Model space selection

Fig. 4 Stereo views (in transparent, space-filling models) of selected hydrophobic interactions in the Fab-peptide and -pentasaccharide complexes. A Trp P3 buried in the hydrophobic cavity formed by CDR L3, His H58 and Trp H47. B Interactions of the side chain of Met P5 with Trp H33. C and D Interactions of the peptide and pentasaccharide, respectively, with His L27D and Tyr L32 (the pocket which accommodates the GlcNAc D methyl group). In (C) the hydrogen bond between the Asp P2 side chain and His L27D NE2 is also shown. A corresponding hydrogen bond between Rha Al 3-OH and His L27D NE2, shown in (D), represents an element of structural mimicry. Reproduced from [80]. 2003 by The National Academy of Sciences of the USA...

The principal component space has several advantages over selected X-variable spaces. Concerns about redundancy are eliminated because the PCs are orthogonal to one another. In addition, because each PC explains the most remaining variance in the X-data, it is often the case that fewer PCs than original X-variables are needed to capture the same information in the X-data. This leads to simpler classification models, less susceptibility to overfitting through the use of too many dimensions in the model space, and less noise in the model. Nonetheless, it is still a good idea to use validation techniques even when PCs are used to define the space. 

The microscopic derivation of a boson hamiltonian from a fermion one is basically a two step process. In the first step, one has to select the collective subspace of the shell model space. For the IBM this means truncating the shell model space to the space of collective S-D pairs. In the second step, this space has to be mapped onto the s-d boson space. 

Both the subspace angle and the expected prediction difference can be used to evaluate the model discrimination capability of a design d over a model space T. In Section 4, the selection of orthogonal designs using criteria based on these measures is discussed. 

Fernandez A, Rogale K (2004) Sequence-space selection of cooperative model proteins. J Phys A Math Gen 37 197-202... 

The similarity of shape of these two curves is an indication of the goodness-of-tit of the selected minimal model space. One should keep in mind that as one extends this space, the computed curves should converge towards the experimental curve. 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

Crucial is the selection of the active spinor space that determines the model space. It should be large enough to take all important interactions between nearby states into account, but also be small enough to keep the formalism computationally tractable and numerically stable. Direct diagonalization of this model space would be equivalent to a small Cl expansion and does of course not take dynamic electron correlation into account. Correlation is included by defining the so-called wave operator U... 

A major advantage of the intermediate Hamiltonian approach is the flexibility in selecting the model space. This has been a major problem in applying the Fock-space scheme, as described at the beginning of this section. While in the Fock-space coupled cluster method one may feel lucky to find any partitioning of the function space into P and Q with convergent CC iterations, the intermediate Hamiltonian method makes it possible for the first time to vary the model space systematically and study the effect upon calculated properties. An example is given in Table 3, which shows the dependence of the calculated electron affinity of Cs on the model spaces Pm and Pi [55]. 

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