The Basis Set Superposition Error

As ab initio calculations were pushed to greater accuracy and researchers began tackling problems such as the weak van der Waals interaaion between rare gas atoms or the hydrogen bonding in water, it became apparent that the 

For example, each atom in a simple diatomic molecule, such as CO, may experience two sources of energy lowering as they approach each other. Not only will there be a lowering due to bond formation, but an artificial lowering may arise as a result of an improved description of each atom. Even in the absence of the other nucleus and its electrons, the energy of either carbon or oxygen by itself may show a decrease when its own basis is augmented by the colleaion of basis functions associated with the other center. 

Several schemes have been proposed for estimating the BSSE. The most widely used is the counterpoise method, which defines the BSSE for a dimer as the sum over both monomers of (monomer in monomer basis) — (monomer in full dimer basis). Some authors caution against allowing the electrons on monomer A to utilize the funaion space corresponding to the occupied orbitals on monomer More recently diis controversy seems 

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Xantheas, S.S. (1996) On the importance ofthe fragment relaxation energy terms in the estimation of the basis set superposition error correction to the intermolecular interaction energy, J. Chem. Phys., 104, 8821-8824. 

Sordo, J. A., T. L. Sordo, G. M. Fernandez, R. Gomperts, S. Chin, and E. Clementi. 1989. A Systematic Study on the Basis Set Superposition Error in the Calculation of Interaction Energies of Systems of Biological Interest. J. Chem. Phys. 90,6361-6370. 

For an explicit examination of the basis set superposition error on weakly bound cationic structures at various theoretical levels, we performed full counterpoise calculations as outlined in Section IIB (Table 5). Table 5 demonstrates nicely that the magnitude of the BSSE is indirectly proportional to the size of the basis set at TZ2P, the error is on the average only about a quarter of that at DZP. This effect becomes more apparent with increased incorporation of electron correlation. At our best theoretical level, the BSSE amounts to only 0.1 kcal mol" (Table 5), which is insignificant for the dissociation energy of about 1.5 kcal mol". ... 

Table 5. The Basis Set Superposition Error (BSSE) for CH5(H2). The Monomers in Parentheses are Ghost Molecules. " AE Indicates the Difference between the Energy of the Monomer within the Monomer vs. the Dimer Basis...

The molecular orbitals in the nonrelativistic and one-component calculations and the large component in the Dirac-Fock functions were spanned in the Cd s Ap9d)l[9slp6d basis of [63] and the H (5s 2p)/[35 l/>] set [61]. Contraction coefficients were taken from corresponding atomic SCF calculations. The basis for the small components in the Dirac-Fock calculations is derived by the MOLFDIR program from the large-component basis. The basis set superposition error is corrected by the counterpoise method [64]. The Breit interaction was found to have a very small effect and is therefore not included in the results. 

As it is well known, the Basis Set Superposition Error (BSSE) affects calculations involving hydrogen bonds [1] and, more generally, intermolecular interaction investigations [2,3], This issue of consistency, as first pointed out in 1968 [4], arises from the use of an incomplete basis set but it does not correspond to the basis set truncation error and it is due to the use of diffuse functions on neighbouring interacting particles, which leads to a non physical contribution to the interaction energy within the complex. 

Special attention has been dedicated to the study of the basis set superposition error (BSSE). The SCF-Ml algorithm which excludes the BSSE from the SCF function was employed. A multi configuration version of it, particularly suited to study proton transfer effects, has been formulated. The use of these techniques has led to binding energy values which show a better stability against variation of the basis set, when compared with standard SCF results. For a more complete evaluation of the advantages of the a priori strategy to avoid BSSE see references [47-50], where applications to the study of the water properties are reported, and reference [51], where the Spin Coupled Valence Bond calculations for the He-LiH system are presented. 

The basis set superposition error [13] has been ehminated from the calculation of charge transfer energies. 

The parameters of the model potential (52) have been fitted (separately for Vs and V3) to ab initio energy surfaces calculated at the MP4 level and corrected for the basis set superposition error. The parameters of the additive potential Vs were fitted to 65 calculated points on the potential curve of Bes, the parameters of the 3-body potential V3 were fitted to the EalBea) potential surface with total number of calculated points equal to 108. It is important to note that in the fitting procedure we find the sum (or difference) of exchange and dispersion terms for each m-body potentials V . This sum is correct for all distances considered. But we cannot expect the same from the absolute... 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

H2-H2 dipole. Early attempts to calculate the induced dipole moments from first principles were described elsewhere [281]. Only in recent times could the substantial problems of such computations be controlled and precise data be generated by SCF and Cl calculations, so that that the basis set superposition errors were small enough and the Cl excitation level is adequate for the long-range effects. The details of the computations are given elsewhere [282, 281],... 

Turning to the barriers we note that they are defined as the difference in enthalpy between the transition state and the separated reactants. This is important because in gas-phase ion-molecule reactions the transition state is typically preceded by an ion-dipole complex131 133 formed between the reactants, and the term barrier is sometimes used for the enthalpy difference between the transition state and the ion-dipole complex. However, these ion-dipole complexes have little relevance to the main topic discussed in this chapter and hence the chosen definition of AH is more appropriate. For reasons explained elsewhere,118 the barriers reported in Table 16 have not been corrected for the basis set superposition error (BSSE),134 although such corrected values are available.118... 

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