Basis set superposition error in N-body clusters

We present a new scheme for calculating the basis set superposition error (BSSE) in N-body clusters. It is based on the assumption that each n-body term can be expressed as a sum of only two-body contributions. The conventional Boys–Bernardi method can be used thus for calculating BSSE-corrected energy terms. The scheme is illustrated by some calculations on the hydrogen fluoride trimers and tetramers. The results are compared to the ones obtained with the site–site function counterpoise (SSFC), the hierarchical (Valiron–Mayer) function counterpoise (VMFC), the pairwise additive function counterpoise (PAFC), and the successive reaction counterpoise (SRCP) schemes.     

On the basis set superposition error in supermolecular calculations of interaction-induced electric properties: many-body components

In the present paper we analyze basis set superposition error (BSSE) removal methods from many-body components of interaction-induced electric properties. The Valiron–Mayer function counterpoise (VMFC), site–site function counterpoise (SSFC) and TB methods have been employed in order to obtain the incremental optical components of linear hydrogen fluoride clusters (HF)n, where n = {3,4}. Following Mierzwicki and Latajka, who have performed similar calculations for the interaction energy, we compare those three methods of eliminating BSSE using several Dunning’s correlation consistent basis sets.     

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.     

Improved SCF interaction energy decomposition scheme corrected for basis set superposition effect

A modified scheme for SCF interaction energy decomposition has been proposed where the nonphysical basis set superposition error (BSSE ) has been corrected by means of the counterpoise method. A new procedure to separate the exchange and induction energy terms free of nonphysical BSSE has been tested in the case of the H₂O dimer. The first order BSSE appears to be non-negligible for strong hydrogen bonded complexes. In addition the scheme allows separation of the long-controversial charge-transfer contribution within the induction term, which has been considerably overestimated in previous studies.     

Ab initio and analytic intermolecular potentials for Ar-CF4

Ab initio calculations at the CCSD(T) level of theory were performed to characterize the Ar + CF4 intermolecular potential. Potential energy curves were calculated with the aug-cc-pVTZ basis set, and with and without a correction for basis set superposition error (BSSE). Additional calculations were performed with other correlation consistent basis sets to extrapolate the Ar-CF4 potential energy minimum to the complete basis set (CBS) limit. Both the size of the basis set and BSSE have substantial effects on the Ar + CF4 potential. Calculations with the aug-cc-pVTZ basis set, and with a BSSE correction, appear to give a good representation of the BSSE corrected potential at the CBS limit. In addition, MP2 theory is found to give potential energies in very good agreement with those determined by the much higher level CCSD(T) theory. Two model analytic potential energy functions were determined for Ar + CF4. One is a fit to the aug-cc-pVTZ calculations with a BSSE correction. The second was derived by fitting an average BSSE corrected potential, which is an average of the CCSD(T)/aug-cc-pVTZ potentials with and without a BSSE correction. These analytic functions are written as a sum of two-body potentials and excellent fits to the ab initio potentials are obtained by representing each two-body interaction as a Buckingham potential.     

Explicit correlation and basis set superposition error: the structure and energy of carbon dioxide dimer

We have investigated the slipped parallel and t-shaped structures of carbon dioxide dimer [(CO(2))(2)] using both conventional and explicitly correlated coupled cluster methods, inclusive and exclusive of counterpoise (CP) correction. We have determined the geometry of both structures with conventional coupled cluster singles doubles and perturbative triples theory [CCSD(T)] and explicitly correlated cluster singles doubles and perturbative triples theory [CCSD(T)-F12b] at the complete basis set (CBS) limits using custom optimization routines. Consistent with previous investigations, we find that the slipped parallel structure corresponds to the global minimum and is 1.09 kJ mol(-1) lower in energy. For a given cardinal number, the optimized geometries and interaction energies of (CO(2))(2) obtained with the explicitly correlated CCSD(T)-F12b method are closer to the CBS limit than the corresponding conventional CCSD(T) results. Furthermore, the magnitude of basis set superposition error (BSSE) in the CCSD(T)-F12b optimized geometries and interaction energies is appreciably smaller than the magnitude of BSSE in the conventional CCSD(T) results. We decompose the CCSD(T) and CCSD(T)-F12b interaction energies into the constituent HF or HF CABS, CCSD or CCSD-F12b, and (T) contributions. We find that the complementary auxiliary basis set (CABS) singles correction and the F12b approximation significantly reduce the magnitude of BSSE at the HF and CCSD levels of theory, respectively. For a given cardinal number, we find that non-CP corrected, unscaled triples CCSD(T)-F12b/VXZ-F12 interaction energies are in overall best agreement with the CBS limit.     

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.     

Is spin-component scaled second-order Møller-Plesset perturbation theory an appropriate method for the study of noncovalent interactions in molecules?

Testing of the spin-component scaled second-order M?ller-Plesset (SCS-MP2) method for the computation of noncovalent interaction energies is done with a database of 165 biologically relevant complexes. The effects of the spin-scaling procedure (i.e., MP2 vs SCS-MP2), the basis set size, and the corrections for basis set superposition error (BSSE) are systematically examined. When using two-point basis set extrapolations for the correlation energy, augmentation of the atomic orbital basis with computationally costly diffuse functions is found to be obsolete. In general, SCS-MP2 also improves results for noncovalent interactions statistically on MP2, and significant outliers are removed. Moreover, it is shown that effects of BSSE and one-particle basis set incompleteness almost cancel each other in the case of triple-zeta sets (SCS-MP2/TZVPP or SCS-MP2/cc-pVTZ without counterpoise correction), which opens a practical route to efficient computations for large systems. We recommend SCS-MP2 as the preferred quantum chemical wave function based method for the noncovalent interactions in large biologically relevant systems when reasonable coupled-cluster with single and double and perturbative triple excitations (CCSD(T)) calculations cannot be performed anymore. A comparison to MP2 and CCSD(T) interaction energies for n-alkane dimers, however, indicates (and this also holds to a lesser extent for hydrogen-bonded systems) limitations of SCS-MP2 when treating chemically "saturated" interactions. The different behavior of second-order perturbation theory for saturated and for stacked pi-systems is discussed.     

Counterpoise corrected interaction energies are not systematically better than uncorrected ones: comparison with CCSD(T) CBS extrapolated values

The effect of the inclusion of counterpoise corrections (CP) on the accuracy of interaction energies has been studied for different systems accounting for (1) intermolecular interactions, (2) intramolecular interactions and (3) chemical reactions. To minimize the error associated with the method of choice, the energy calculations were performed using CCSD(T) in all the cases. The values obtained using aug-cc-pVXZ basis sets are compared to CBS-extrapolated values. It has been concluded that at least for the tested systems CP corrections systematically leads to results that differ from the CBS-extrapolated ones to a larger extension than the uncorrected ones. Accordingly, from a practical point of view, we do not recommend the inclusion of such corrections in the calculation of interaction energies, except for CBS extrapolations. The best way of dealing with basis set superposition error (BSSE) is not to use CP corrections, but to make a computational effort for increasing the basis set. This approach does not eliminate BSSE but significantly decreases it, and more importantly it proportionally decreases all the errors arising from the basis set truncation.     

BSSE‐corrected geometry and harmonic and anharmonic vibrational frequencies of formamide–water and formamide–formamide dimers

The basis set superposition error (BSSE) influence in the geometry structure, interaction energies, and intermolecular harmonic and anharmonic vibrational frequencies of cyclic formamide–formamide and formamide–water dimers have been studied using different basis sets (6‐31G, 6‐31G**, 6‐31++G**, D95V, D95V**, and D95V++**). The a posteriori “counterpoise” (CP) correction scheme has been compared with the a priori “chemical Hamiltonian approach” (CHA) both at the Hartree–Fock (HF) and second‐order Møller–Plesset many‐body perturbation (MP2) levels of theory. The effect of BSSE on geometrical parameters, interaction energies, and intermolecular harmonic vibrational frequencies are discussed and compared with the existing experimental data. As expected, the BSSE‐free CP and CHA interaction energies usually show less deep minima than those obtained from the uncorrected methods at both the HF and MP2 levels. Focusing on the correlated level, the amount of BSSE in the intermolecular interaction energies is much larger than that at the HF level, and this effect is also conserved in the values of the force constants and harmonic vibrational frequencies. All these results clearly indicate the importance of the proper BSSE‐free correlation treatment with the well‐defined basis functions. At the same time, the results show a good agreement between the a priori CHA and a posteriori CP correction scheme; this agreement is remarkable in the case of large and well‐balanced basis sets. The anharmonic frequency correction values also show an important BSSE dependence, especially for hydrogen bond stretching and for low frequencies belonging to the intermolecular normal modes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

The Weak Interaction on CO2 …CO van der Waals Complex : A Theoretical Study

The geometries of van der Waals complex CO2…CO were optimized at DFT and second-order Moller-Plesset perturbation(MP2) levels with the large basis set,three stable structures were found.The most stable structure has a T-shape geometry in which the CO lies along the C2 axis of CO2,with the two C atoms direct contact and R(C…C)=0.3227nm.The corresponding energies of the most stable structure were calculated by means of MP2,MP4D,MP4DQ,MP4SDTQ,MP4SDQ,CCSD and CCSD(T) methods,The BSSE (basis set superposition error) wads eliminated by the Boys-Bernardi counterpoise correction(CP) method.According to thermodynamics data.van der Waals complex CO2…CO can found at a low temperature and or a high pressure,There is a little charge transferred between the two interacted subunits.In the most stable structure,CO2 is the acceptor and CO is the donor.     

Counterpoise correction is not useful for short and Van der Waals distances but may be useful at long range

This article investigates the errors in supermolecule calculations for the helium dimer. In a full CI calculation, there are two errors. One is the basis set superposition error (BSSE), the other is the basis set convergence error (BSCE). Both of the errors arise from the incompleteness of the basis set. These two errors make opposite contributions to the interaction energies. The BSCE is by far the largest error in the short range and larger than (but much closer to) BSSE around the Van der Waals minimum. Only at the long range, the BSSE becomes the larger error. The BSCE and BSSE largely cancel each other over the Van der Waals well. Accordingly, it may be recommended to not include the BSSE for the calculation of the potential energy curve from short distance till well beyond the Van der Waals minimum, but it may be recommended to include the BSSE correction if an accurate tail behavior is required. Only if the calculation has used a very large basis set, one can refrain from including the counterpoise correction in the full potential range. These results are based on full CI calculations with the aug-cc-pVXZ (X = D, T, Q, 5) basis sets.     

Theoretical analysis on the hydrogen bonding and reactivity that associated with the proton transfer reaction of carboxylic acid dimers and their monosulfur derivatives

Carboxylic acid dimers and their monosulfur derivatives are investigated by density functional theory calculations. Basis set superposition error (BSSE) counterpoise correction is included to compare the influence of BSSE on the interaction energies as well as on the geometries. The nature of hydrogen bond is determined on the basis of atoms in molecules (AIM) and natural bond orbital (NBO) analyses. Good correlations have been established between H‐bond length versus AIM topological parameter, orbital interaction, and barrier height for proton transfer. The reactivity behavior along the reaction path of the double proton transfer reaction has also been studied. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Ne-CO2的从头算势能面及微波光谱

采用三重激发校正的耦合簇[CCSD(T)]方法和大基组计算了范德华复合物Ne-CO2的分子间势能面. 分子间相互作用能的计算采用考虑了基组重叠误差修正的超分子方法. 计算结果表明, 该势能面有两个极小值点, 分别对应T形构型和线性Ne-OCO构型. 采用离散变量表象(DVR)方法及Lanczos算法计算了Ne-CO2的振转能级. 计算结果表明, 体系势能面支持22个振动束缚态. 计算得到的微波光谱的跃迁频率与实验值吻合得很好.     

Parameterization of a B3LYP specific correction for non-covalent interactions and basis set superposition error on a gigantic dataset of CCSD(T) quality non-covalent interaction energies

A vast number of non-covalent interaction energies at the counterpoise corrected CCSD(T) level have been collected from the literature to build a diverse new dataset. The whole dataset, which consists of 2027 CCSD(T) energies, includes most of the published data at this level. A large subset of the data was then used to train a novel, B3LYP specific, empirical correction scheme for non-covalent interactions and basis set superposition error (abbreviated as B3LYP-MM). Results obtained with our new correction scheme were directly compared to benchmark results obtained with B3LYP-D3(1) and M06-2X(2) (two popular density functions designed specifically to accurately model non-covalent interactions). For non-covalent complexes dominated by dispersion or dipole-dipole interactions all three tested methods give accurate results with the medium size aug-cc-pVDZ(3-6) basis set with MUE's of 0.27 (B3LYP-MM), 0.32 (B3LYP-D3) and 0.47 kcal/mol (M06-2X) (with explicit counterpoise corrections). These results validate both B3LYP-D3 and M06-2X for interactions of this type using a much larger data set than was presented in prior work. However, our new dispersion correction scheme shows some clear advantages for dispersion and dipole-dipole dominated complexes with the small LACVP* basis set, which is very popular in use due to its low associated computational cost: The MUE for B3LYP-MM with the LACVP* basis set for this subset of complexes (without explicit counterpoise corrections) is only 0.28 kcal/mol, compared to 0.65 kcal/mol for M06-2X or 1.16 kcal/mol for B3LYP-D3. Additionally, our new correction scheme also shows major improvements in accuracy for hydrogen-bonded systems and for systems involving ionic interactions, for example cation-π interactions. Compared to B3LYP-D3 and M06-2X, we also find that our new B3LYP-MM correction scheme gives results of higher or equal accuracy for a large dataset of conformer energies of di- and tripeptides, sugars, and cysteine.     

Hydrogen bonding in the urea dimers and adenine–thymine DNA base pair: anharmonic effects in the intermolecular H-bond and intramolecular H-stretching vibrations

The equilibrium structures, binding energies, vibrational harmonic frequencies, and the anharmonic corrections for two different (cyclic and asymmetric) urea dimers and for the adenine–thymine DNA base pair system have been studied using the second-order Møller–Plesset perturbation theory (MP2) method and different density functional theory (DFT) exchange–correlation (XC) functionals (BLYP, B3LYP, PBE, HCTH407, KMLYP, and BH and HLYP) with the D95V, D95V**, and D95V++** basis sets. The widely used a posteriori Boys–Bernardi or counterpoise correction scheme for basis set superposition error (BSSE) has been included in the calculations to take into account the BSSE effects during geometry optimization (on structure), on binding energies and on the different levels of approximation used for calculating the vibrational frequencies. The results obtained with the ab initio MP2 method are compared with those calculated with different DFT XC functionals; and finally the suitability of these DFT XC functionals to describe intermolecular hydrogen bonds as well as harmonic frequencies and the anharmonic corrections is assessed and discussed.     

A geometrical correction for the inter- and intra-molecular basis set superposition error in Hartree-Fock and density functional theory calculations for large systems

A semi-empirical counterpoise-type correction for basis set superposition error (BSSE) in molecular systems is presented. An atom pair-wise potential corrects for the inter- and intra-molecular BSSE in supermolecular Hartree-Fock (HF) or density functional theory (DFT) calculations. This geometrical counterpoise (gCP) denoted scheme depends only on the molecular geometry, i.e., no input from the electronic wave-function is required and hence is applicable to molecules with ten thousands of atoms. The four necessary parameters have been determined by a fit to standard Boys and Bernadi counterpoise corrections for Hobza's S66×8 set of non-covalently bound complexes (528 data points). The method's target are small basis sets (e.g., minimal, split-valence, 6-31G*), but reliable results are also obtained for larger triple-ζ sets. The intermolecular BSSE is calculated by gCP within a typical error of 10%-30% that proves sufficient in many practical applications. The approach is suggested as a quantitative correction in production work and can also be routinely applied to estimate the magnitude of the BSSE beforehand. The applicability for biomolecules as the primary target is tested for the crambin protein, where gCP removes intramolecular BSSE effectively and yields conformational energies comparable to def2-TZVP basis results. Good mutual agreement is also found with Jensen's ACP(4) scheme, estimating the intramolecular BSSE in the phenylalanine-glycine-phenylalanine tripeptide, for which also a relaxed rotational energy profile is presented. A variety of minimal and double-ζ basis sets combined with gCP and the dispersion corrections DFT-D3 and DFT-NL are successfully benchmarked on the S22 and S66 sets of non-covalent interactions. Outstanding performance with a mean absolute deviation (MAD) of 0.51 kcal/mol (0.38 kcal/mol after D3-refit) is obtained at the gCP-corrected HF-D3/(minimal basis) level for the S66 benchmark. The gCP-corrected B3LYP-D3/6-31G* model chemistry yields MAD=0.68 kcal/mol, which represents a huge improvement over plain B3LYP/6-31G* (MAD=2.3 kcal/mol). Application of gCP-corrected B97-D3 and HF-D3 on a set of large protein-ligand complexes prove the robustness of the method. Analytical gCP gradients make optimizations of large systems feasible with small basis sets, as demonstrated for the inter-ring distances of 9-helicene and most of the complexes in Hobza's S22 test set. The method is implemented in a freely available FORTRAN program obtainable from the author's website.     

Basis sets for molecular interactions. 2. Application to H3N?HF,H3N?HOH,H2O?HF, (NH3)2, and H3CH?OH2

Modifications of the standard 6-31G** basis set as recommended in the accompanying paper are found to markedly lower the basis set superposition error (BSSE) in the title complexes, in contrast to enlargement to a triple-ζ scheme or by addition of a diffuse sp shell or a second set of d-functions without prior optimization, all of which lead to BSSE increase. After appropriate correction for correlation and superposition effects, all basis sets (with the exception of the standard 6-31G** and 6-311G** with their very large BSSE) predict the cyclic geometry of NH₃ dimer to be more stable than the linear arrangement. Correlation and BSSE can shift the equilibrium intermolecular distance in H₃CH-OH₂ by up to 0.4 Å. Failure to correct for superposition error leads to a drastic exaggeration of both the SCF and MP2 components of the interaction energy in this complex. Much better estimates are furnished by our recommended basis sets with their smaller superposition errors.     

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.