Basis set superposition error and counterpoise corrections

This is called the basis set superposition (BSS) error. Why superposition error When the two components of the dimer are at large distance, Xa and %b are well separated, i.e., they have small superposition (or overlap), and so the relative error we are considering is modest, zero at infinity. When the two monomers are at shorter distances the superposition of the two basis sets increases (the basis functions are always centered on the pertinent nuclei) as well as the error. 

There is a simple recipe to correct this error it consists of performing all the necessary calculations with the same basis set, the dimeric basis Xab which depends on the geometry of the dimer. This means that the energy of the monomers must be repeated for each position in the R configuration space (see Section 8.4 for its definition). We add a superscript CP (coimterpoise) to denote quantities modified in such a way and we also add the specification of the basis set. We replace eq. [8.11] with the following one  

Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli 

The same holds for calculations performed at higher levels of the QM theory, beyond the HF approximation, with decreasing effects of the CP corrections, however. We may leave this last subject to the attention of the specialists because for the determination of interaction potential for liquids, the HF approximation is in general sufficient in some cases it may be supplemented by applying simple levels of description for electron correlation, as we have already said. 

The physical definition of ES (electrostatic interactions among rigid partners) clearly indicates that there is no room here for CP corrections, which would involve some shift of the monomer s charge on the ghost basis ftmctions of the partner. This physical observation corresponds to the structure of the blocks of the Hamiltonian matrix used to compute ES no elements regarding the BS of the partner are present in conclusion and thus no corrections to ES are possible. 

F igure 8.4. Location of the minimum energy for HjO -HF dimer according to the various basis sets and the various methods. Crosses (x) refer to SCF calculations without CP corrections, full circles ( ) to CP eorreeted calculations. The ovoidal area refer to an estimate of the location of the minimum at the HF limit (including an estimate of the error). Left side minimal basis sets right side double valence-shell basis sets. 

AE is by far more corresponding to die exact potential energy functions in calculations performed at the HF level with a basis set of a small-medium size. Passing to calculations with larger basis sets, the BSS error is obviously smaller, but the CP correction is 

As noted when introducing the Boys-Bernardi CP correction above, BSSE is a concern whenever the supermolecule method is used to compare the energies of fragments to the energy of the entire cluster [i.e., when computing the dissociation (Dg) or interaction (Tint) energy]. 

Although many arguments for and against CP corrections have been made over the years, this part of the chapter will briefly illustrate three important concepts related to this issue. 

The CP procedure can lead to unphysical descriptions of PESs that are not easily corrected. 

CP corrections often do not improve calculated interaction energies. In fact, the procedure can even make results worse. 

BSSE is a poor diagnostic for the quality of a computed Dg or Eint- The crucial quantity is the basis set completeness error (BSCE). 

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It was recognized early on that the binding energy of the benzene dimer is quite sensitive to computational method. The potential energy surface of 77 is completely repulsive at the HF level, as well as with any DFT that does not include a dispersion correction, such as B3LYP. " Furthermore, the weakly bound benzene dimer is susceptible to basis set superposition error and counterpoise corrections are likely to be necessary. ... 

Alagona, G., Ghio, C., Latajka, Z., and Tomasi, J., Basis set superposition errors and counterpoise corrections for some basis sets evaluated for a few X --M dimers, J. Rhys. Chem. 94, 2267-2273 (1990). 

In Fig. 6 the various contribution to the cohesive energy of magnesium are presented. For the embedded eluster the cohesive contribution of the one-body increment is defined as the difference between the correlation energy of the embedded atom (Aa ) and the free atom fre" As = ea The basis set superposition error (BSSE) was corrected applying the counterpoise (CP) method the free atom, described with cc-pVTZ was surrounded by two shells of ghost atoms. The one-body contribution is... 

The Basis Set Superposition Error (BSSE) is corrected by means of the counterpoise procedure, an important aspect that will be discussed later. In the figure energies are referred to two groimd-state cytosine molecules separated about 20 a.u. In the asymptotic limit. Si and S2 become degenerate, which is consistent with the fact that they are related to the equivalent situations C-tC and C -tC, where C and C stand for the ground-state cytosine and its lowest singlet excited state, respectively. The absorption So -> Si calculated at 20 a.u. (4.41 eV) corresponds to the monomer absorption. On the other hand, the vertical emission from the minimum of Si is calculated at 3.40 eV, and it can be considered the source of emission observed in DNA or oligomer samples. In particular, the fluorescence maximum was reported in aqueous solution for the dimer d(C)2 and the 15-mer d(C)is at 3.22 eV, considerably red-shifted as compared to that of the monomer (3.96 eV). 

The corrections for basis set superposition errors (BSSE) with the counterpoise correction are still a controversial issue [120-122], However, some type of correction is needed for calculations on systems with weak interactions. It has been proved that only considering the BSSE meaningful results can be obtained on large systems [122], Of course, by using good quality basis sets the BSSE is less troublesome and the results obtained can be accepted, readily, as reliable. The BSSE in DFT calculations is less important than in conventional ab-initio calculations. We compute the counterpoise correction as ... 

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. 

The effect of basis set superposition error (BSSE) was estimated for this series using the counterpoise method of Boys and Bemardi (16), The isotropic shielding value of methane by itself was calculated to be 31.8276 ppm. Isotropic shielding values corrected for BSSE for the proximal proton of methane were calculated for each distance from ethene (Table I). The difference between the isotropic shielding value calculated for methane alone and the isotropic shielding values at each geometry for the proximal proton of methane obtained for the methane-ethene pair in a counterpoise calculation including the basis functions of ethene (but no electrons) is... 

The above equation is very conveniently used as the computation of the total energy is the standard quantum-chemical procedure. However, a purely theoretical problem arises when using monomer-centered basis set for evaluation of EA and EB according to (20.1) The intermolecular interaction energy will suffer from what is known as basis set superposition error (BSSE) [3], In order to overcome this unphysical effect which usually manifests itself in too negative interaction energies, one frequently applies the so-called counterpoise correction [4],... 

Basis set superposition error (BSSE) is a particular problem for supermolecule treatments of intermolecular forces. As two moieties with incomplete basis sets are brought together, there is an unavoidable improvement in the overall quality of the supermolecule basis set, and thus an artificial energy lowering. Various approximate corrections to BSSE are available, with the most widely used being those based on the counterpoise method (CP) proposed by Boys and Bemardi [3]. There are indications that potential energy surfaces corrected via the CP method may not describe correctly the anisotropy of the molecular interactions, and there have been some suggestions of a bias in the description of the electrostatic properties of the monomers (secondary basis set superposition errors). 

Collins, J. R. and Gallup, G. A., The full ver.sus the virtual counterpoise correction for basis set superposition error in self-consistent field calculations, Chem. Phys. Lett. 123, 56-61 (1986). 

Cammi, R, Del Valle, F. J. O., and Tomasi, J., Decomposition of the interaction energy with counterpoise corrections to the basis set superposition error for dimers in solution. Method and application to the hydrogen fluoride dimer, Chem. Rhys. 122, 63-74 (1988). 

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