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.     

Theoretical Study on Intermolecular Interactions and Thermodynamic Properties of Nitroamine Dimers

Ab initio self-consistent field(SCF) and Mφller-Plesset correlation correction methods employing 6-31G^** basis set have been applied to the optimizations of nitroamine dimers.The binding energies have been corrected for the basis set superposition error (BSSE) and the zero-point energy.Theree optimized dimers have been obtained.The BSSE corrected binding energy of the most stable dimer is predicted to be -31.85kJ/mol at the MP4/6-31G^**//MP2/6-31G^** level.The energy barriers of the Walden conversion for -NH2 group are 19.7kJ/mol and 18.3kJ/mol for monomer and the most stable dimer,respectively.The molecular interaction makes the internal rotation around N1-N2 even more difficult.The thermodynamic properties of nitroamine and its dimers at different temperatures have been calculated on the basis of vibrational analyses.The change of the Gibbs free energy for the aggregation from monomer to the most stable dimer at standard pressure and 298.2 K is predicted to be 14.05kJ/mol.     

Basis sets for molecular interactions. 1. Construction and tests on (HF)2 and (H2O)2

Basis set superposition error (BSSE) remains one of the major difficulties besetting current ab initio calculations of molecular interactions. Despite the widespread notion that lowering of the BSSE to negligible magnitude requires extremely large basis sets, we show that simple modifications of basis sets of only moderate size (e.g., 6-31G**) can accomplish the same end at much reduced computational expense. These modifications include reoptimization of the orbital exponents within the framework of the relevant molecules, plus addition of a single diffuse shell of sp orbitals on nonhydrogen centers. Subsequent addition of a second set of d-functions further lowers the SCF BSSE, bringing it below 0.1 kcal/mol for both (HF)₂ and (H₂O)₂. It is notable that addition of the latter d-functions without prior reoptimization of the valence orbitals produces the opposite effect of an increase in the BSSE. Although the MP2 BSSE is also substantially decreased by the above modifications, it appears difficult to reduce this quantity below about 0.4 kcal/mol.     

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.     

A quantum-chemical study of the LiH + LiH⋅− ⇄ LiH2− + Li⋅ reaction

Parts of the potential energy surface of the title process and related processes have been investigated at the SCF /6-31G **, SCF /6-31++G **, and MP 2/6-31++G ** levels. The investigated reaction is exothermic (?6.23 kcal/mol, MP 4/6-31++G **//MP 2/6–31++G** level, ZPE included): A linear intermediate radical anion, Li? H? Li? H??, is significantly stabilized with respect to LiH + LiH?? (?38.74 kcal/mol, the same level as above). The BSSE at MP 2/6–31++G **//MP 2/6–31++G ** amounts to 1.8 kcal/mol. The title process seems to be suitable for experimental study in molecular beams.     

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.     

Combined bond-polarization basis sets for accurate determination of dissociation energies. II. Basis set superposition error as a function of the parent basis set

The effect of the parent basis set on the basis set superposition error caused by bond functions is investigated systematically. An important difference between BSSE at the SCF and correlated levels is pointed out. Three new basis sets are defined, denoted 6-311 + G(d,p)B, 6-311 + G(2d,p)B, and 6-311 + G(2df,p)B. BSSE for the first-row hydrides seems to increase uniformly with increasing atomic number of the central atom. Expansion of the valence part of the basis set from 6-31G to 6-311G, as well as adding f functions, has a significant effect on the BSSE. Additional BSSEs incurred by bond functions are less than or equal to 1 kcal/mol for the 6-311 + G(2df,p)B basis set. For the dissociation energies of the first-row hydride species, agreement with experiment within only a few kcal/mol can be obtained even without resorting to isogyric reaction cycles. For high-quality calculations, adding bond functions seems to have definite advantages over expanding the polarization space beyond the [2d1f] level.     

Ab initio investigation of structure and cohesive energy of crystalline urea

The structure and cohesive energy of crystalline urea have been investigated at the ab initio level of calculation. The performance of different Hamiltonians in dealing with a hydrogen-bonded molecular crystal as crystalline urea is assessed. Detailed calculations carried out by adopting both HF and some of the most popular DFT methods in solid-state chemistry are reported. Local, gradient-corrected, and hybrid functionals have been adopted: SVWN, PW91, PBE, B3LYP, and PBE0. First, a 6-31G(d,p) basis set has been adopted, and then the basis set dependence of computed results has been investigated at the B3LYP level. All calculations were carried out by using a development version of the periodic ab initio code CRYSTAL06, which allows full optimization of lattice parameters and atomic coordinates. With the 6-31G(d,p) basis set, structural features are well reproduced by hybrid methods and GGA. LDA gives lattice parameters and hydrogen-bond distances that are too small relative to experiment, while at the HF level the opposite trend is observed. Results show that hybrid methods are more accurate than HF and both LDA and GGA functionals, with a trend in the computed properties similar to that of hydrogen-bonded molecular complexes. When BSSE and ZPE are taken into account, all methods, except LDA, give computed cohesive energies that are underestimated with respect to the experimental sublimation enthalpy. Dispersion energy, not properly taken into account by DFT methods, plays a crucial role. Such a deficiency also affects dramatically the computed crystalline structure, especially when large basis sets are adopted. We show that this is an artifact due to the BSSE. Indeed, with small basis sets the BSSE gives an extra-binding that compensates for the missing dispersion forces, thus yielding structures in fortuitous agreement with experiment.     

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.     

Counterpoise estimates of the BSSE in the evaluation of protonation energies

Counterpoise estimates of the BSSE in the evaluation of protonation energies have been calculated for basis sets ranging from minimal to split-valence plus polarization quality. Three-, five- and six-membered-ring heterocycles have been chosen as suitable model compounds for this study. Counterpoise corrections are significant, at the minimal basis set and 3–21G levels, when considering both, absolute and relative protonation energies and depend on the nature of the centre which undergoes protonation. In general, second- and third-order counterpoise corrections to the protonation energies are comparable to the corresponding SCF values. BSSE depend not only on the size of the basis sets but also on their quality. The presence in the basis of quite diffuse functions (either sp or d) leads to lower protonation energies and greater BSSE. Relative protonation energies are not substantially affected by BSSE or correlation effects.     

平面五元水分子簇的量子化学研究

选用Gaussian03的B3LYP/6-31G(d,p)、DMol3的BLYP/DNP和deMon的BLYP/TZVP等方法计算了甲烷水合物(结构-1)中平面五元水分子簇的结合能和氢键能,作了基组重叠误差(BSSE)和色散能(dispersion)的修正,估算了次级相互作用的贡献.在DMol3程序中使用了大型数值基组DNP,将基组重叠误差降至最低.在Gaussi-an03的B3LYP/6-31G(d,p)计算中,采用平衡法(Counterpoise)校正基组重叠误差.两种计算方法给出了一致的结果,证实了在使用6-31G(d,p)基组时,一对水分子在平衡距离的基组重叠误差高达8 kJ/mol.为估算色散能的贡献,使用了新近发展的包含色散能的密度泛函的DFT程序deMon计算了五元水分子簇.用多种方法计算出了经基组重叠误差和色散能修正的五元水分子簇的分子间结合能和氢键能的较为精确的势能超曲面,为甲烷和其他气体水合物的分子动力学模拟提供了依据.     

A BSSE-free SCF algorithm for intermolecular interactions. II. Sample calculations on hydrogen-bonded complexes

The usefulness and reliability of the recent BSSE -free SCF algorithm based on the “chemical Hamiltonian approach” (CHA /F ) is demonstrated by calculating potential curves for several hydrogen-bonded complexes with 4-31G , 6-31G , and 6-31G ** basis sets. It is concluded that the CHA /F scheme gives results that are numerically close to those of the Boys–Bernardi a posteriori correction scheme but are free from the “overcompensation” characteristic of the latter at smaller distances and given basis sets. © 1992 John Wiley & Sons, Inc.     

Ab initio base-pairing energies of an oxidized thymine product, 5-formyluracil, with standard DNA bases at the BSSE-free DFT and MP2 theory levels

Oxidation of the thymine methyl group produces two stable products, non-mutagenic 5-hydroxymethyluracil and highly mutagenic 5-formyluracil. We have calculated the interaction energy of base-pair formation involving 5-formyluracil bound to the natural DNA bases adenine (A), cytosine (C), guanine (G), and thymine (T), and discuss the effects of the 5-formyl group with respect to similar base-pairs containing uracil, 5-hydroxyuracil, thymine (5-methyluracil), and 5-hydroxycytosine. The interaction geometries and energies were calculated four ways: (a) using density functional theory (DFT) without basis set super-position error (BSSE) corrections, (b) using DFT with BSSE correction of geometries and energies, (c) using M?ller-Plesset second order perturbation theory (MP2) without BSSE correction, and (d) using MP2 with BSSE geometry and energy correction. All calculations used the 6-311G(d,p) basis set. Notably, we find that the A:5-formyluracil base-pair is more stable than the precursor A:T base-pair. The relative order of base-pair stabilities is A:5-Fo-U > G:5-Fo-U > C:5-Fo-U > T:5-Fo-U.     

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.     

Ghost orbitals in semiempirical methods. Estimation of basis set superposition error

Based on the all-valence ZDO SCF approximation a procedure for estimating the basis set superposition error (BSSE ) in semiempirical CNDO /INDO methods has been proposed. The results of the calculation show that the BSSE effect may improve the results obtained from the standard CNDO /INDO supermolecule calculation. The estimated BSSE effect enables one to explain some recently reported artificial structures for water and ethylene dimers.     

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.     

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     

Adsorption of polycyclic aromatic hydrocarbons and inversion barriers of curved conjugated systems inside the molecular cage ExCage6+

We investigated the hosting of planar and curved π systems by ExCage⁶⁺. The M06‐2X/6‐311G* method and including basis set superposition error (BSSE) corrections showed good agreement with the experimental free energy changes in solution. The mean absolute deviation (MAD) with respect to experiment was 1.3 kcal/mol. The M06‐2X/6‐31G* method exhibited a MAD of 2.1 kcal/mol, so it may be useful to investigate large systems. The good agreement between the M06‐2X/6‐311G*+BSSE results and the M06‐2X/6‐31G* ones was due to a fortuitous error cancelation between basis set incompleteness and BSSE. The interaction energies decreased linearly with the number of CC double bonds present in the PAH, but the shape of the PAH is an important factor in determining the binding strength. Finally, we studied how effective is ExCage⁶⁺ in reducing the inversion barriers of buckybowls. For corannulene, sumanene and dibenzo[a,g]corannulene they are reduced by 2.0, 2.7, and 2.0 kcal/mol, respectively. Although these values indicate an induced fit catalysis by ExCage⁶⁺, they are far from those values calculated inside bilayer graphene. Therefore, much work is necessary to replicate the reduction of inversion barriers observed for graphene.