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.     

Comparison of a posteriori and a priori BSSE correction schemes for SCF intermolecular energies

A systematic SCF study has been undertaken to compare the conventional a posteriori Boys–Bernardi BSSE correction scheme with our recent CHA/F method in which BSSE is excluded in a priori manner. Potential curves have been obtained for nine simple hydrogen-bonded systems by using nine different basis sets for each. It is concluded that the difference between the a posteriori BB and the a priori CHA schemes diminishes much faster when the basis set improves than BSSE disappears from the uncorrected SCF results. This fact gives an additional confidence in the CHA results, permitting one to draw the explicit conclusion that, at the SCF level of theory, the a priori CHA/F scheme can be considered the ultimate solution of the BSSE problem for weakly bonded systems. © 1993 John Wiley & Sons, Inc.     

On basis set superposition error corrected stabilization energies for large n-body clusters

In this contribution, we propose an approximate basis set superposition error (BSSE) correction scheme for the site-site function counterpoise and for the Valiron-Mayer function counterpoise correction of second order to account for the basis set superposition error in clusters with a large number of subunits. The accuracy of the proposed scheme has been investigated for a water cluster series at the CCSD(T), CCSD, MP2, and self-consistent field levels of theory using Dunning's correlation consistent basis sets. The BSSE corrected stabilization energies for a series of water clusters are presented. A study regarding the possible savings with respect to computational resources has been carried out as well as a monitoring of the basis set dependence of the approximate BSSE corrections.     

A new approach to theab initio energy of the homodesmic reaction for the resonance energy of benzene

Summary A scheme of the basis set superposition error (BSSE) correction is first proposed and introduced to determine theab initio energy of the homodesmic reaction for the resonance energy of benzene. Calculations with 6-31G*(5D) and 6-31G*(6D) basis sets at the complete fourth-order Møller-Plesset perturbation level furnish the energy value of 21.35 kcal/mol after the correction, which is in complete agreement with the experimental value of 21.3±0.2 kcal/mol. The energy values at the lower theoretical levels are generally underestimated but they are superior to the uncorrected values. The inclusion of triple excitations displays the dominant contribution of the correlation energy. Detailed analysis of the results reveals some of the similarities between the homodesmic reaction of benzene and the interaction of van der Waals molecule, which provides further justification of the BSSE correction scheme presented in this study.     

Procedure supplementing SCF interaction energies by dispersion term evaluated in dimer basis set within variation-perturbation approach

A variation-perturbation procedure for the evaluation of dispersion interaction, originally proposed by Jeziorski and van Hemert, has been reformulated to include basis set extension effects on an equal footing with the SCF interaction energy, corrected for basis set superposition error (BSSE). This approach has been tested for He₂, (H₂)₂, (H₂O)₂, and (C₂H₄)₂ complexes.The research was supported by ONR under contract N00014-80-C-0003 and in part by PAN within MR-I-9 project.     

A scheme for rapid prediction of cooperativity in hydrogen bond chains of formamides,acetamides, and N‐methylformamides

A scheme is proposed in this article to predict the cooperativity in hydrogen bond chains of formamides, acetamides, and N‐methylformamides. The parameters needed in the scheme are derived from fitting to the hydrogen bonding energies of MP2/6‐31+G** with basis set superposition error (BSSE) correction of the hydrogen bond chains of formamides containing from two to eight monomeric units. The scheme is then used to calculate the individual hydrogen bonding energies in the chains of formamides containing 9 and 12 monomeric units, in the chains of acetamides containing from two to seven monomeric units, in the chains of N‐methylformamides containing from two to seven monomeric units. The calculation results show that the cooperativity predicted by the scheme proposed in this paper is in good agreement with those obtained from MP2/6‐31+G** calculations by including the BSSE correction, demonstrating that the scheme proposed in this article is reasonable. Based on our scheme, a cooperativity effect of almost 240% of the dimer hydrogen bonding energy in long hydrogen bond formamide chains, a cooperativity effect of almost 190% of the dimer hydrogen bonding energy in long hydrogen bond acetamide chains, and a cooperativity effect of almost 210% of the dimer hydrogen bonding energy in long hydrogen bond N‐methylformamide chains are predicted. The scheme is further applied to some heterogeneous chains containing formamide, acetamide, and N‐methylformamide. The individual hydrogen bonding energies in these heterogeneous chains predicted by our scheme are also in good agreement with those obtained from Møller‐Plesset calculations including BSSE correction. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Implementation of gradient-optimization algorithms and force constant computations in BSSE-free direct and conventional SCF approaches

A modification of the Roothaan equations was described in a previous work, which aimed to avoid the BSSE at the Hartree–Fock level of theory. The resulting scheme was called the self-consistent field for molecular interactions (SCF-MI) to underline its special usefulness in the computation of intermolecular interactions. The method provides a complete a priori elimination of the BSSE, while taking into account the natural nonorthogonality of the MOs of the two interacting fragments. Compatibility with the usual formulation of the analytic derivatives of the SCF energy is also guaranteed. This allowed the implementation of gradient-optimization algorithms and force constant matrix computations in both the direct and conventional SCF approaches. The SCF-MI method has been incorporated into the GAMESS-US package. Tests have been performed at the Department of Chemistry of the Iowa State University. Increases in the complication and computation time are minimal if compared to standard SCF codes and the method shows much less basis-set dependence in the predicted molecular properties. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 69: 151–158, 1998     

Improved intermolecular SCF theory and the BSSE problem

Analytical and numerical studies are performed concerning the exclusion of the basis set superposition error (BSSE ) from the SCF calculations of intermolecular interactions. Based on these studies a new procedure is proposed, which consists of the following steps: (1) determine the orbitals by the SCF scheme based on the recent “chemical Hamiltonian approach” (CHA-SCF method), i.e., excluding the delocalization effects caused by BSSE , and then (2) calculate the usual energy expectation value. (This gives results superior to those obtained by the previous nonsymmetric CHA energy formula.) The actual numerical calculations performed for different simple systems (He₂, water dimer) by using various basis sets indicate that the CHA/CE (CHA with “conventional energy” formula) potential curves are well-balanced and are close to those obtained by the Boys–Bernardi (BB ) method and usually (but not necessarily) go slightly beyond the latter. So our method gives results better than (or close to) those given by the BB method by performing only a single ～N⁴ calculation at each geometrical arrangement of the system.     

Theoretical study on the (O2-HF) complex

The potential energy surface of the (O₂-HF)⁺ complex has been investigated theoretically. Equilibrium structure has been calculated at the UCCSD/6-311++G(2d,2p) and UCCSD/aug-cc-pVTZ levels of theory. Two stable complexes, T-shape and collinear, have been found on the surface whose interaction energies are calculated to be −12.36 and −10.19 kcal mol⁻¹, respectively, at the UCCSD(T)/aug-cc-pVTZ level after correcting for the basis set superposition error (BSSE). Natural bond orbital (NBO) analysis revealed intermolecular charge transfers occur followed by intramolecular charge rearrangement. A large contribution from the n_F to π* _OO is found.     

F2 dimer: Improved intermolecular potential energy surface using ab initio calculations

A computational study on the intermolecular potential energy of 44 different orientations of F₂ dimers is presented. Basis set superposition error (BSSE) corrected potential energy surface is calculated using the supermolecular approach at CCSD(T) and QCISD(T) levels of theory. The interaction energies obtained using the aug‐cc‐pVDZ and aug‐cc‐pVTZ basis sets are extrapolated to the complete basis set limit using the latest extrapolation scheme. The basis set effect is checked and it is found that the extrapolated intermolecular energies provide the best compromise between the accuracy and computational cost. Among 1320 energy points of F₂–F₂ system covering more relative orientations, the most stable structure of the dimers was obtained with a well depth of ?146.62 cm^?1 that related to cross configuration, and the most unstable structure is related to linear orientation with a well depth of ?52.63 cm^?1. The calculated second virial coefficients are in good agreement with experimental data. The latest extrapolation scheme of the complete basis set limit at the CCSD(T) level of theory is used to determine the intermolecular potential energy surface of the F₂ dimer. Comparing the results obtained by the latest scheme with those by older schemes show that the new approach provides the best compromise between accuracy and computational cost.     

Ab initio base-pairing energies of uracil and 5-hydroxyuracil with standard DNA bases at the BSSE-free DFT and MP2 theory levels

Oxidized cytosine product 5-hydroxyuracil has been shown to be the major chemical precursor for the GC to AT transition, the most frequent substitution mutation observed in aerobic organisms. We have calculated the interaction energy of base-pair formation involving uracil or 5-hydroxyuracil, which is formed in cells by oxidative deamination of cytosine, bound to any of the natural DNA bases, A, C, G, and T, and discuss the effects of the hydroxyl group in this respect. The base-pair geometries and energies were calculated using the 6-311G(dp) basis set under four conditions: using density functional theory (DFT) without out basis set super-position error (BSSE) correction, using DFT with BSSE correction of geometries and energies, using M?ller-Plesset second order perturbation theory (MP2) without BSSE correction, and using MP2 with BSSE geometry and energy correction. We find that the hydroxyl group of 5-HO-U (relative to U) has little effect on the base-pairs with A, C or one conformation of T, while making a substantial energy difference in base-pairs involving G or a different conformation of T. For most of the complexes studied, the BSSE-corrected energies at the DFT and MP2 levels of theory agreed to within 0.5 kcal.     

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     

On the effect of the BSSE on intermolecular potential energy surfaces. Comparison of a priori and a posteriori BSSE correction schemes

A comparative study of geometrical parameters is performed on the complexes HF–HF, H₂O–H₂O, and HF–H₂O using 12 different basis sets at the RHF, MP2, and DFT (BLYP and B3LYP) levels of theory. The equilibrium geometries were obtained from uncorrected, a posteriori (counterpoise, CP) and a priori (Chemical Hamiltonian Approach, CHA) BSSE‐corrected potential energy surfaces. The calculation of equilibrium geometries using the CP and CHA schemes is described in details. The effect of the BSSE on various intermolecular parameters is discussed and the performance of the applied theoretical models is critically evaluated from the BSSE point of view. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 765–786, 2001     

A new approach to counterpoise correction to BSSE

In the present work the intermolecular BSSE, associated to the A-B interaction, is obtained by subtracting the intramolecular BSSE of the fragments from the intramolecular BSSE of the supermolecule, and considering every atom as a fragment in the calculation of each intramolecular BSSE. This atom by atom scheme (CP(aa)) is based on the consideration that the proximity of the fragments may affect the intramolecular BSSE of every involved species, and artificially influences the value of the BSSE associated to the supermolecule formation. It drastically decreases the reported counterpoise overcorrection of the A-B interaction, even though it does not deal with all the overcorrection because it includes all the orbitals, and not only the unoccupied ones. This new approach has been tested on the water dimer, some hydrogen fluoride weakly bonded complexes, the conformational analysis of 1,2-dichloroethane, and the reaction profile of formaldehyde + OH reaction.     

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     

Blue-shifting hydrogen bond in the benzene-benzene and benzene-naphthalene complexes

Ab initio complete optimizations at MP2/6-31++G** level have been performed in the T-shaped geometry of the benzene-benzene and benzene-naphthalene complexes. To check the effect of the basis set superposition error (BSSE), optimizations have been done in the BSSE corrected and BSSE uncorrected potential energy surfaces. The BSSE effect in the calculation of the Hessian has also been evaluated to check its influence in the frequency values. Quantum theory atoms in molecules (QTAIM) calculations have also been performed on both dimers. Intermolecular energies differ around a 25% when the optimization is performed with or without counterpoise corrected gradients. The influence of BSSE is also noticeable in the distances. Frequency shifts show big changes because of the BSSE. Thus, uncorrected values are up 350% larger than corrected ones. The hypotheses given in the literature to explain the origin of the blue-shifting hydrogen bond do not seem to give a suitable explanation for all characteristics of the behavior found in the studied systems.     

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.     

Calculation of the interaction energy in a localized representation for a trimer (Ne3) system

We studied the transferability of the localized orbitals (LOs) of interacting Ne atoms using several basis sets. Both at SCF and at MP2 and MP3 levels, the contributions of the LOs have been calculated and discussed for the Ne₂ and Ne₃ systems. It was shown that for the LOs the transferability is satisfied to a good extent and due to the transferability the interaction energy at the correlated level can be calculated by using only the LOs of the supermolecule. The basis set superposition error (BSSE) is simply extracted from the intramolecular parts of the correlation energy. The two- and three-body interaction energies have been investigated for the studied systems. © John Wiley & Sons, Inc.     

Ab initio study of substituent effect on the addition of hydrogen fluoride to fluoroethylenes

An ab initio 3-21G study of the direct addition of HF to C₂H_nF_(4–n), with n = 0 to 4, has been performed to investigate the effect of the substituent on the reaction. Geometry optimization of all charge-transfer complexes and transition states has been done. Standard analysis of activation energies of addition reactions, vibrational and thermodynamical analysis, as well as Morokuma energy decomposition, BSSE correction, PMO analysis, and Pauling bond orders were used to explain the results. A subset of the reactions, including that of C₂H₄ as reference one and the two most favorable cases, was also studied at the MP2/6–31G(d,p)//HF/6–31G(d,p) level. The barriers so obtained are in agreement with the indirectly found from experimental data. It was found that the effect of the substituent is not monotonic for the additions. Decomposition of the interaction energy is shown to be adequate to explain this nonmonotonic behavior. The implications for laser chemistry of the addition of hydrogen halides to fluorosubstituted olefins is briefly discussed.     

Accurate ab initio determination of the van der Waals interaction in the X 2Σ+ ground state of LiHe

Accurate quantum-chemical ab initio calculations have been performed at the SCF and CEPA (coupled electron pair approximation) levels for the van der Waals interaction in the X ² Σ ⁺ ground state of LiHe. An extended basis set has been used and the counterpoise correction for the basis set superposition error (BSSE) has been applied. The calculated potential energy curve has a very shallow minimum at 11.56 a ₀ with a well depth of only 1.49 cm^?1. This is too small to allow for a bound vibrational level. The analysis of the results shows that the interaction mainly consists of the Pauli repulsion between Li(1s ²2s) and He (1s ²), which is decaying exponentially, and the attractive London dispersion energy. Van der Waals coefficients C₆, C₈, and C₁₀ have been determined by a least squares fit to the long-range part of the calculated potential curve.