The cc-pVnZ Basis Sets

A variety of extrapolation algorithms have been applied to the sequences generated by the correlation-consistent cc-pVnZ basis sets [12, 51-55], Dunning and his colleagues had initially suggested fitting their calculations to an exponentially decaying function [12, 51, 52], 

However, as definitive values for E (CBS) became available from the MP2-R12 calculations of Klopper [48], it became clear that Eq. (6.1) seriously underestimates the magnitude of the basis set truncation error. Wilson and Dunning therefore examined [53] a wide variety of extrapolations (24 variations) based on generalizations of Eq. (1.1). They obtained RMS deviations from Klopper s results of less than 1 mEh using several different extrapolation schemes. We arrived at comparable results (Table 4.4) using just two points, E (lmax2) and E(2)(Zmaxi), so 

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The denominator shift of 1/2 was chosen as a compromise between the situation for hydrogen and helium (where n = 1 + 1 for the cc-pVnZ basis set) and main-group elements (where n = 1). As is immediately obvious upon series expansion, there is considerable coupling between the denominator shift and the exponent. As a result, the three-point extrapolation generally leads to exponents well in excess of three [34],... 

Figure 4.6 The RMS errors in MP2 correlation energies obtained with the cc-pVnZ basis set (n = Zroax = 2, 3, 4, 5, and 6).

As a first try, we have elected to follow our treatment of the SCF and second-order correlation energies described above, and employ Eq. (6.2) to provide a linear extrapolation of the cc-pVDZ and cc-pVTZ total CBS-CCSD(T) energies obtained with Eq. (2.2), including the interference correction. These total energies reproduce the CCSD(T) limits estimated by Martin [55] via an (lmax + 5)-3 extrapolation of the CCSD(T)/cc-pVDZ, TZ, QZ, 5Z, and 6Z basis sets to within 0.96 kcal/mol RMS error. The agreement with Martin s energies for a small set of chemical reactions is even better (Table 4.8). The use of the cc-pVnZ basis sets for PNO-(Zmax + 5)-3 double extrapolations is indeed promising. 

The orthogonal subspace used in Table I is spanned by the large basis 0=19sl4p8d6f4g3h2i, H=9s6p4d3f2g. The number of basis functions of the large basis that are (nearly) linearly dependent on the cc-pVnZ basis is drastically increased with the cc-pVnZ basis sets. We observe that the externally contracted MP2 calculations converge faster to the MP2 limit. As expected, the energies from the externally contracted MP2 method lie between the standard... 

Fig. S. Counterpoise-corrected and uncorrected D, R, and o) for N2, computed with the cc-pVnZ basis sets. The dashed lines represent the estimated CBS limit.

The greater dependence on basis-set quality of correlated calculations compared to those of the HF variety has prompted many developers of basis sets to optimize contractions via some scheme that includes evaluating results from the former. For instance, the correlation consistent prefix of the cc-pVnZ basis sets discussed in Chapter 6 highlights this feature. 

Basis set effects have also been examined systematically using CIS(D) and a serious of the cc-pVnZ basis sets. The results are plotted in Supporting Information. When the energy is plotted as a function of 1/n, an extrapolation limit... 

As mentioned in Section 15.4, the cc basis sets can be used to extrapolate to the complete-basis-set (CBS) limit for molecular properties. Energies in hartrees for SCF MO H2O optimized-geometry calculations with the cc-pVnZ basis sets for n = 2,3,4,5,6 are -76.02705, -76.05777, -76.06552, -76.06778, -76.06810, respectively. The following empirical formula has been found to work rather well for extrapolating cc-pVnZ SCF MO energies to the CBS limit ... 

Table 9 Total Energies and Spectroscopic Constants for the Hydrogen Fluoride and Nitrogen Molecules Obtained from Hartree-Fock Calculations Using the cc-pVnZ Basis Sets (Experimental Results taken from Ref. 72)...

Table 21 Calculated Anharmonicity Corrections (cm for HCN from CCSD(T) Calculations with the cc-pVnZ Basis Sets...

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