The Complete Basis Set Extrapolation

As we noted in Chapter 7, the CBS family of methods all include a component which extrapolates from calculations using a finite basis set to the estimated complete basis set limit. In this section, we very briefly introduce this procedure. 

The extrapolation to the complete basis set energy limit is based upon the MoUer-Plesset expansion E= + E + E + E + E +. .. as described earlier in this appendix. Recall that E + E is the Hartree-Fock energy. We will denote E and all higher terms as E , resulting in this expression for E  

Remember that the CBS models begin with a large enough SCF calculation to obtain the desired level of accuracy (see Chapter 7) therefore, no explicit extrapolation of the SCF energy is included. CBS extrapolation involves computing the second-order and infinite-order corrections to the energy. 

Schwartz has shown that for a helium-like ion, the contribution to the second-order MoUer-Plesset energy from the 1 angular momentum component can be approximated by the following expression  

This expression describes how the energy converges as we add successive s functions, p functions, d functions, f functions, and so on, to spherical atoms. 

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The filled and hollow circles indicate the contributions of each successive natural orbital. The filled circles correspond to complete shells. Only these points are useful for extrapolating to the complete basis set limit. 

The slow convergence of the correlation energy with the one-electron basis set expansion has provided the motivation for several attempts to extrapolate to the complete basis set limit [6-13], Such extrapolations require a well defined sequence of basis sets and a model for the convergence of the resulting sequence of approximations to the... 

In contrast, a linear size-consistent (Zmax + )-3 extrapolation of just the MP2/cc-pVTZ and MP2/cc-pVQZ energies is accurate to 0.63 kcal/mol (Table 4.4). If we try to further reduce the basis sets to cc-pVDZ and cc-pVTZ, the error in the extrapolation increases to 6.0 kcal/mol. However, the new double extrapolation provides the complete basis set MP2 limit with an absolute accuracy of 0.86 kcal/mol without recourse to basis sets larger than cc-pVTZ [4s3p2dlf/3s2pld] (Table 4.6). 

The main feature of the CBS (complete basis set) methods (e.g. CBS-Q [15] and CBS-QB3 [20]) is extrapolation to the complete basis set limit at the UMP2 level. Additional calculations [UMP4 and UQ-CISD(T) or UCCSD(T)] are performed to estimate higher-order effects. A scaled ZPVE, together with a size-consistent empirical correction and a spin-contamination correction, are added to yield the total CBS energy of the molecule. 

In addition to the Gaussian-n methods, there is a variety of CBS (complete basis set) methods, including CBS-Q, CBS-q, and CBS-4, which have also enjoyed some popularity among computational chemists. In these methods, special procedures are designed to estimate the complete-basis-set limit energy by extrapolation. Similar to the Gn methods, single-point calculations are also... 

Hypoiodous acid and iodine nitrate have been investigated with coupled cluster theory and the results extrapolated to the complete basis set limit. Together with revised thermochemistry for several ancillary molecules, the enthalpy changes of working reactions yields new thermochemistry for HOI and IONO2. The latter data, employed in unimolecular rate theory, appear to be consistent with kinetic measurements on the lO -l- NO2 reaction to within the uncertainties of the kinetic analysis. 

RI-MP2/TZVPP, extrapolating to the complete basis set and including corrections fromCCSD(T). 

Extrapolated to the complete basis set limit and corrected for CCSD(T). 

Jurecka et al have published a database of accurate benchmark quality interaction energies of small model complexes, DNA base pairs, and amino acid pairs obtained from MP2 and CCSD(T) calculations together with estimates of the complete basis set limit. They examine interaction energies and geometries for more than 100 DNA base pairs, amino acid pairs and model complexes. Extrapolation to the complete basis set limit was carried out by these researchers using a two-point method in conjunction with correlation consistent basis sets. 

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