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that our single term extrapolation is linear [50],
and thus rigorously size-consistent. In addition, Eq. (6.2) provides a basis for easily obtaining analytical derivatives of the extrapolated MP2 CBS energies.
7.NEW DEVELOPMENTS
The Dunning cc-pVnZ basis sets can be used with our PNO extrapolations to form a potent new combination. We shall consider the SCF energy first, then the MP2 correlation energy, and finally higherorder correlation energy through CCSD(T).
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7.1.The SCF Limit
One might expect a three-point extrapolation based on Eq. (6.1) to apply to the SCF energy, since the exponential convergence of the SCF energy with the number of Gaussian primitives is well documented [54]. Unfortunately, although the number of contracted basis functions increases in a completely smooth and systematic fashion, the number of primitive Gaussian functions does not increase in a uniform pattern for the cc-pVnZ basis sets (Fig. 4.7). The DZ and TZ basis sets include (9s, 4p) and (10s, 5p) sets of Gaussian primitives, respectively. The QZ basis set then increments the s primitives by 2 but continues the pattern of single increments to the p primitives, yielding (12s, 6p). The 5Z and 6Z basis sets increment both the s and the p by 2, providing (14s, 8p) and (16s, 10p) sets of primitives. Thus, Eq. (6.1) applies to the (QZ, 5Z, 6Z) sequence of basis sets (for first-row atoms, the second-row pattern is different), but not to the (DZ, TZ, QZ) or the (TZ, QZ, 5Z) sequence.
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A linear extrapolation circumvents this problem. If we assume that the exponent a in Eq. (6.1) is universal, we need only two consecutive points to extrapolate,
which would make extrapolations based on the relatively inexpensive DZ and TZ calculations possible. Empirically, we find that setting a equal to 
in Eq. (7.1) and using 
(i.e. employing the cc-pVDZ and cc-pVTZ SCF energies to extrapolate to the SCF limit) reduces the error in the cc-pVTZ SCF energies by more than an order of magnitude (Table 4.5). Thus, SCF energies with absolute errors of less than 0.5 kcal/mol are available from relatively inexpensive calculations.
7.2.The CBS Limit for the MP2 Correlation Energy
We have recently employed the Dunning correlation-consistent basis sets for our pair natural orbital CBS extrapolation algorithm, Eqs. (2.1) and (2.2) [50]. The results produced a substantial improvement over the raw second-order energies, but were inferior to the 
extrapolations listed in Table 4.4. The residual underestimation of
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the magnitude of the second-order energy component after pair natural orbital extrapolations has two possible origins. Either the number of PNOs employed for the extrapolation was too small for the asymptotic formulae in Eqs. (2.1) and (2.2) to be applicable, or the correlationconsistent basis sets did not describe these PNOs to a sufficient accuracy. The former was less likely since the relative performance of the PNO extrapolations did not improve with increasing 
In either case, we expected (and actually found) a correlation of this residual error with 
as indicated in Fig. 4.8 for the example of the formaldehyde molecule. One extrapolation was good, but two were better. The agreement in Fig. 4.8 between the two types of extrapolation and the MP2-R12 limit is striking.
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The numerical results of this double extrapolation are presented in Table 4.6. The improvement is dramatic for the 
extrapolation of the cc-pVDZ and cc-pVTZ PNO extrapolated results, giving an absolute accuracy of better than 1 kcal/mol with the largest calculation again using just a [4s3p2df/3s2pd] basis set. These calculations are quite routine for molecules as large as naphthalene! Application to several 
species required one to two days each (depending on the specific example) on an SGI Origin 2000 with 8 193 MHz R10000 processors running Gaussian 98 [50].
The PNO extrapolations in Fig. 4.8 and Table 4.6 require localization of the occupied SCF orbitals to ensure size-consistency. In order to preserve this size-consistency for the CBS PNO extrapolations, we have restricted these 
extrapolations to a linear form, Eq. (6.2). The new double extrapolation employs this linear extrapolation of pairs of CBS2/cc-pVnZ calculations and thus is rigorously size-consistent. Note that the nonlinear N-parameter 
extrapolations using least-squares fits to more than N cc-pVnZ energies are not size-consistent [53, 55].
Without any extrapolation, energies computed with even the very large [7s6p5d4f3g2h1i/6s5p4d3f2g1h] cc-pV6Z basis sets are still 5.3 kcal/mol from the MP2-R12 limit for our test set of 12 small molecules.
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In contrast, a linear size-consistent 
extrapolation of just the MP2/cc-pVTZ and MP2/cc-pVQZ energies is accurate to 
(Table 4.4). If we try to further reduce the basis sets to ccpVDZ and cc-pVTZ, the error in the extrapolation increases to 
However, the new double extrapolation provides the complete basis set MP2 limit with an absolute accuracy of 
without recourse to basis sets larger than cc-pVTZ [4s3p2dlf/3s2pld] (Table 4.6).
7.3.The Higher-Order Correlation Energy
The higher-order contributions to the correlation energy [such as CCSD(T)-MP2] are more than an order of magnitude smaller than their second-order counterparts. However, the basis set convergence to the CCSD(T)-R12 limit does not follow the simple linear behavior found for the second-order correlation energy. This is a consequence of the interference effect described in Eq. (2.2). The full CI or CCSD(T) basis set truncation error is attenuated by the interference factor (Fig. 4.9). The CBS correction to the higher-order components of the correlation energy is thus the difference between the left-hand sides of Eqs. (2.2) and