Coupled cluster accuracy

The accuracy of these two methods is very similar. The advantage of doing coupled cluster calculations is that it is a size extensive method (see chapter 26). Often, coupled-cluster results are a bit more accurate than the equivalent... 

There is a variation on the coupled cluster method known as the symmetry adapted cluster (SAC) method. This is also a size consistent method. For excited states, a Cl out of this space, called a SAC-CI, is done. This improves the accuracy of electronic excited-state energies. 

Correlated calculations, such as configuration interaction, DFT, MPn, and coupled cluster calculations, can be used to model small organic molecules with high-end workstations or supercomputers. These are some of the most accurate calculations done routinely. Correlation is not usually required for qualitative or even quantitative results for organic molecules. It is needed to obtain high-accuracy quantitative results. 

Figures 11.9 and 11.10 compare the performance of the CCSD and CCSD(T) methods, based on either an RFIF or UHF reference wave function. Compared to the RMP plot (Figure 11.7), it is seen that the infinite nature of coupled cluster causes it to perform somewhat better as the reference wave function becomes increasingly poor. While the RMP4 energy curve follows the exact out to an elongation of 1.0A, the CCSD(T) has the same accuracy out to - 1.5 A. Eventually, however, the wrong dissociation limit of the RHF wave also makes the coupled cluster methods break down, and the energy starts to decrease.

If affordable, there is a range of very accurate coupled-cluster and symmetry-adapted perturbation theories available which can approach spectroscopic accuracy [57, 200, 201]. However, these are only applicable to the smallest alcohol cluster systems using currently available computational resources. Near-linear scaling algorithms [192] and explicit correlation methods [57] promise to extend the applicability range considerably. Furthermore, benchmark results for small systems can guide both experimentalists and theoreticians in the characterization of larger molecular assemblies. 

From these expressions, it is evident that the steep increase of the computational cost with X (the basis-set hierarchy) and m (the coupled-cluster hierarchy) severely restricts the levels of theory that can be routinely used for large systems or even explored for small systems. Nevertheless, we shall see that, with current computers, it is possible to arrange the calculations in such a manner that chemical accuracy (of the order of 1 kcal/mol 4 kJ/mol) can be achieved - at least for molecules containing not more than ten first-row atoms. 

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... 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

The field of quantum chemistry has seen tremendous development over the last thirty years. Thanks to high-accuracy models such as coupled-cluster theory and standardized, widely available program packages such as Gaussian 98, what was once merely an esoteric tool of a few specialists has evolved into an indispensable source of knowledge for both the prediction and the interpretation of chemical phenomena. With the development of reduced scaling algorithms for coupled cluster... 

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

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]. 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

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