Wavefunction methods, CCSD

Figure 1. The shape of the potential curve for nitrogen in a correlation-consistent polarized double-zeta basis set is presented for the variational 2-RDM method as well as (a) single-reference coupled cluster, (b) multireference second-order perturbation theory (MRPT) and single-double configuration interaction (MRCl), and full configuration interaction (FCl) wavefunction methods. The symbol 2-RDM indicates that the potential curve was shifted by the difference between the 2-RDM and CCSD(T) energies at equilibrium.

Figure 4. Ground-state potential energy curves of Hs from 2-RDM and wavefunction methods are shown. MP2 and MP4 denote second- and fourth-order perturbation theories, while CCSD and CCSD) represent coupled cluster methods.

Ground-State Energies from the ACSE with V, NY, and M 3-RDM Reconstructions Compared with the Energies from Several Wavefunction Methods, Including Hartree-Fock (HF), Second-Order Many-Body Perturbation Theory (MP2), Coupled-Cluster Singles-Doubles (CCSD), and Full Configuration Interaction (FCI), for Molecules in Valence Double-Zeta Basis Sets."... 

In this chapter, I have highlighted how modem F12-based wavefunction methods can provide complete basis set results for molecular energies and properties. The basis-set requirements are lowered dramatically as compared to conventional approaches, while the extra computational cost for the evaluation of new, F12- specific terms, remains low. This dramatically extends the range of application of correlated methods, particularly for CCSD(T) calculations. 

The computational complexity of the coupled-cluster method truncated after a given excitation level m - for example, m = 2 for CCSD - may be discussed in terms of the number of amplitudes (Nam) in the coupled-cluster operator and the number of operations (Nop) required for optimization of the wavefunction. Considering K atoms, each with Nbas basis functions, we have the following scaling relations ... 

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). 

While CCSD and especially CCSD(T) are known [36] to be less sensitive to nondynamical correlation effects than low-order perturbation theoretical methods, some sensitivity remains, and deterioration of W1 and W2 results is to be expected for systems that exhibit severe nondynamical correlation character. A number of indicators exist for this, such as the T diagnostic of Lee and Taylor [64], the size of the largest amplitudes in the converged CCSD wavefunction, and natural orbital occupations of the frontier orbitals. 

Several of the methods referred to in this chapter use the URCCSD(T) procedure in which a spin-unrestricted CCSD(T) calculation is performed on a high-spin RHF reference wavefunction, as implemented in the MOLPRO program. H. J. Werner, P. J. Knowles, R. D. Amos, A. Bemhardsson, A. Beming, P. Celani, D. 

In the SOPPA(CCSD) method [36] the Mqller-Plesset correlation coefficients and are replaced in all SOPPA matrix elements by the corresponding coupled cluster singles and doubles amplitudes and r , whereas in the earlier CCSDPPA method [52,53] only some of the Mqller-Plesset correlation coefficients were replaced. Although SOPPA(CCSD) is based on a CCSD wavefunction, it is still only correct through second order and not the linear... 

In general, the results of the perturbation theory based methods, SOPPA, SOPPA(CCSD), MPn and CCSD become smaller with increasing level of theory, whereas the results of the CASSCF/RASSCF calculations go through a maximum for the or °°°CAS wavefunctions depending on the molecule. 

Figure 4. Hierarchy of the SF models. Similar to the non-SF SR methods, the SF models converge to the exact n-electron wavefunction when the spin-flipping operator 0 includes up to n-tuple excitations. For example, the SF-CCSD model...

In the optimized orbitals CCD (00-CCD or OD) model, the orbitals are optimized variationally to minimize the total energy of the 00-CCD wavefunction. This allows one to drop single excitations from the wavefunction. Conceptually, 00-CCD is very similar to the Brueckner CCD (B-CCD) method. Both 00-CCD and B-CCD perform similarly to CCSD in most cases. 

A different way to approximate SS-MRCC is the so-called externally corrected CCSD (ec-CCSD) [13-17], The ec-CCSD method is based on the Coupled Cluster Approach (CCA). In CCA the exact wavefunction is written in an exponential form,... 

SC) CAS-SDCI is a size-extensive method, while ec-CCSD is almost size-extensive (CC procedure is itself size-extensive, but the corrections are not, provided they come from a truncated variational Cl wavefunction). 

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