CCSD+T

Figure B3.1.9 [83] displays the errors (in pieometres eompared to experimental findings) in the equilibrium bond lengths for a series of 28 moleeules obtained at the FIF, MP2-4, CCSD, CCSD(T), and CISD levels of theory using three polarized eorrelation-eonsistent basis sets (valenee DZ tlu-ough to QZ).

Clearly, the HF method, independent of basis, systematically underestimates the bond lengdis over a broad percentage range. The CISD method is neither systematic nor narrowly distributed in its errors, but the MP2 and MP4 (but not MP3) methods are reasonably accurate and have narrow error distributions if valence TZ or QZ bases are used. The CCSD(T), but not the CCSD, method can be quite reliable if valence TZ or QZ bases are used. 

Coupled cluster calculations are similar to conhguration interaction calculations in that the wave function is a linear combination of many determinants. However, the means for choosing the determinants in a coupled cluster calculation is more complex than the choice of determinants in a Cl. Like Cl, there are various orders of the CC expansion, called CCSD, CCSDT, and so on. A calculation denoted CCSD(T) is one in which the triple excitations are included perturbatively rather than exactly. 

CCSD(T)/aug-cc-pVDZ Atomization energy 18 kcal/mol mean abs. dev. 

Coupled Cluster methods, including doubles (energies and optimizations) or singles and doubles (energies only), and optional triples terms (CCD, CCSD, CCSD(T)). 

The concept has been generalized in the ONIOM method to include several layers, for example using high level ab initio (e.g. CCSD(T)) in the central part, lower-level electronic structure theory (e.g. MP2) in an intermediate layer and a force field to treat the outer layer. 

It has later been shown that the resulting equations are identical to CCSD where some of the terms have been omitted. The omitted terms are computationally inexpensive, and there appears to be no reason for using the less complete QCISD over CCSD (or QCISD(T) in place of CCSD(T)), although in practice they normally give very similar results. There are a few other methods which may be considered either as CISD with addition of extra terms to make them approximately size extensive, or as approximate versions of CCSD. Some of the methods falling into this category are Averaged... 

Analogously to MP methods, coupled cluster theory may also be based on a UFIF reference wave function. The resulting UCC methods again suffer from spin contamination of the underlying UHF, but the infinite nature of coupled cluster methods is substantially better at reducing spin contamination relative to UMP. Projection methods analogous to those of the PUMP case have been considered but are not commonly used. ROHF based coupled cluster methods have also been proposed, but appear to give results very similar to UCC, especially at the CCSD(T) level. 

Specifically, if T] < 0.02, the CCSD(T) metliod is expected to give results close the full Cl limit for the given basis set. If is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should preferentially be employed. Such methods are being developedbut have not yet seen any extensive use. 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

A comparison between Gl, G2, G2(MP2) and G2(MP2,SVP) is shown in Table 5.2 for the reference G2 data set the mean absolute deviations in kcal/mol vary from 1.1 to 1.6 kcal/mol. There are other variations of tlie G2 metliods in use, for example involving DFT metliods for geometry optimization and frequency calculation or CCSD(T) instead of QCISD(T), with slightly varying performance and computational cost. The errors with the G2 method are comparable to those obtained directly from calculations at the CCSD(T)/cc-pVTZ level, at a significantly lower computational cost. ... 

We need to look at the convergence as a function of basis set and amount of electron correlation (Figure 4.2). For the former we will use the correlation consistent basis sets of double, triple, quadruple, quintuple and, when possible, sextuple quality (Section 5.4.5), while the sensitivity to electron correlation will be sampled by the HF, MP2 and CCSD(T) methods (Sections 3.2, 4.8 and 4.9). Table 11.1 shows how the geometry changes as a function of basis set at the HF level of theory. 

The variation at the CCSD(T) level is shown in Table 11.3, with the ehange relative to the MP2 level given as A values. Additional eorrelation with the CCSD(T) method gives only small changes relative to the MP2 level, and the effeet of higher-order eorrelation diminishes as the basis set is enlarged. For H2O the CCSD(T) method is virtually indistingable from CCSDT. ... 

Table 11.3 H2O geometry as a function of basis set at the CCSD(T) level of theory...

The MP2 and CCSD(T) values in Tables 11.2 and 11.3 are for correlation of the valence electrons only, i.e. the frozen core approximation. In order to asses the effect of core-electron correlation, the basis set needs to be augmented with tight polarization functions. The corresponding MP2 results are shown in Table 11.4, where the A values refer to the change relative to the valence only MP2 with the same basis set. Essentially identical changes are found at the CCSD(T) level. 

The correlation energy is expected to have an inverse power dependence once the basis set reaches a sufficient (large) size. Extrapolating the correlation contribution for n = 3-5(6) with a function of the type A + B n + I) yields the cc-pVooZ values in Table 11.8. The extrapolated CCSD(T) energy is —76.376 a.u., yielding a valence correlation energy of —0.308 a.u. 

The magnitude of the core correlation can be evaluated by including the oxygen Is-electrons and using the cc-pCVXZ basis sets the results are shown in Table 11.9. The extrapolated CCSD(T) correlation energy is —0.370 a.u. Assuming that the CCSD(T) method provides 99.7% of the full Cl value, as indicated by Table 11.7, the extrapolated correlation energy becomes —0.371 a.u., well within the error limits on the estimated experimental value. The core (and core-valence) electron correlation is thus 0.063 a.u.. 

Inclusion of electron correlation normally lowers the force constants, since the correlation energy increases as a function of bond length. This usually means that vibrational frequencies decrease, although there are exceptions (vibrational frequencies also depend on off-diagonal force constants). The values calculated at the MP2 and CCSD(T) levels are shown in Tables 11.14 and 11.15. 

The MP2 treatment recovers the majority of the correlation effect, and the CCSD(T) results with the cc-pVQZ basis sets are in good agreement with the experimental values. The remaining discrepancies of 9cm , 13cm and lOcm are mainly due to basis set inadequacies, as indicated by the MP2/cc-pV5Z results. The MP2 values are in respectable agreement with the experimental harmonic frequencies, but of course still overestimate the experimental fundamental ones by the anharmonicity. For this reason, calculated MP2 harmonic frequencies are often scaled by 0.97 for comparison with experimental results. ... 

H2O CCSD(T) harmonic frequencies as a function of basis set (valence electrons only)... 

---