CCSDT, CCSD

Nitrogen pentafluoride represents an interesting contrast to oxirene. Oxirene is, on paper, a reasonable molecule there is no obvious reason why, however unstable it might be because of antiaromaticity [4] or strain, it should not be able to exist. On the other hand, NFs defies the hallowed octet rule why should it be more reasonable than, say, CHfi Yet a comprehensive computational study of this molecule left tittle doubt that it is a (relative) minimum on its potential energy surface [5]. The full armamentarium of post-HF methods, CASSDF, MRCl, CCSDT, CCSD(T), MP2 (section 5.4) and DFT (chapter 7) was employed here, and all agreed that Dan (section 2.6) NF5 is a minimum. 

The CCSD energies were obtained at the fc-CCSD(F12)/cc-pVQZ-F12 level. The corrections for connected triple excitations (T) were obtained at the fc-CCSD(T)/aug-cc-pwCVQZ level. The correction (Q) for connected quadruples contains the difference CCSDT-CCSD(T) calculated at the fc-CCSDT/aug-cc-pVDZ level and the (Q) term obtained at the fc-CCSDT(Q)/cc-pVDZ level. The correction for core valence correlation (CV) was obtained at the ae-CCSD(T)/aug-cc-pwCVQZ level. The harmonic zero-point vibrational energy (ZPVE) correction is supplemented by a zero-point-energy correction for hindered rotation (HR), and both scalar relativistic (MVD) and spin-orbit (SO) effects are taken into account. 

Correlation methods that can use ROHF reference determinants include MBPT(2), CCSD, CCSDT, CCSD(T), CCSDT-1, CCSDT-2, and CCSDT-3. 

The reader might wonder why not use %TAE[post-CCSD(7)], that is, %TAE[T4 -I- Tg] -I- %TAE[CCSDT -CCSD(T)], as a benchmark instead. However, the good performance of CCSD(T) itself results [32, 33] from a (mildly erratic) error compensation between T4 (which universally increases TAE) and Tg — (T) (which almost universally decreases it), and hence highly problematic molecules like singlet C2 look much more well-behaved on this criterion than they actually are. 

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. 

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. 

Among the various approximate methods for including the connected triple excitations, the CCSD(T) method is the most popular [19]. In this approach, the CCSD calculation is followed by the calculation of a perturbational estimate of the triple excitations. In addition to reducing the overall scaling with respect to the number of atoms K from K8 in CCSDT [see Eq. (2.5)] to K7 in CCSD(T), the CCSD(T) method avoids completely the storage of the triples amplitudes. 

CCSD(T) instead of CCSDT amounts to no more than 10 % of the total triples correction and 1 % of the total correlation energy, thus fulfilling our requirement for an acceptable approximate triples theory. 

The prerequisites for high accuracy are coupled-cluster calculations with the inclusion of connected triples [e.g., CCSD(T)], either in conjunction with R12 theory or with correlation-consistent basis sets of at least quadruple-zeta quality followed by extrapolation. In addition, harmonic vibrational corrections must always be included. For small molecules, such as those contained in Table 1.11, such calculations have errors of the order of a few kJ/mol. To reduce the error below 1 kJ/mol, connected quadruples must be taken into account, together with anhar-monic vibrational and first-order relativistic corrections. In practice, the approximate treatment of connected triples in the CCSD(T) model introduces an error (relative to CCSDT) that often tends to cancel the... 

It has been well known for some time (e.g. [36]) that the next component in importance is that of connected triple excitations. By far the most cost-effective way of estimating them has been the quasiper-turbative approach known as CCSD(T) introduced by Raghavachari et al. [37], in which the fourth-order and fifth-order perturbation theory expressions for the most important terms are used with the converged CCSD amplitudes for the first-order wavefunction. This account for substantial fractions of the higher-order contributions a very recent detailed analysis by Cremer and He [38] suggests that 87, 80, and 72 %, respectively, of the sixth-, seventh-, and eighth-order terms appearing in the much more expensive CCSDT-la method are included implicitly in CCSD(T). 

Would the use of full CCSDT [65] energies, instead of their quasi-perturbative-triples CCSD(T) counterparts, solve the problem Our experience has taught us that this generally leads to a deterioration of the results it has been shown (e.g. [66]) that the excellent performance... 

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]. 

Eq (16) can be derived in several different ways. The original derivation of eq (16), presented in ref 9, has been based on the analysis of the mathematical relationships between multiple solutions of nonlinear equations representing different CC approximations (CCSD, CCSDT, etc.). An elementary derivation of eq (16), based on applying the resolution of identity to an asymmetric energy expression. 

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. 

R Energy (H) HE CISD CISDT CCSD CCSDT 2POS 2POS + T1T2 3POS... 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

Table X shows the total energies as a function of bond length Tnn using several CT methods, as well as those using the HF, FCl, CASSCF, MRMP, CCSD, and CCSDT methods. Figure 9 plots the energy differences from the FCl results.

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