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.     

The effect of basis set superposition error on the convergence of intermolecular interaction energies for deprotonated complexes

The intermolecular interaction energies of the deprotonated hydrogen-bonded complexes F(-)(HF), F(-)(H(2)O), F(-)(NH(3)), Cl(-)(HF), SH(-)(HF), H(2)P(-)(HF), OH(-)(H(2)O), OH(-)(H(2)O)(2), OH(-)(NH(3)), Cl(-)(H(2)O), SH(-)(H(2)O), H(2)P(-)(H(2)O), Cl(-)(NH(3)), SH(-)(NH(3)), H(2)P(-)(NH(3)), Cl(-)(HCl), Cl(-)(H(2)S), Cl(-)(PH(3)), SH(-)(H(2)S), SH(-)(PH(3)), and H(2)P(-)(PH(3)) were calculated with correlation consistent basis sets at the MP2, MP4, QCISD(T), and CCSD(T) levels. When the basis set is smaller, the counterpoise-uncorrected intermolecular interaction energies are closer to the complete basis set limit than the counterpoise-corrected intermolecular interaction energies. The counterpoise-uncorrected intermolecular interaction energies obtained at the MP2/aug-cc-pVDZ level of theory are close to the interaction energies obtained at the extrapolated complete basis set limit in most of the complexes. Also, we investigate the accuracy of the other levels.     

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.     

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

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

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.     

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.     

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.     

Syntheses and properties of enantiomerically pure higher (n > or = 7) [n-2]triangulanedimethanols and sigma-[n]helicenes

(P)-(+)-Hexaspiro[2.0.0.0. 0.0.2.1.1.1.1.1]pentadecane [(P)-17] as well as (M)-(-)- and (P)-(+)-octaspiro[2.0.0.0.0.0.0.0.2.1.1.1.1.1.1.1]nonadecanes [(M)- and (P)-25]-enantiomerically pure unbranched [7]- and [9]triangulanes-have been prepared starting from racemic THP-protected (methylenecyclopropyl)methanol 6. The relative configurations of all important intermediates as well as the absolute configurations of the key intermediates were established by X-ray crystal structure analyses. This new convergent approach to enantiomerically pure linear [n]triangulanes for n=7, 9 was also tested in two variants towards [15]triangulane. Some of the most prominent and unexpected features of the newly prepared compounds are the remarkable modes of self-assembly of the diols (P)-14, (E)-(3S,3'S,4S,4'S,5R,5'R)-21, (P)-(+)-22, and (E)-31 in the solid state through frameworks of intermolecular hydrogen bonds leading to, depending on the respective structure, nanotube- [(P)-14, (P)-(+)-22, and (E)-31], honeycomb-like structures [(E)-(3S,3'S,4S,4'S,5R,5'R)-21] or a supramolecular double helix [(P)-(+)- and (M)-(-)-22]. Liquid crystalline properties of the esters and ethers of the diols (P)-14, (P)-, and (M)-22 have also been tested. Although all of these [n]triangulanes have no chromophore which would lead to significant absorptions above 200 nm, they exhibit surprisingly high specific rotations even at 589 nm with [alpha](20)(D)=+672.9 (c=0.814 in CHCl(3)) for (P)-(+)-17, +909.9 (c=0.96 in CHCl(3)) for (P)-(+)-25, -890.5 (c=1.01 in CHCl(3)) for (M)-(-)-25, and -1302.5 (c=0.36 in CHCl(3)) for (M)-(-)-39, and the specific rotations increase drastically on going to shorter wavelengths. This outstanding rotatory power is in line with their rather rigid helical arrangement of sigma bonds, and accordingly these helically shaped unbranched [n]triangulanes may be termed "sigma-[n]helicenes", as they represent the sigma-bond analogues of the aromatic pi-[n]helicenes. Density functional theory (DFT) computations at the B3 LYP/6-31+G(d,p) level of theory for the geometry optimization and time-dependent DFT for determining optical rotations with a triplet-zeta basis set (B3 LYP/TZVP) reproduce the optical rotatory dispersions (ORD) very well for the lower members (n=4, 5) of the sigma-[n]helicenes. For the higher ones (n=7, 9, 15) the computed specific rotations turn out increasingly larger than the experimental values. The remarkable increase of the specific rotation with an increasing number of three-membered rings is proportional neither to the molecular weight nor to the number of cyclopropane rings in these sigma-[n]helicenes.     

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.     

Theoretical determination of the differential polarizability and anisotropy of alkaline earth oxide nanoclusters (BeO)n [n = 2–9]: The basis set and electron correlation effects

As an important lineage of metal oxides, alkaline earth oxides have attracted significant interest due to their unique properties and potential applications in material science and industry. In this article, we present the first ab initio (HF, MP2, and CCSD(T)) and density functional theory (TPSSh functional) investigation on the optical properties such as polarizabilities per atom (PPA), differential polarizability per unit (DPPU), and anisotropies of (BeO)_n [n = 2–9] nanoclusters as an illustrative example of alkaline earth oxides nanostructures. Basis set augmentation effects on the studied properties of BeO nanoclusters have been explored by using basis sets of triple‐zeta quality starting from 6‐311G with increasing completeness of the diffuse and polarization functions to the 6‐311+G(3df) basis. Checking carefully the basis set effects, it is shown that the 6‐311+G(3d) basis set provides the best compromise between the accuracy and computational cost. We found a decreasing trend for PPA values of BeO nanoclusters using all considered methods, indicating the strong electron delocalization with increasing cluster size. Moreover, in accordance with the energetic analysis of stability of BeO nanoclusters, the values of PPA show that the (BeO)₄ and (BeO)₆ clusters are the most stable magic numbers compared to the neighbors, satisfying the minimum polarizability principle. The computed values of DPPU demonstrate a strong binding effect in BeO nanoclusters. Taking into account the electron correlation correction (ECC), it is observed that the variations of ECC on dipole polarizabilities are almost smooth for clusters under study. © 2013 Wiley Periodicals, Inc.     

Carbo[n]helicenes Restricted to Enantiomerize: An Insight into the Design Process of Configurationally Stable Functional Chiral PAHs

The most important stereodynamic feature of carbo[n]helicenes is the interconversion of their enantiomers. The Gibbs activation energy (ΔG^≠(T)) of this process, which determines the rate of enantiomerization, dictates the configurational stability of [n]helicenes. High values of ΔG^≠(T) are required for applications of functional chiral molecules incorporating [n]helicenes or helicene substructures. This minireview provides an overview of the mechanism, recent developments, and factors affecting the enantiomerization of [n]helicenes, which will accelerate the design process of configurationally stable functional chiral molecules based on helicene substructures. Additionally, this minireview addresses the misconception and irregularities in the recent literature on how the terms “racemization” and “enantiomerization” are used as well as how the activation parameters are calculated for [n]helicenes and related compounds.     

Importance of electron correlation effects and basis set superposition error in calculations of interaction energies and interaction-induced electric properties in hydrogen-bonded complexes: a model study

A detailed study of the interaction energies and interaction-induced electric dipole properties in model linear hydrogen cyanide complexes (HCN) _m (m?=?2?C4) is carried out within the finite field HF SCF, MP2, CCSD and CCSD(T) approximations using the recently developed LPol-n (n?=?ds, fs, dl, fl) basis sets. The importance of high-order correlation effects and the basis set superposition error is evaluated. To correct for the latter is crucial for obtaining accurate interaction energy values, but the error can safely be neglected in the estimation of induced electric properties when the LPol-n (n?=?ds, fs, dl, fl) basis sets are used. Correlation effects are important in the evaluation of both the interaction energies and the induced electric properties of the systems.     

Binding affinities and spectra of complexes formed by dehydrotetrapyrido[20]annulene and small molecules

Theoretical study on the binding affinities of dehydrotetrapyrido[20] annulene to the alkene and aromatic molecules was performed using the AM1 and DFT methods. It indicated that the host possesses the ability to bind the above molecules since the binding energies of the complexes were negative. The complexes were stabilized via the hydrogen bonding, static effect, and π — π stacking interaction between the host and guest molecules. Based on the B3LYP/3-21G optimized geometries, the electronic, IR, and NMR spectra were calculated using the INDO/CIS, AM1, and B3LYP/3-21G methods, respectively. Due to the hydrogen bonding, the first absorption maxima in the electronic spectra of studied complexes were blue-shifted, whereas the main IR frequencies for some of the complexes were red-shifted. At the same time, the chemical shifts of carbon atoms forming the bonds in the complexes were lower, compared to those of the host.     

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

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

Role and effective treatment of dispersive forces in materials: Polyethylene and graphite crystals as test cases

A semiempirical addition of dispersive forces to conventional density functionals (DFT-D) has been implemented into a pseudopotential plane-wave code. Test calculations on the benzene dimer reproduced the results obtained by using localized basis set, provided that the latter are corrected for the basis set superposition error. By applying the DFT-D/plane-wave approach a substantial agreement with experiments is found for the structure and energetics of polyethylene and graphite, two typical solids that are badly described by standard local and semilocal density functionals.     

Estimation of isotropic nuclear magnetic shieldings in the CCSD(T) and MP2 complete basis set limit using affordable correlation calculations

A linear correlation between isotropic nuclear magnetic shielding constants for seven model molecules (CH₂O, H₂O, HF, F₂, HCN, SiH₄ and H₂S) calculated with 37 methods (34 density functionals, RHF, MP2 and CCSD(T)), with affordable pcS‐2 basis set and corresponding complete basis set results, estimated from calculations with the family of polarization‐consistent pcS‐n basis sets is reported. This dependence was also supported by inspection of profiles of deviation between CBS estimated nuclear shieldings and shieldings obtained with the significantly smaller basis sets pcS‐2 and aug‐cc‐pVTZ‐J for the selected set of 37 calculation methods. It was possible to formulate a practical approach of estimating the values of isotropic nuclear magnetic shielding constants at the CCSD(T)/CBS and MP2/CBS levels from affordable CCSD(T)/pcS‐2, MP2/pcS‐2 and DFT/CBS calculations with pcS‐n basis sets. The proposed method leads to a fairly accurate estimation of nuclear magnetic shieldings and considerable saving of computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

The basis set superposition error in multilevel methods: a test on the H2O and HF dimer

The basis set superposition error (BSSE) is often very important in the accurate calculation of dimerization energies. Accurate prediction of thermochemical properties requires appropriate consideration of the basis set incompleteness error. Multilevel methods introduce adjustable parameters to reproduce experimental data, that is, the higher level correction (HLC) in G3 and adjustable coefficients in the MCCM. However, the HLC term is cancelled out and the adjustable coefficients in the MCCM do not remove the BSSE completely. We have calculated the BSSE involved in the multilevel methods. The H₂O and HF dimer systems were used as a test case. This study shows that empirical coefficients do reduce the BSSE in some cases and an MCCM with good parameters can be used to reproduce dimerization energies within chemical accuracy without the BSSE correction.     

Intermolecular bond lengths: extrapolation to the basis set limit on uncorrected and BSSE‐corrected potential energy hypersurfaces

Geometry optimizations were carried out for the (HF)₂, (H₂O)₂, and HF–H₂O intermolecular complexes using the MP2/aug‐cc‐pVXZ {X=2, 3, 4, and 5} theoretical models on both the uncorrected and counterpoise (CP) corrected potential energy hypersurfaces (PES). Our results and the available literature data clearly show that extrapolation of intermolecular distances to the complete basis set (CBS) limit is satisfactory on PESs corrected for BSSE. On the other hand, one should avoid such extrapolations using data obtained from uncorrected PESs. Also, fixing intramolecular parameters at their experimental values could cause difficulties during the extrapolation. As the available literature data and our results clearly show, the MP2/aug‐cc‐pVXZ {X=2, 3, 4} data series of intermolecular distances obtained from the CP‐corrected surfaces can be safely used for the purpose of CBS extrapolations. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 196–207, 2001     

Accuracy and efficiency of atomic basis set methods versus plane wave calculations with ultrasoft pseudopotentials for DNA base molecules

Recent results from Preuss et al. (J Comput Chem 2004, 25, 112) on DNA base molecules, obtained by plane wave density functional calculations using ultrasoft pseudopotentials, are compared with calculations using Gaussian basis sets. Bond lengths and angles agree closely, but dihedral angles and vibrational frequencies show significant differences. The Gaussian basis calculations are at least an order of magnitude more efficient than the plane wave/ultrasoft pseudopotential calculations at a similar level of accuracy; the advantage is even larger if the Fourier Transform Coulomb method is used. To obtain definite benchmark values, the geometries of the four DNA bases were optimized at the MP2 level with large basis sets, up to cc-pVQZ and aug-cc-pVTZ.