BSSE-free SCF theories: a comment

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Abstract

Some aspects of treating the basis set superposition error (BSSE) problem at the SCF level are briefly discussed. It is stressed that—contrary to the recurring propositions in the literature—the BSSE-problem cannot be properly handled by restricting each molecular orbital to be expanded in the basis of only one of the monomers, because this also excludes the physically important, true charge-transfer effects. The discussion is illustrated by some calculations performed for water dimer and for two pairs of nucleic bases (thymine–adenine and uracil–adenine) which are compared with those in the recent paper of Gianinetti et al. [E. Gianinetti, M. Raimondi, E. Tornaghi, Int. J. Quant. Chem. 60 (1996) 157].

Introduction

Basis set superposition error (BSSE) is a notorious problem when studying weakly bonded molecular complexes: the extension of the molecular basis set with the basis functions of the partner molecule leads to a spuriously improved (in energetic sense) description of the internal electronic structure of the constituting molecules (“monomers”) and this results in too deep minima on the computed potential energy surfaces. The importance of intermolecular interactions in most different areas of science motivated new and new attempts of finding computational schemes permitting to get rid of BSSE. However, BSSE poses some delicate theoretical problems and, as a consequence, not all the methods proposed are correct.

In a recent paper Gianinetti et al. [1] have repeated the mistake, time by time recurring in the literature—see Ref. [2] for another recent example—according to which one can perform BSSE-free SCF calculations in such a way that each molecular orbital (MO) is expanded by using the basis orbitals of only one of the monomers.1 The use of such a constraint (often called “polarization only approximation”) means that one excludes from consideration any delocalizations between the monomers: not only the BSSE-caused delocalizations are avoided but also the true, physical intermolecular charge transfer (CT) effects are omitted. In a specific case of the He+p interaction the results of the “SCF–MI” method of [1] will be completely independent of the hydrogenic basis set; the same result will be obtained if there are no basis functions on the proton or if there is a (nearly) complete basis on it, although at smaller distances there will be some electronic charge near the proton too. CT has an important role in hydrogen bonding, too, which makes “SCF–MI” approach unacceptable also for such systems.2

As to our knowledge, two conceptually different, main avenues have been proposed, which can be considered straightforward approaches in treating the BSSE problem: the a posteriori counterpoise (CP) method of Boys and Bernardi (BB) [3] (first proposed by Jansen and Ross [4]) and the a priori scheme of excluding BSSE by using the “Chemical Hamiltonian Approach” (CHA) [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. We agree with Gianinetti et al. [1] that—contrary to the common belief—the CP scheme does not necessarily overcorrect for BSSE, but the degree of undercorrection they have assumed [1] is quite unrealistic. From the other hand, Gianinetti et al. [1] criticize the CHA method because it uses a non-Hermitian Fock matrix. This criticism is, however, unfounded: BSSE is not a physical phenomenon, it is not connected with a physical observable to which a Hermitian operator could be assigned. Therefore, a theoretically correct a priori treatment of the BSSE problem is possible only by using non-Hermitian operators and/or Fock matrices.3 (Hermitian Fock matrices may appear applicable as a special approximation for the limit of weak interactions [16].) It is to be stressed, that—except some early applications [14]—the energy in the CHA method is calculated as the expectation value of the conventional (Hermitian) Hamiltonian over the BSSE-free CHA wave functions; this guarantees the energy to be real even if complex wave functions are admitted.

The aim of the present note is to discuss briefly some aspects of the BSSE-corrected CP and the BSSE-free CHA–SCF methods and to repeat the calculations (the water dimer and two pairs of nucleic bases) discussed by Gianinetti et al. [1] in order to illustrate the uppermost importance of the physical CT terms omitted by them.

Section snippets

Total energy in the Boys–Bernardi's CP scheme

In the BB framework one usually considers interaction energies only. The interaction energy in this scheme is defined as the difference of supermolecule and monomer energies, all calculated in the supermolecule basis corresponding to the given geometry of the system:ΔEABCP=EAB(AB)−EA(AB)−EB(AB)where subscripts indicate the molecule in question and the basis applied is shown in parentheses. For different intermolecular geometries4

Water dimer calculations

We have repeated the water dimer SCF and CP calculations of Gianinetti et al. [1] by using the same 4-31G basis set and the geometry as specified in [18]. All SCF and CP energies quoted in [1] have been reproduced with the accuracy of 0.01–0.02 kcal/mol, except the point RO–O=2.8Å for which simple inspection of the curve in [1] also shows that the CP energy is obviously in error by some 0.15 kcal/mol.

With this correction, the data displayed in [1] are reproduced on our Fig. 1 by dotted lines; we

Conclusions

In this article we have performed a short comparison of the counterpoise and CHA schemes of solving the BSSE problem at the SCF level of theory and performed some comparative calculations for the water dimer and for two pairs of nucleic bases—the same model systems as were considered by Gianinetti et al. For all the systems studied, we have found that the difference between the BSSE corrected (CP) and BSSE-free (CHA/F) curves is not much significant, while the SCF–MI method of Gianinetti et al.

Acknowledgements

This work was supported in part by the Hungarian Research Fund (grants OTKA no. C0020, T15838 and T25369). Á.V. also acknowledges the Széchenyi scholarship to the Hungarian Ministry of Education.

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