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.     

The supermolecule method,as applied to studies of liquid-phase reaction mechanisms taking cyclocarbonate aminolysis in dioxane as an example: specific features

The supermolecule method was used to describe the mechanism of liquid-phase processes taking the reaction of ethylene carbonate with methylamine as an example. Specific features of the approach are considered. The problem of choosing the reference point for calculating the relative energies of individual reaction steps was solved by introducing the idea of the structure of noninteracting solvated reactants. In this case, no basis set superposition error (BSSE) correction is required because the solvated reactants, the pre-reaction complex, and the transition state have the same atomic composition and are calculated in the same basis set. To calculate the title reaction in dioxane by the supermolecule method with acceptable accuracy, it is sufficient to consider one solvent molecule.     

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.     

On the non-additivity of the basis set superposition error and how to prevent its appearance

An analytical consideration is made for the simplest possible model in which the BSSE problem may appear. The results demonstrate that BSSE cannot be corrected in any consistent manner by readjusting the monomer energies to the enlarged basis, because the energy effects caused by BSSE and by the true interactions are not additive. The way out is to correct BSSE, or prevent its appearance by an appropriate analysis and special treatment at the supermolecule level, permitting to keep the supermolecule problem consistent with the monomer calculations, as provided by the chemical Hamiltonian approach recently introduced.     

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.     

Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations

The Roothaan equations have been modified to compute molecular interactions between weakly bonded systems at the SCF level of theory without the basis set superposition error (BSSE). The increase in complication with respect to the usual SCF algorithm is negligible. Calculation of the SCF energy on large systems, such as nucleic acid pairs, does not pose any computational problem. At the same time, it is shown that a modest change in basis-set quality from 3-21G to 6-31G changes the binding energy by about 50% when computed according to standard SCF “supermolecule” techniques, while remaining practically constant when computed without introducing BSSE. Bader analysis shows that the amount of charge transferred between the interacting units is of the same order of magnitude when performed on standard SCF wave functions and those computed using the new method. The large difference between the corresponding computed energies is thus ascribed to the BSSE. © 1996 John Wiley & Sons, Inc.     

Localized orbitals for the description of molecular interaction

The intermolecular interaction between the molecules CH₂O and NH₃ was investigated by the supermolecule method. The interaction energies were first calculated at the ab initio SCF level, and the electron correlation was included via second-order Møller-Plesset perturbation theory (MP 2). The basis set superposition error (BSSE ) was taken into account by the counter-poise (CP ) method. The occupied and the virtual canonical molecular orbitals (CMOS ) of the supermolecule were separately localized by the Boys' procedure. The correlation correction was calculated by the many-body perturbation theory (MBPT ) in the localized representation. Contributions of the third- and fourth-order localized diagrams were added to those of the second-order canonical diagram. This procedure gives a correction nearly equivalent to that of MP 2. The possibility to separate LMO contributions responsible for the dispersion interaction was investigated.     

Model for the fast estimation of basis set superposition error in biomolecular systems

Basis set superposition error (BSSE) is a significant contributor to errors in quantum-based energy functions, especially for large chemical systems with many molecular contacts such as folded proteins and protein-ligand complexes. While the counterpoise method has become a standard procedure for correcting intermolecular BSSE, most current approaches to correcting intramolecular BSSE are simply fragment-based analogues of the counterpoise method which require many (two times the number of fragments) additional quantum calculations in their application. We propose that magnitudes of both forms of BSSE can be quickly estimated by dividing a system into interacting fragments, estimating each fragment's contribution to the overall BSSE with a simple statistical model, and then propagating these errors throughout the entire system. Such a method requires no additional quantum calculations, but rather only an analysis of the system's interacting fragments. The method is described herein and is applied to a protein-ligand system, a small helical protein, and a set of native and decoy protein folds.     

SCF theory of intermolecular interactions without basis set superposition error

Special SCF LCAO MO type equations are derived, permitting “supermolecule” calculations for intermolecular interactions, excluding basis set superposition error (BSSE) from the beginning on the basis of the “chemical Hamiltonian approach”. (No additional “monomer” calculations are necessary to correct for BSSE.) The formalism excluding the BSSE results in a non-Hermitean Fock matrix; an algorithm is proposed to obtain the required molecular orbitals, in which no integral transformation is needed.     

On the application of the counterpoise correction for the basis set superposition error in geometry optimization calculations of molecular systems: some inconsistent results

It is shown that the conjecture that the total energy for a given molecular or supermolecular system is affected by basis set superposition error (BSSE) leads to inconsistent results. While the calculations of interaction energies, dissociation energies, or energy barriers depend on the fragments (reactants, products) involved in their definitions and, consequently, are affected by BSSE, the total energies of molecular or supermolecular systems do not depend on any virtual fragment partition and are, therefore, BSSE free. Contribution to the Serafin Fraga Memorial Issue.     

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.     

Kinetic study of H‐atom transfer in imidazoline‐, imidazolidine‐, and pyrrolidine‐based alkoxyamines: Consequences for nitroxide‐mediated polymerization

The H‐atom transfer reaction was studied for a series of imidazoline, imidazolidine, and pyrrolidine‐based alkoxyamines containing either isobutyrate‐2‐yl or 1‐phenylethyl alkyl fragments. The C O bond homolysis rate constants and activation energies were determined by ¹H NMR product analysis as a function of temperature. Inter‐ and intramolecular H‐atom transfer reactions were distinguished by examination of alkoxyamine thermolysis products in the absence and in the presence of a radical scavenger (thiophenol or deuterated styrene). A correlation between the structure of the nitroxyl fragment of alkoxyamine and the H‐transfer reaction was found. The high steric demands of the substituents on the nitroxyl part of the isobutyrate‐2‐yl alkoxyamine decrease both types of reaction. For alkoxyamines containing the 1‐phenylethoxyamines, neither inter‐ nor intramolecular H‐atom transfer was observed. Controlled polymerization of methylmethacrylate initiated with imidazoline‐based alkoxyamine was observed, although the polymer obtained was not living. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 6579–6595, 2009     

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.     

Intramolecular basis set superposition error effects on the planarity of benzene and other aromatic molecules: a solution to the problem

Recently, the surprising result that ab initio calculations on benzene and other planar arenes at correlated MP2, MP3, configuration interaction with singles and doubles (CISD), and coupled cluster with singles and doubles levels of theory using standard Pople's basis sets yield nonplanar minima has been reported. The planar optimized structures turn out to be transition states presenting one or more large imaginary frequencies, whereas single-determinant-based methods lead to the expected planar minima and no imaginary frequencies. It has been suggested that such anomalous behavior can be originated by two-electron basis set incompleteness error. In this work, we show that the reported pitfalls can be interpreted in terms of intramolecular basis set superposition error (BSSE) effects, mostly between the C-H moieties constituting the arenes. We have carried out counterpoise-corrected optimizations and frequency calculations at the Hartree-Fock, B3LYP, MP2, and CISD levels of theory with several basis sets for a number of arenes. In all cases, correcting for intramolecular BSSE fixes the anomalous behavior of the correlated methods, whereas no significant differences are observed in the single-determinant case. Consequently, all systems studied are planar at all levels of theory. The effect of different intramolecular fragment definitions and the particular case of charged species, namely, cyclopentadienyl and indenyl anions, respectively, are also discussed.     

Solvent effects on glycine. I. A supermolecule modeling of tautomerization via intramolecular proton transfer

The relative stabilities of glycine tautomers involved in the intramolecular proton transfer are investigated computationally by considering glycine-water complexes containing up to five water molecules. The supermolecule results are compared with continuum calculations. Specific solute-solvent interactions and solvent induced changes in the solute wave function are considered using the natural bond orbitals (NBO) method. The stabilization of the zwitterion upon solvation is explained by the changes in the wave functions localized on the forming and breaking bonds as well as by the different interaction energies in the zwitterionic and neutral clusters. Only the neutral species exist in mono- and dihydrated clusters and in the gas phase. In the smaller clusters, zwitterions are mainly stabilized by conformational effects, whereas in larger clusters, in particular when glycine is solvated on both sides of its heavy atom backbone, polarization effects dominate the stability of a given tautomer. Generally, the strength of the solute-solvent interactions is governed by the intermolecular charge transfer interactions. As the solvation progresses, the hypothetical gaseous zwitterion is better solvated than the gaseous neutral, making zwitterion to neutral tautomerization progressively less exothermic for clusters containing up to three water molecules, and endothermic for larger clusters. The neutral isomer does not exist for some solvent arrangements with five water molecules. Only solvent arrangements in which water molecules do not interact with the reactive proton are considered. Hence, the experimentally observed double well potential energy surface may be due to such an interaction or to a different reaction mechanism.     

Direct intramolecular gas‐phase transfer reactions during fragmentation of sildenafil and thiosildenafil analogs in electrospray ionization mass spectrometry

A study is made of the mass spectral fragmentation pathways of sildenafil, thiosildenafil, and analogous compounds. A prominent gas‐phase reaction that occurs during collision‐induced dissociation (CID) of thiosildenafil compounds is the transfer of an alkyl group from the piperazine nitrogen atom to the sulfur atom of the thiocarbonyl group. This phenomenon is clearly demonstrated through a comparison of electrospray ionization mass spectral fragmentation patterns of four sildenafil‐type compounds and three related thiosildenafil derivatives. Molecular modeling and fragmentation patterns support a direct intramolecular alkyl transfer mechanism rather than an ion‐neutral complex mechanism. CID of thiohydroxyhomosildenafil results in a facile hydroxyethyl migration to the sulfur atom followed by a second intramolecular reaction to form a spiro‐1,3‐oxathiolane ring, which fragments in two directions to generate both carbonyl and thiocarbonyl product ions from this thiocarbonyl compound. While methyl migration to the thiocarbonyl sulfur atom of thiosildenafil is dominant, methyl migration to the carbonyl oxygen atom of sildenafil may occur to a small extent. Published in 2008 by John Wiley & Sons, Ltd.     

O(1D)+Si(CH3)3Cl反应生成振动激发的OH(ν≤3)

The vibrationally excited OH(v) from the reaction of O(~1D)+Si(CH_3)_3Cl was observed by UV laser photolysis/FTIR emission spectroscopy. The vibrational number was only up to 3 with a ratio of 1:0.8:0.1 for v=1:2:3. Comparing this result with the similar reaction of O(~1D)+Si(CH_3)_4 which OH(v) vibrational number was high up to 4 with a ratio of 1:1:1:0.3 for v=1:2:3:4 under the same experimental conditions, it was found that the substitution of Cl for CH_3 affected the extent of heavy Si atom blocking the energy migration in a molecule. This results identifies the prediction that the characteristic of electrons in central atom plays an important role in intramolecular energy transfer.     

Performance of numerical atom‐centered basis sets in the ground‐state correlated calculations of noncovalent interactions: Water and methane dimer cases

Numerical atom‐centered basis sets (orbitals) (NAO) are known for their compactness and rapid convergence in the Hartree–Fock and density‐functional theory (DFT) molecular electronic‐structure calculations. To date, not much is known about the performance of the numerical sets against the well‐studied Gaussian‐type bases in correlated calculations. In this study, one instance of NAO [Blum et al., The Fritz Haber Institute ab initio Molecular Simulations Package (FHI‐aims), 2009] was thoroughly examined in comparison to the correlation‐consistent basis sets in the ground‐state correlated calculations on the hydrogen‐bonded water and dispersion‐dominated methane dimers. It was shown that these NAO demonstrate improved, comparing to the unaugmented correlation‐consistent based, convergence of interaction energies in correlated calculations. However, the present version of NAO constructed in the DFT calculations on covalently‐bound diatomics exhibits enormous basis‐set superposition error (BSSE)—even with the largest bases. Moreover, these basis sets are essentially unable to capture diffuse character of the wave function, necessary for example, for the complete convergence of correlated interaction energies of the weakly‐bound complexes. The problem is usually treated by addition of the external Gaussian diffuse functions to the NAO part, what indeed allows to obtain accurate results. However, the operation increases BSSE with the resulting hybrid basis sets even further and breaks down the initial concept of NAO (i.e., improved compactness) due to the significant increase in their size. These findings clearly point at the need in the alternative strategies for the construction of sufficiently‐delocalized and BSSE‐balanced purely‐numerical bases adapted for correlated calculations, possible ones were outlined here. For comparison with the considered NAOs, a complementary study on the convergence properties of the correlation‐consistent basis sets, with a special emphasis on BSSE, was also performed. Some of its conclusions may represent independent interest. © 2013 Wiley Periodicals, Inc.     

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.