CCSDT corrections

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... 

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.

The most challenging and therefore the most telling example for excitation theories is C2, whose ground state has a severe multi-determinant wave function. It is known that, to obtain quantitative results (errors < 0.1 eV), one must resort to EOM-CCSDTQ [134], Figure 2-11 compares EOM-CCSD, CCSDT, and various perturbation corrections to EOM-CCSD with FCI for three excited states of C2 [126], EOM-CCSD, which is usually highly accurate, is inadequate for the two states A and B with errors approaching 2 eV. All variants of the perturbation corrections are... 

Watts JD, Bartlett RJ (1996) Iterative and non-iterative triple excitation corrections in coupled-cluster methods for excited electronic states The EOM-CCSDT-3 and EOM-CCSD(r) methods. Chem Phys Lett 258 581-588. 

As pointed out in the preceding section, the CCSD energy contains contributions identical to those of the MBPT(2) and MBPT(3) energy, but lacks triple-excitation contributions necessary for MBPT(4). Thus, a natural approach to the triples problem is to correct the CCSD energy for the missing MBPT(4) terms,using the CCSDT similarity-transformed Hamiltonian,... 

J. D. Watts and R. J. Bartlett, Chem. Phys. Lett., 258, 581 (1996). Iterative and Noniterative Triple Excitation Corrections in Coupled-Cluster Methods for Excited Electronic States— The EOM-CCSDT-3 and EOM-CCSD(T) Methods. 

The analysis of the several different CC approaches in terms of the fifth-order energy contributions points out that within an n6 dependent scheme, i.e., LCCD to CCSD in Table I, the CCSD is much preferred since it accounts for nearly one-third of all terms and avoids potential singularities in LCCD.42 It may also be observed that it pays off to include, even partially, the triple contribution, as was done in the CCSDT-1 method.10 In this model the number of terms is nearly doubled as compared to CCSD, and, of course, this method is correct through the fourth-order energy and the second-order wave function. Also, the connected T contributions are numerically important.11-34... 

We may conclude that the recommended CC approximation would be CCSDT, which accounts for 80% of the fifth-order terms, but the basis set dependence is still ns. The inclusion of T4 in the coupled-cluster scheme would appear to cause a significant increase in computational time for what is normally considered to be a fairly small correction since T4< T for most nonmetallic cases. Of course, if a reference function is sufficiently poor, T4 and even higher clusters could be important. However, as was shown in the preceding section, the missing, i.e., beyond CCSDT, fifth-order MBPT energy terms that arise from T may be calculated by supplementing the CCSDT code with a few additional diagrams, the basis set dependence of which is n6 or less, to introduce most of the correction due to T4. 

For nonlinear (magneto-) optical properties, calculations of an accuracy close to that of modern gas phase experiments require - similar to what has also been found for other properties like structures [79, 109], reaction enthalpies [79, 110, 111], vibrational frequencies [112, 113], NMR chemical shifts [114], etc. - at least an approximate inclusion of connected triple excitations in the wavefunction. This has been known for years now from calculations of static hyperpolarizabilities with the CCSD(T) approximation [9-13]. CCSD(T) accounts rather efficiently for connected triples through a perturbative correction on top of CCSD. For the reasons pointed out in Section 2.1 CCSD(T) is, as a two-step approach, not suitable for the calculation of frequency-dependent properties. Therefore, the CC3 model has been proposed [56, 58] as an alternative to CCSD(T) especially designed for use in connection with response theory. CC3 is an approximation to CCSDT - alike CCSDT-la and related methods - where the triples equations are truncated such that the scaling of the computational efforts with system size is reduced to as for CCSD(T),... 

If we further consider only these T3 terms in CCSDT-1 noniteratively (i.e., we do not let the T3 coefficients change), we obtain highly accurate but considerably less expensive methods. This is done as follows. The fourth-order triple excitation correction may be written as... 

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