CCSD /6 calculations

Since the singly excited determinants effectively relax the orbitals in a CCSD calculation, non-canonical HF orbitals can also be used in coupled cluster methods. This allows for example the use of open-shell singlet states (which require two Slater determinants) as reference for a coupled cluster calculation. 

As these expressions correspond to the CC energy derivative, they must give size-extensive results. However, the price we pay is that the energy of a given order requires wave function contributions of the same order. Furthermore, these non linear terms are difficult to evaluate. The quadartic in term in second-order, requires comparable difficulty to the quadratic terms in a CCSD calculation... 

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. 

In the second row of Table 1.4, we have listed the corrections to the Hartree-Fock energies that are obtained from CCSD calculations. Clearly, we now have a better description of the atomization process, the error in the calculated AE being only -19.6 kJ/mol (2 %). Still, we are far away from the prescribed target accuracy of 1 kJ/mol. 

As a result, one may justifiably extrapolate the (T) contribution from smaller basis sets than its CCSD counterpart in W1 theory, we extrapolate from the small and medium basis sets, and in W2 theory from the medium and large basis sets. This means that the most extensive basis sets in the calculations, namely large in W1 theory and extra large in W2 theory only require CCSD calculations, which are both much less expensive than CCSD(T) and much more amenable to direct algorithms such as those described in Refs. 40-41. 

We have compared our MCSCE results for the vibrational ground state with CCSD, SOPPA, and SOPPA(CCSD) calculations. In particular we have investigated the importance of the PEC on the ZPVCs and find that there are significant differences between LiH and HF. In LiH the CCSD results for the ZPVC are very close to the MCSCF results independent on whether the CCSD or MCSCF PEC was employed. Similarly, the differences between SOPPA(CCSD) calculations with either the CCSD or the MCSCF energy surface are very small. In HF, on the other hand somewhat larger differences are found if the CCSD polarizabilities are averaged over the CCSD PEC and the difference between... 

SOPPA(CCSD) calculations with the CCSD or MCSCF PEC are also larger. In general the differences in the ZPVC are larger between the different PEC than between the different linear response methods. The SOPPA(CCSD) results for the equilibrium geometry as well as the vibrationally averaged polarizabilities are in both molecules in better agreement with the MCSCF results than the pure SOPPA values. 

SOPPA(CCSD) and RAS332 SD is interchanged. One can therefore conclude that for the molecules studied here the results of SOPPA(CCSD) calculations are comparable to results of other state-of-the-art methods. 

Finally, the results for CH4 show quite a different pattern. The correlation corrections are positive and the results of MPn/CCSD calculations differ more from the results of SOPPA/SOPPA(CCSD) and the RASSCF calculations than for the other molecules. Further investigations of the correlation effects in this molecule are necessary. 

The fact that the Tx and Ti clusters obtained by solving the QECCSD and ECCSD equations are significantly better than the T and clusters resulting from the standard CCSD calculations can also be seen by calculating the overlaps of the normalized CCSD, QECCSD, and ECCSD wave functions, J,CCSD) jvpQEccsD pEccsD respectively, as defined by eq (60), with... 

In consequence, the most expensive steps of the ground- and excited-state calculations using methods based on the MMCC(2,3) approximation are essentially identical to the n nf noniterative steps of the ground-state CCSD(T) calculations uo and are the numbers of occupied and unoccupied correlated orbitals, respectively). Similar remarks apply to the memory and disk-space requirements. Clearly, these are great simplifications in the computer effort, compared to the higher-level EOMCC approaches, such as EOMCCSDT [43,44,55,56], particularly if we realize that we only have to use the Ti and T2 clusters, obtained in the CCSD calculations, to construct matrix elements of that enter 9Jt (2), Eqs. (58) and (59). In... 

The CR-EOMCCSD(T) approach is a purely single-reference, blackbox method based on the MMCC(2,3) approximation, in which the wave function jH/ ) entering Eq. (67) is designed by using the singly and doubly excited cluster amplitudes tl and defining Ti and T2, respectively, obtained in the CCSD calculations, and the zero-, one- and two-body amplitudes ro p), rl pL) and r pi), defining R, 1, and respectively. 

The CCSD calculations are similar, both in methodology and in the accuracy of the results obtained, to calculations performed with Pople s quadratic Cl method. Like CCSD calculations, QCISD calculations also explicitly include single and double excitations and the effects of quadruple excitation in QCISD are obtained from quadrature of the effects of double excitations. However, CCSD does contain terms for the effects of excitations beyond quadruples, which are absent from QCISD. 

Table 7.2 Valence correlation energies (—Econ, hiEa) from standard and R12 CCSD calculations and from extrapolation using Eq. (7.57) for seven closed-sheU singlet molecules...

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