CCSDT level

In Table 1.2, we have listed the valence cc-pVDZ electronic energies and AEs of N2 and HF at different levels of coupled-cluster theory. The energies are given as deviations from the FCI values. Comparing the different levels of theory, we note that the error is reduced by one order of magnitude at each level. In particular, at the CCSDT level, there is a residual error of the order of a few kJ/mol in the calculated energies and AEs, suggesting that the CCSDTQ model is usually needed to reproduce experimental measurements to within the quoted errors bars (often less than 1 kJ/mol). 

Static value estimated at CCSDT level frequency dependence estimated using CC3 analytic response (and found to be negligible). 

Along the ordinate, a sequence of correlation-consistent cc-pVXZ basis sets with X > 2 is depicted. Along the abscissa, the FCI limit is approached—beginning with Hartree-Fock theory and followed by the first correlated level, at which the single and double excitations are described by MP2 perturbation theory. The same excitations are subsequently treated by coupled-cluster theory at the CCSD level, which is then further improved upon by a perturbation treatment of the triple excitations at the CCSD(T) level. At the CCSDT level, the triple excitations are fully treated by coupled-cluster theory, and so on. In this manner, the hierarchy Hartree-Fock -> MP2 — CCSD -> CCSD(T) > CCSDT —---------------> FCI is established. 

In this section, we investigate the accuracy of ab initio electronic-structure predictions of bond distances [21. In Section 15.3.1, we review the experimental data on which the investigation is based. Next, in Sections 15.3.2-15.3.5, we consider the statistical measures of errors, piecing together a picture of the performance of the standard models with respect to the calculation of equilibrium bond distances. After a discussion of higher-order effects at the CCSDT level in Section 15.3.6 and core correlation in Section 15.3.7, we attempt to rationalize the behaviour of the different models in Section 15.3.8. We conclude our discussion in Section 15.3.9, which... 

The simplest treatment of the triples occurs at the MP4 and CCSD(T) levels, the two methods differing in that, at the CCSD(T) level, the doubles are fiilly relaxed (in the absence of the triples), whereas no such relaxation occurs at the MP4 level. In agreement with this observation, we find that the MP4 and CCSD(T) models both lengthen the bonds (relative to MP3 and CCSD) but that the MP4 bonds are the longest since neither the doubles nor the triples have been fully relaxed. Finally, relaxation of the triples at the CCSDT level contracts the bonds somewhat relative to CCSD(T). 

It is noteworthy that the inclusion of the full triples correction does not improve the agreement with the experimental atomization energies, indicating that there is an element of error cancellation at the CCSD(T) level which is absent at the CCSDT level. For better agreement with experiment, other corrections must also be included — most important, the contributions from the connected quadruples. 

In the preceding sections, we discussed the energy differences associated with atomizations and chemical reactions. In the present section, we consider the smaller differences associated with conformational changes [101 the barrier to linearity of water in Section 15.9.1, the inversion barrier of ammonia in Section 15.9.2 and the torsional barrier of ethane in Section 15.9.3. All barriers have been studied at the Hartree-Fock, MP2, CCSD, CCSD(T) and CCSDT levels of theory in the cc-pVXZ, aug-cc-pVXZ and cc-pCVXZ basis sets, with the valence electrons correlated in the valence... 

As for water, the Hartree-Fock model provides the dominant contribution to the barrier, with an estimated basis-set limit of 19.3 kJ/mol. The correlation contribution is small and positive at all levels but decreases with the cardinal number - see Figure 15.20. At the cc-pCV5Z level, this reduction leads to a near-zero MP2 correlation correction. By contrast, the CCSD correction of 1.7 kJ/mol is close to the CCSD(T) correction of 2.3 kJ/mol. The triples correction is quite stable, in particular for the augmented sets. Moreover, a full optimization of the triples at the CCSDT level hardly changes the CCSD(T) results. From the smallness of the triples correction, we expect the contributions from higher connected excitations to be negligible. 

As with atomic EAs, comparisons to higher-level calculations suggest that correlation effects on VDEs beyond the CCSD(T) level are quite small for molecular anions.Consider, for example, the notoriously challenging HNC and HCN anions, " whose binding energies are only 0.004 eV and w 0.002 eV, respectively, with the VDE for HCN arising almost entirely from electron correlation effects. For these two species, VDEs computed at the CCSD(T) level and the CCSDT level agree to within 0.001 eV. The (H20) anion provides another example here, the VDE computed at the CCSD(T) leveP lies within the statistical error bars of a quantum Monte Carlo (QMC) calculation, the latter of which is free of basis-set artifacts and does not require truncation of the excitation level. 

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. ... 

The HF level as usual overestimates the polarity, in this case leading to an incorrect direction of the dipole moment. The MP perturbation series oscillates, and it is clear that the MP4 result is far from converged. The CCSD(T) method apparently recovers the most important part of the electron correlation, as compared to the full CCSDT result. However, even with the aug-cc-pV5Z basis sets, there is still a discrepancy of 0.01 D relative to the experimental value. 

The CCS, CC2, CCSD, CC3 hierarchy has been designed specially for the calculation of frequency-dependent properties. In this hierarchy, a systematic improvement in the description of the dynamic electron correlation is obtained at each level. For example, comparing CCS, CC2, CCSD, CC3 with FCI singlet and triplet excitation energies showed that the errors decreased by about a factor 3 at each level in the coupled cluster hierarchy [18]. The CC3 error was as small as 0.016 eV and the accuracy of the CC3 excitation energies was comparable to the one of the CCSDT model [18]. 

As has been pointed out in the past (e.g. concerning the linear-cyclic equilibrium in Ceand Cio carbon clusters (40)), Hartree-Fock underestimates the resonance stabilization of aromatic relative to non-aromatic systems (in the case at hand, between the N- and / -protonated isomers) and MP2 overcorrects. The structures are found to be nearly isoenergetic at the CCSD level inclusion of connected triple excitations favors the N-protonated ion. The direction of the effect of connected quadruples is somewhat unclear, and a CCSD(TQ) or CCSDT(Q) calculation impossible on systems this size, but the contribution will anyhow be much smaller in absolute magnitude than that of connected triple excitations, particularly for systems like these which are dominated by a single reference determinant. We may therefore infer that at the full Cl limit, the N-protonated species will be slightly more stable than its / -protonated counterpart. 

Even CCSDT is not capable of adequately describing certain doubly excited states, and several extensions that incorporate connected quadruple excitations (i.e. methods that include T4 in the ground state) have been implemented. Unless some restrictions are placed on the subspaces for which quadruple excitations are possible, methods such as EOM-CCSDTQ will not be practical in other than benchmark model calculations. Such calculations are, of course, of some importance since one can calibrate approximate treatments of quadruple excitations by comparisons with the full EOM-CCSDTQ method, for example. Even higher excitation levels have been implemented and compared with FCI results [48-51], Again, these methods are not expected to be generally applicable to anything other than a model system, but they are of great value as benchmarks. 

In their CCSDT work, Kucharski et al. [53] obtained vertical excitation energies for three excited states of N2 and four excited states of CO with up to the aug-cc-pVTZ basis set. The CCSDT errors in the excitation energies with this basis set for the three states of N2 ( ng, lXu, 1 Au) are 0.11, 0.10, and 0.16 eV, respectively. These errors are somewhat less than those obtained at the CCSD level (0.19, 0.20, and 0.30 eV). For CO there is less uniformity, but for three of the four states, the CCSDT treatment provides somewhat closer agreement with experiment. The CCSDT errors for the 1n, A, and 1X+ states are 0.03, 0.17, -0.05, and... 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

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