavior of functional/basis set combinations for hydrogen-bonding in the water dimer with emphasis on basis set superposition error

We evaluate the performance of ten functionals (B3LYP, M05, M05-2X, M06, M06-2X, B2PLYP, B2PLYPD, X3LYP, B97D, and MPWB1K) in combination with 16 basis sets ranging in complexity from 6-31G(d) to aug-cc-pV5Z for the calculation of the H-bonded water dimer with the goal of defining which combinations of functionals and basis sets provide a combination of economy and accuracy for H-bonded systems. We have compared the results to the best non-density functional theory (non-DFT) molecular orbital (MO) calculations and to experimental results. Several of the smaller basis sets lead to qualitatively incorrect geometries when optimized on a normal potential energy surface (PES). This problem disappears when the optimization is performed on a counterpoise (CP) corrected PES. The calculated interaction energies (ΔEs) with the largest basis sets vary from -4.42 (B97D) to -5.19 (B2PLYPD) kcal/mol for the different functionals. Small basis sets generally predict stronger interactions than the large ones. We found that, because of error compensation, the smaller basis sets gave the best results (in comparison to experimental and high-level non-DFT MO calculations) when combined with a functional that predicts a weak interaction with the largest basis set. As many applications are complex systems and require economical calculations, we suggest the following functional/basis set combinations in order of increasing complexity and cost: (1) D95(d,p) with B3LYP, B97D, M06, or MPWB1k; (2) 6-311G(d,p) with B3LYP; (3) D95++(d,p) with B3LYP, B97D, or MPWB1K; (4) 6-311++G(d,p) with B3LYP or B97D; and (5) aug-cc-pVDZ with M05-2X, M06-2X, or X3LYP.     

Combined bond-polarization basis sets for accurate determination of dissociation energies

Summary The effect of bond functions on the basis set superposition error (BSSE) is investigated at both SCF (self consistent field) and correlated levels for a number of basis sets using the pairwise additive function counterpoise (PAFC), the site-site function counterpoise (SSFC), and the newly proposed successive reaction counterpoise method (SRCP). BSSEs using bond functions are shown to be roughly twice those without bond functions, whereas the latter may still be quite sizeable. The addition of f functions dramatically decreases the bond function BSSE. The results obtained support the empirical decision in our earlier papers to neglect BSSE altogether.     

On the influence of the basis set superposition error on calculated vibrational frequencies

The relation between the so called basis set superposition error and intramolecular vibrational frequencies calculated at the Hartree Fock SCF level of approximation was investigated. A linear conformation of HF dimer was chosen as test system for the investigation. It was found that the direct basis set superposition error for the studied system is rather small. It was further found that the shifts are mainly determined by the geometry parameters of the system. AcknowledgementsJ. M. H.-R. wishes to thank the Ministerio de Educación, Cultura y Deporte for the award of a research grant.     

HCN(HNC)与NH3, H2O和HF分子间相互作用的理论研究

在MP2/aug-cc-pVTZ水平上, 对HCN(HNC)与NH3, H2O和HF分子间可能存在的氢键型复合物进行了全自由度能量梯度优化, 通过在相同水平上的频率验证分析发现了稳定的分子间相互作用形式是HCN(HNC)作为质子供体或作为质子受体形成的复合物. 基组重叠误差对总相互作用能的影响均小于3.34 kJ/mol. 通过自然键轨道(NBO)分析, 研究了单体和复合物中的原子电荷和电荷转移对分子间相互作用的影响. 对称性匹配微扰理论(SAPT, Symmetry Adapted Perturbation Theory)能量分解结果表明, 在分子间相互作用中, 静电作用与诱导作用占主导地位, 而诱导作用与复合物的电荷转移之间具有良好的正相关性.     

Modification of Guest and Saunders open shell SCF equations to exclude BSSE from molecular interaction calculations

The self-consistent field (SCF) for molecularinteractions algorithm, particularly devised to compute intermolecular interactions, is extended to the case in which one of the two interacting fragments is an open shell system. The method excludes the basis set superposition error in an a priori fashion. To preserve the simplicity of the standard SCF procedure, Guest and Saunders equations concerning the open shell fragment are modified at the cost of a negligible complication with respect to the usual algorithm. Received: 27 May 1997 / Accepted: 29 May 1998 / Published online: 18 September 1998     

A systematic ab initio study of the water dimer in hierarchies of basis sets and correlation models

A systematic, high-level ab initio investigation of the water dimer has been performed. The oxygen-oxygen bond distance has been estimated to be around 2.90 ?, about 0.05 ? shorter than the experimentally estimated distance, challenging the accuracy of the latter. The interaction energy has been obtained at −5.0±0.1 kcal/mol, which compares favourably with the experimentally estimated value of −5.4±0.7 kcal/mol. The importance of employing basis sets that include diffuse functions in correlated calculations on hydrogen-bonded systems is confirmed. In correlated calculations on the water dimer and the hydrogen fluoride dimer, the counterpoise-corrected interaction energies converge considerably slower towards the basis set limit than do the uncorrected energies, provided that the correlation-consistent basis sets are augmented with diffuse functions. Received: 12 February 1997 / Accepted: 5 June 1997     

Dressed Second-order Epstein–Nesbet Perturbation Theory and Consequences of Orbital Delocalization for the BSSE Correction in Dimer Systems (in Honor of J.-P. Malrieu)

The dressed diagonal approximation to the self-consistent – size consistent CI, corrected for off-diagonal Fock matrix elements in localized orbitals is developed and applied to the ammoniac dimer system. A quite correct correlation energy can be obtained for this system, with a significantly reduced dependence of the results on the choice of the localization procedure. When calculating an interaction energy, the choice of the monomer orbitals and the application of the Boys–Bernardi counterpoise procedure shows in this case an unusual behavior: the correlation energy does not increase with the size of the atomic basis sets. Nevertheless a reasonable potential curve can be obtained.     

用GGA密度泛函及其长程、色散校正方法计算各类氢键的结合能

应用广义梯度近似(GGA) (PW91和PBE)、含动能密度的广义梯度近似(meta-GGA) (M06-L)、杂化泛函(hyper-GGA)(M06-2X、X3LYP和B3LYP)及其长程校正泛函LC-DFT(CAM-B3LYP、LC-ωPBE和ωB97X)和色散校正密度泛函(DFT-D)(ωB97X-D和B97-D),用多种基函数对15种不同强度的传统氢键和非传统氢键体系的结合能进行了系统的计算与分析.并与高精度的CCSD(T)/aug-cc-pVQZ结果比较发现:在上述各类泛函中,对于氢键结合能的计算M06-2X和ωB97X-D泛函较为精确与可靠,且没有必要使用过大的基函数,6-311++G(2d,2p)或aug-cc-pVDZ水平的基组就已足够,各类泛函所计算结合能的基组重叠误差(BSSE)均较小,除ωB97X和ωB97X-D外,其它9种泛函不经BSSE校正也能得到同样甚至更准确的结果.     

水和甲醇混合溶剂氢键相互作用及对高分子链溶解能力的理论研究

用密度泛函理论对水和甲醇混合溶剂体系的氢键结构进行了详细研究.通过构象和频率分析发现在水团簇中五聚体和六聚体环状结构最为稳定,同时发现一个全新的特征,即甲醇分子能与水五聚体和六聚体形成双氢键.根据各相互作用的稳定化能,分析了水和甲醇混合溶剂对PNIPAM溶解能力的影响,并对实验现象给予了合理解释.     

Evaluation of the intramolecular basis set superposition error in the calculations of larger molecules: [n]helicenes and Phe-Gly-Phe tripeptide

Correlated ab initio calculations on large systems, such as the popular MP2 (or RI-MP2) method, suffer from the intramolecular basis set superposition error (BSSE). This error is typically manifested in molecules with folded structures, characterized by intramolecular dispersion interactions. It can dramatically affect the energy differences between various conformers as well as intramolecular stabilities, and it can even impair the accuracy of the predictions of the equilibrium molecular structures. In this study, we will present two extreme cases of intramolecular BSSE, the internal stability of [n]helicene molecules and the relative energies of various conformers of phenylalanyl-glycyl-phenylalanine tripeptide (Phe-Gly-Phe), and compare the calculated data with benchmark values (experimental or high-level theoretical data). As a practical and cheap solution to the accurate treatment of the systems with large anticipated value of intramolecular BSSE, the recently developed density functional method augmented with an empirical dispersion term (DFT-D) is proposed and shown to provide very good results in both of the above described representative cases.     

Hydrogen bonds: a comparison of semiempirical and ab initio treatments

H-bonding interactions calculated using the AM1, PM3 and SAM1 semiempirical molecular orbital methods are compared with the best available ab initio calculations for several intermolecular interactions of interest: acetic acid dimers, water/ acetylene, water/HCN, formaldehyde/acetylene, formaldehyde/HCN, ozone/acetylene, ozone/HCN, acetylacetone, melamine/ cyanuric acid, and nitromethane/ammonia. Experimental values are also presented where available. The energetic comparisons are based upon enthalpies of interaction from the ab initio calculations after counterpoise and vibrational corrections have been applied. Overall, AM1 seems to do best, except for O---H…O interactions, where none of the three methods excel.     

Ab initio investigation on stability and properties of XYCO... HZ complexes. II: Post hartree-fock studies on H2CO... HF

Ab initio MP2 level of theory in conjunction with three basis sets of a triple-zeta quality was applied to study the molecular geometry and stability of the H₂CO... HF complex. An interaction energy predicted for this system at the highest, MP4(SDTQ)/6-311 + +G(2df, 2pd)//MP2/6-311 + +G(2df, 2pd), level corrected for the BSSE and ZPE contributions amounts to -4.85 kcal/ mol. BSSE contributes significantly to the interaction energies at all applied levels. Reliable MP2/ 6-311 + +G(2df, 2pd) level harmonic vibrational frequencies, IR intensities, and the predicted isotopic shifts upon deuteration and¹⁸O substitution are presented in order to facilitate experimental studies on the IR spectrum of the title complex.