Full CC limit

It can be easily verified that the stationarity of AfJ(E,T) with respect to E yields Eq. (226). In the exact (full CC) limit, the system of equations defined by Eq. (226) becomes equivalent to standard CC equations for T, Eq. (35), and AE(E,T) becomes identical to AE c [75]. It can be further demonstrated that the stationarity of AE(E, T) with respect to T yields (again, in the full CC limit) the lambda equation of the analytic gradient SRCC theory,... 

Properties computed using Eq. (11) are not size-extensive, however, because of the appearance of unlinked diagrams arising from disconnected terms implicit in Eq. (3). Such terms naturally cancel if the T and operators are not truncated, implying that the EOM-CC method is formally exact in the full-CC limit. 

Clearly, in the exact, full Cl or full CC, limit, there exists an excitation operator S, such that (1 + A) = so that one can always give the dual CC state ( ol a completely exponential form. 

Approximate many-electron wave functions are then constructed from the Hartree-Fock reference and the excited-state configurations via some sort of expansion (e.g., a linear expansion in Cl theory, an exponential expansion in CC theory, or a perturbative power series expansion in MBPT). When all possible excitations have been incorporated (S, D, T,. .., for an -electron system), one obtains the exact solution to the nonrelativistic electronic Schrodinger equation for a given AO basis set. This -particle limit is typically referred to as the full Cl (FCI) limit (which is equivalent to the full CC limit). As Figure 5 illustrates, several WFT methods can, at least in principle, converge to the FCI limit by systematically increasing the excitation level (or perturbation order) included in the expansion technique. 

We will now look at how different types of wave functions behave when the O-H bond is stretched. The basis set used in all cases is the aug-cc-pVTZ, and the reference curve is taken as the [8, 8J-CASSCF result, which is slightly larger than a full-valence Cl. As mentioned in Section 4.6, this allows a correct dissociation, and since all the valence electrons are correlated, it will generate a curve close to the full Cl limit. The bond dissociation energy calculated at this level is 122.1 kcaPmol, which is comparable to the experimental value of 125.9 kcal/mol. 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

Truncation of C at the single- and double-excitation level (CISD) leads to a wavefunction with exactly the same number of amplitudes (cf and c j ) as that needed for the CCSD wavefunction (tf and However, the latter implicitly includes higher excitation levels (triples and quadruples) by the inclusion of T products in the power series expansion of e. Such products are commonly referred to in the literature as disconnected wavefunction contributions/ Both the Cl and CC methods will produce exact wavefunctions if one does not truncate C (full Cl) or T (full CC). In fact, in the limit of exact linear and exponential wavefunction expansions, a relationship between the Cl and CC amplitudes may be developed that reveals the factorization of each level of Cl excitation into connected and disconnected components, for example,... 

Which arise from the so-called B matrix of response theory. It is easily shown that B vanishes in the full B-CC or full OO-CC limits. 

The exponential operator T creates excitations from Po according to T = Ti + Ti + T-i -F , where the subscript indicates the excitation level (single, double, triple, etc.). This excitation level can be truncated. If excitations up to Tn (where N is tile number of electrons) were included, cc would become equivalent to the full configuration interaction wave function. One does not normally approach this limit, but higher excitations are included at lower levels of coupled-cluster calculations, so that convergence towards the full Cl limit is faster than for MP calculations. 

The magnitude of the core correlation can be evaluated by including the oxygen Is-electrons and using the cc-pCVXZ basis sets the results are shown in Table 11.9. The extrapolated CCSD(T) correlation energy is —0.370 a.u. Assuming that the CCSD(T) method provides 99.7% of the full Cl value, as indicated by Table 11.7, the extrapolated correlation energy becomes —0.371 a.u., well within the error limits on the estimated experimental value. The core (and core-valence) electron correlation is thus 0.063 a.u.. 

CC) methods, which have largely superseded Cl methods, in the limit can also be used to give exact solutions but again with same prohibitive cost as full Cl. As with Cl, CC methods are often truncated, most commonly to CCSD (N cost), but as before these can still only be applied to systems of modest size. Finally, Moller-Plesset (MP) perturbation theory, which is usually used to second order (MP2 has a cost), is more computationally accessible but does not provide as robust results. 

The operation of Eq. (3.3) is illustrated by the results given in Table 2 out of 48 molecules of the cc-pVTZ set. They are listed in order of increasing correlation energy. The first column of the table lists the molecule. The next 6 columns show how many orbitals and orbital pairs of the various types are in each molecule, i.e. the numbers Nl, Nb, Nu, Nlb etc. The seventh column lists the CCSD(T)/triple-zeta correlation energy and the eight column lists the difference between the latter and the prediction by Eq. (3.3). The mean absolute deviation over the entire set of cc-pVTZ data set is 3.14 kcal/mol. For the 18 molecules of the CBS-limit data set it is found to be 1.57 kcal/mol. The maximum absolute deviations for the two data sets are 11.29 kcal/mol and 4.64 kcal/mol, respectively. Since the errors do not increase with the size of the molecule, the errors in the estimates of the individual contributions must fluctuate randomly within any one molecule, i. e. there does not seem to exist a systematic error. The relative accuracy of the predictions increases thus with the size of the system. It should be kept in mind that CCSD(T) results can in fact deviate from full Cl results by amounts comparable to the mean absolute deviation associated with Eq. (3.3). 

Fig. 8 Osmotic coefficient as a function of counterion concentration cc for the poly(p-phenylene) systems described in the text. The solid line is the PB prediction of the cylindrical cell-model, the dashed curve is the prediction from the correlation corrected PB theory from Ref. [58]. The full dots are experiments with iodine counterions and the empty dots are results of MD simulations described in ref. [29,59]. The Manning limiting value of l/2 is also indicated...

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

In Table 3, the MP2, MP3, EN2 and EN3 energies for the lowest 1Hi state of CH2 and NHj are compared with the SCF (ROHF), UGA-CC [at both linear and quadratic levels of the first interacting space (is) approximation, referred to as L-CCSD(is) and CCSD(is), respectively], and various Cl [including full Cl (FCI)] results, considering both DZ and DZP basis sets. (For full SD space CCSD results and other limited Cl results, see Table 2 of (13)). These results indicate that MP2 underestimates the exact correlation energy by about 12-15%, yielding 84.8 and 88.5% of the correlation energy for the 1H/ state of CH2 and NH, respectively, with... 

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