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This equation contains two unknowns and we can thus extrapolate to the basis-set limit from two separate calculations with different cardinal numbers X and Y. This gives us the following simple expression for the energy at the basis-set limit [32, 33]:
For example, if we carry out calculations with 
and 
using optimized numerical orbitals (i.e., no longer simple STOs), we obtain errors in the energy of 4.9 and 2.1 kJ/mol, respectively. The error in the energy extrapolated from these two results using Eq. (5.14) is less than 0.1 kJ/mol, which would require a FCI principal expansion with 
or more.
6.CALIBRATION OF THE EXTRAPOLATION TECHNIQUE
6.1.Valence-Shell Correlation Energy
In section 4, we established that the orbital truncation error represents a serious obstacle to the accurate calculation of AEs. Next, in section 5, we found that this problem may be solved in two different ways: we may either employ wavefunctions that contain the interelectronic distance explicitly (in particular the R12 model), or we may try to extrapolate to the basis-set limit using energies obtained with finite basis sets. In the present section, we shall apply both methods to a set of small molecules, to establish whether or not these techniques are useful also for systems of chemical interest.
It is important to realize that molecular electronic systems differ from the He atom in the sense that the uncorrelated Hartree-Fock description cannot be expressed in terms of a single, doubly occupied 1s orbital. In particular, in sequences of calculations using the correlationconsistent orbitals, we observe not only changes related to the improved description of electron correlation, but also changes in the uncorrelated Hartree-Fock description. Within the Hartree-Fock model, effects such as polarization of the atomic charge distributions upon molecular formation require the use of flexible basis sets, albeit convergence is usually reached much more rapidly than for the description of electron correlation. Therefore, in order to study the asymptotic convergence of the short-range correlation problem, we must first subtract from the total electronic energy the Hartree-Fock energy. In passing, we note that the
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Hartree-Fock convergence does not present an insurmountable difficulty; there are clear indications that the molecular Hartree-Fock energy con-
verges as exp |
and thus rather rapidly |
[51-53]. |
Table 1.6 displays the CCSD valence correlation energies of six small |
||
molecules and the Ne atom in units of |
The |
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first five rows contain the energies calculated in the standard manner in the cc-pVXZ basis sets with 
As expected, the convergence is slow, with errors relative to the R12 energies (contained in the last row [54]) of about 10 kJ/mol or more, even for the largest basis sets. This can also be seen from Fig. 1.2, which shows how the calculated CCSD/cc-pVXZ energies converge slowly but smoothly towards the R12 valence-shell correlation energies.
This convergence is significantly accelerated by applying the extrapolation formula (5.14), the errors being reduced to 1 - 3 kJ/mol for all extrapolations except for the cc-pV(DT)Z energies. (Here and elsewhere we shall use the notation cc-pV(X - 1,X)Z for the energy obtained by extrapolation from the cc-pV(X - 1)Z and cc-pVXZ correlation energies.)
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Fig. 1.3 shows the normal distributions of the errors of the cc-pV(X - 1,X)Z extrapolations (note that the scale is different from Fig. 1.2). In comparison with the cc-pV6Z results, which are included in Fig. 1.3 as a broad distribution on its scale, the agreement between the R12 energies and the extrapolated energies is excellent, confirming our earlier conclusions from the discussion of the He atom.
6.2.Total Electronic Energy
Having observed the agreement between R12 and extrapolation for molecular systems, let us now compare directly with experiment. In Table 1.7, we compare the all-electron CCSD(T)/cc-pCVXZ energies of the seven systems with a set of empirically estimated nonrelativistic total electronic energies, obtained by combining the atomic energies compiled by Chakravorty and Davidson [55] with the experimental equilibrium AEs of Bak et al. [9]. The sextuple-zeta results were obtained from the valence-electron cc-pV6Z energies by adding the differences between the all-electron cc-pCV5Z and valence-electron cc-pV5Z energies. The resulting energies are denoted cc-pcV6Z and should be close to the true ccpCV6Z energies.
As expected from our previous discussion of the CCSD valence correlation energies, the convergence towards the experimental energies is slow, with mean absolute errors of 511, 163, 61, 28, and 17 kJ/mol as
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we go from cc-pCVDZ to cc-pcV6Z. In particular, even at the cc-pcV6Z level, the mean absolute error of 17 kJ/mol is significantly larger than the intrinsic error of the CCSD(T) model. Without the benefit of a systematic cancellation of errors, calculations carried out with these basis sets would not be sufficiently accurate.
Let us now compare the R12 energies [54] and the extrapolated energies with the experimental energies for the same systems; see Table 1.8. The R12 energies are between 2 and 4 kJ/mol above the experimental estimates – a significant improvement over the cc-pcV6Z results in Table 1.7. Clearly, the precision of the R12 method is sufficiently high to make the intrinsic error of the CCSD(T) model become an important consideration. Moreover, heats of reaction can be calculated accurately without having to rely on a large cancellation of errors among the products and reactants.
Turning our attention to the extrapolated CCSD(T) energies, we find that the accuracy of the R12 method is attained already at the ccpCV(TQ)Z level, although the standard deviation in the cc-pCV(TQ)Z errors is somewhat larger than for the R12 errors. The cc-pCV(DT)Z energies are considerably less accurate, but still as good as the cc-pVQZ energies at a much reduced cost. Finally, with a mean absolute deviation of 1 kJ/mol and a maximum deviation of 5 kJ/mol, the cc-pCV(Q5)Z and cc-pcV(56)Z energies agree with their experimental counterparts,
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making these levels of theory well suited for computational thermochemistry.
Although the agreement between calculated and experimental correlation and total energies is reassuring, as chemists we are more interested in relative quantities. Let us therefore turn our attention to AEs. In Table 1.9, we compare the calculated all-electron CCSD(T) equilibrium AEs with the corresponding AEs derived from experimental data, see Ref. 9.
Without extrapolation, the errors are reduced by a factor of two to three compared with the errors in the total electronic energies, reflecting the systematic nature of the errors in the total energies. For low cardinal numbers, the same is true for the extrapolated AEs. However, for the cc-pCV(Q5)Z and cc-pcV(56)Z data, the errors in the AEs are similar to the errors in the corresponding total energies, indicating the presence of statistical errors of the order of 1 kJ/mol in the experimental and extrapolated energies. From a practical point of view, we note that the cc-pCV(TQ)Z AEs agree with their experimental counterparts to within 2 kJ/mol, suggesting that chemical accuracy in calculated AEs and heats of reaction should be obtainable at the all-electron CCSD(T)/cc-pCV(TQ)Z level of theory.