Basis set incompletion error

If the basis set is mathematically complete, then the equation holds precisely. In practice, one has to work with an incomplete finite basis set and hence the equality is only approximate. Results close to the basis set limit (the exact HF solutions) can nowadays be found, but for all practical intents and purposes, one needs to live with a basis set incompleteness error that must be investigated numerically for specific applications. 

The calculated dissociation energies reported here have not been corrected for the basis set superposition error (BSSE). There are two types of error in calculations using a truncated basis set the BSSE and the basis set incompletion error (BSIE). These two errors have opposite sign. Both errors can, in principle, be corrected by saturating the basis set, which is not possible in this case. However, correcting for the BSSE would leave the BSIE uncorrected. We think that for a comparison with experimental values, directly calculated bond energies should be used rather than estimated data obtained from correction procedures such as the counterpoise method. For a discussion of the BSSE, see Reference 114. 

Basis Set Incompleteness Error (BSIE). It is known that in electronic structure calculations the basis sets are not complete. As a result, it is of interest to calculate the interaction energy in the Complete Basis Set (CBS) limit to eliminate BSIE. 

The effect of the correction is demonstrated in Table 3. The basis-set incompleteness error of Hartree-Fock is reduced by nearly one order of magnitude. 

BSSE (basis set superposition error) an error introduced when using an incomplete basis set... 

To summarize, the RPPA is a method that can accurately describe relativistic effects, even though the relativistic perturbation operator used in the pseudopotential procedure is acting on the valence space and not the region dose to the nudeus, as this is the case for the correct all-electron relativistic perturbation operator. That is, relativistic effects are completely transferred into the valence space. These effects are also completely transferable from the atomic to the molecular case as the results for Au2 show. If relativistic pseudopotentials are carefully adjusted, they can produce results with errors much smaller than the errors originating from basis set incompleteness, basis set superposition or from the electron correlation procedure applied. 

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. 

Koopmans theorem also implies that the eigenvalue associated with the HF LUMO may be equated with the EA. However, in the case of EAs errors associated with basis set incompleteness and differential correlation energies do not cancel, but instead they reinforce one another, and as a result EAs computed by this approach are usually entirely untrustworthy. 

Another aspect of the ab intio computation of molecular complexes is the basis set superposition error (BSSE). This is due to the use of an incomplete basis set for the description of molecular complexes. As a result, the basis set of one molecule provides a framework for artificial lowering the energy of its partner and vice versa. Consequently, the BSSE introduces a nonphysical attraction between two (or more) subunits which form a molecular complex. In another words, the BSSE decreases to zero as the atomic basis set used for the description of the molecular system becomes complete. 

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). 

Basis Set Superposition Error Since in practice, basis sets must be of some limited size far short of the HF limit, their incompleteness can lead to a spurious result known as basis set superposition error (BSSE). This is readily grasped in the context of the binding of two molecules, A and B, to form the complex AB. The binding energy is evaluated as... 

---