Truncation level

The kernel of pk on a truncated Barsotti-Tate group of level n with n > k is a truncated level k Barsotti-Tate group (see [Me] II 3.3.11). 

At a higher level of complexity, correlation energies are computed assuming tliat effects associated with basis-set incompleteness and, say, truncated levels of perturbation theory,... 

As we stated above, SAPT is formulated in a top down manner. Eq. (6) then forms the top going down to workable equations, one is forced to introduce a multitude of approximations. In practice, i is restricted to the values 1 and 2 interactions of first and second order in Different truncation levels for j + k are applied, depending on the importance of the term (and the degree of complexity of the formula). Working out the equations to the level of one- and two-electron integrals is a far from trivial job. This has been done in a long series of papers that use techniques from coupled cluster theory and many-body PT see Refs. [147,148] for references to this work and a concise summary of the formulas resulting from it. 

The improvements within the hierarchy of A -electron models is probably a more complicated task. There are a couple of quantum-chemistry theories that allow us to approach the exact Schrodinger equation systematically. Among them the coupled-cluster (CC) method represents probably the most successful approach. It can be applied to relatively large systems and the theory is both size-extensive and size-consistent. So far the only way to approach the exact Schrodinger equation, within the hierarchy of the A -electron models (following the horizontal axis on Figure 1), is the systematic extension of the excitation level. In CC theory there is a series of models that refer to the way the cluster operator is truncated (CCS, CCSD, CCSDT, CCSDTQ and so on). In the limit of the untruncated cluster operator the CC wave function becomes equivalent with full Cl, which is the exact solution of the Schrodinger equation within a particular basis set. The truncation level indicates, in some sense, the accuracy of the model which is almost always limited by the available computational resources. 

This is in contrast with both perturbation and Cl methods and therefore CC theory should provide a better description of electron correlation effects at a given truncation level. 

Fig. 4.15 Relative deviation of hydrodynamic radii Ri,.t calculated with MVE for varying truncation levels L of multipole expansion from the correct value (L = 10) left dependency on L for straight chains, hcp DLCA aggregates N = 10-500), right dependency on aggregation number N for DLCA aggregates...

Results of this section are based on MVE method—-mainly because of its high accuracy and because it allows for insight into the velocity field of aggregates. The truncation level was set to L = 2 in order to study aggregates with up to 1000 primary particles. According to Fig. 4.15, that means an error of less than 0.1 %. 

Table 4.1 Reduced FSF model orders corresponding to different truncation levels for the second order system...

The reduced FSF model order n corresponding to maximum truncation levels of 10%, 5% and 1% are listed in Table 4.1 for the above examples, where the neglected frequency response coefficients all have magnitudes less them the indicated percentage of the steady state gain. Note that the reduced orders given in Table 4.1 are independent of the time constant T,... 

Despite the considerable computational effort required by CC methods, even at a low truncation level as EOM-CCSD, the results shown in Section 3 indicate that the effort is certainly worth it. Research in areas of energy and material science, as well as atmospheric science, benefits from predictions obtained from theoretical calculations. Often, in fact, new chromophores can be studied before they are synthetized in the lab, or unknown compounds can be screened according to their photochemical characteristics. In order to do that, however, reliable theoretical paradigms are necessary, which provide a consistent level of accuracy across a variety of different chromophoric moieties. Although DFT still provides the best compromise between cost and accuracy, the examples... 

The accuracy of the MPP is influenced by the FE mesh, the truncation level M of the random parameter field, the partition of E, and the choice of the ansatz functions both in spatial and random domain. These parameters can be gradually adapted such that the MPP is computed with a prescribed accuracy. 

In Figure 11.4, we have plotted the binomial distribution of excitation levels for 1, 20, 40, 60, 80 and 100 noninteracting monomers, assuming IFd = 0.05. According to (11.3.34), the mean numbers of double excitations in these supersystems are 0.05, 1, 2, 3, 4 and 5, corresponding to mean excitation levels of 0.1, 2, 4, 6, 8 and 10, respectively. As the number of monomers increases, the distribution approaches the Gaussian (11.3.36). Clearly, any approximation to the FCl wave function based on the truncation of the expansion at a fixed excitation level can describe accurately only systems in which the typical excitation level is lower than the truncation level. The use of Cl expansions for systems containing many electrons is therefore, at best, laborious. 

Let us now consider the treatment of the allyl radical at the Cl level. Obviously, we may base our Cl calculation on the RHF or UHF orbitals. At the PCI level, the choice of orbitals does not matter since, in a complete expansion, the same solution is recovered in either case. At the truncated level, however, the RHF- and UHF-based Cl wave functions will differ. In Figure 11.8, we have plotted the CISD/cc-pVDZ potential-energy curves obtained using the RHF orbitals (full line) and the UHF orbitals (dotted lines). These curves should be compared with the corresponding Hartree-Fock curves in Figure 10.6. 

Without truncation, the FCI and full coupled-cluster functions contain the same number of parameters since there is then one connected cluster amplitude for each determinant. In this special case, the Cl and coupled-cluster models provide linear and nonlinear parametrizations of the same state and there is then no obvious advantage in employing the more complicated exponential parametrization. The advantages of the cluster parametrization become apparent only upon tmncation and are related to the fact that, even at the truncated level, the coupled-cluster state contains contributions from all determinants in the FCI wave function, with weights obtained from the different excitation processes leading to the determinants. 

As a result, only singles and doubles amplitudes contribute directly to the coupled-cluster energy -irrespective of the truncation level in the cluster operator. Of course, the higher-order excitations contribute indirectly since all amplitudes are coupled by the projected equations (13.2.23). 

In the coupled-cluster calculations, the error decreases by about the same factor at each excitation level. Also, at a given truncation level, the error is significantly smaller than that of the Cl energy, which converges in a less systematic and satisfactory manner because of the neglect of higher-order disconnected clusters. As an example, the CCSD error falls between the CISDT and CISDTQ errors... 

---