Frozen environment approximation

The alignments due to the pairwise potential are computed using the first iteration of the frozen environment approximation (FEA) [22]. That is, when... 

Statistical filters based on local and global Z scores were outlined. We observe that, while using very conservative Z scores that essentially exclude false positives, the new protocol recognizes correctly (without any information about sequences) most of the family members with the RMS distance between the superimposed side chain centers of up to 4 A. We also observe many instances of successful recognition of family members that cannot be confidently recognized by pair energies with the so-called frozen environment approximation. 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

The suitability of light-atom crystals for charge density analysis can be understood in terms of the relative importance of core electron scattering. As the perturbation of the core electrons by the chemical environment is beyond the reach of practically all experimental studies, the frozen-core approximation is routinely used. It assumes the intensity of the core electron scattering to be invariable, while the valence scattering is affected by the chemical environment, as discussed in chapter... 

A justification for this approximation is that the inner-shell electrons of an atom are less sensitive to their environment than the valence electrons. Thus the error introduced by freezing the core orbitals is nearly constant for molecules containing the same types of atoms. In fact, it is often preferable to employ the frozen core approximation as a general rule because most of the basis sets commonly used in ab initio quantum chemistry do not provide sufficient flexibility in the core region to accurately describe the correlation of the core electrons. Recently, Woon and Dunning have attempted to alleviate this problem by publishing correlation consistent core-valence basis sets.125... 

The larger the basis set the more virtual MOs and the more excited Slater determinants can be generated. The quality of a calculation is determined by both the size of the basis set and the number of excited determinants that are considered. If all possible determinants together with an infinite basis set could be used, one would get the exact solution of the nonrelativistic Schrodinger equation within the Born-Oppenheimer approximation. Because a different chemical environment mostly affects the valence electrons, but does not influence the core electrons, the frozen core approximation includes only determinants with excited valence electrons. Also the highest virtual orbitals may be left unoccupied in all determinants (frozen virtuals). 

Abstract We summarize an ab-initio real-space approach to electronic structure calculations based on the finite-element method. This approach brings a new quality to solving Kohn Sham equations, calculating electronic states, total energy, Hellmann-Feynman forces and material properties particularly for non-crystalline, non-periodic structures. Precise, fully non-local, environment-reflecting real-space ab-initio pseudopotentials increase the efficiency by treating the core-electrons separately, without imposing any kind of frozen-core approximation. Contrary to the variety of well established k-space methods that are based on Bloch s theorem... 

Fig. 6.4.5. N powder pattern spectra of [ N ]Trpi3 gramicidin A in a lipid environment as a function of temperature. (A) at 143 K all significant amplitude motions except for methyl and primary amine groups cease. Samples were fast frozen by plunging thin films into liquid propane.

As methods based on dynamic programming cannot account for pairwise-interaction potentials the so-called frozen approximation approach [189] has been proposed. This method performs several iterations of profile environments. In the first iteration, the chemical environment is defined via the contact partners of the template. In subsequent rounds the aligned residues from the previous iteration replace the residues of the template. The idea is that target and template structure are similar enough such that the iterative process converges towards the optimal assignment. 

One possible solution to this problem is a combination of the MO and the empirical molecular mechanics (MM), treating the active and important part of the molecule with the MO method and the remainder, such as bulky substituents or other chemical environments, with the MM method. Such treatments have been made for some organometallic systems [83-86]. However, in these treatments, only the geometry of the MM part is optimized under the assumption that the MO part is frozen at the optimized geometry of the small model system. This frozen assumption can result in a substantial overestimation of the MM energy. Recently, Maseras and Morokuma have proposed a new integrated MO + MM scheme, called IMOMM, in which both the MO part and the MM part of geometry are simultaneously optimized [87]. This method can combine any MO approximation with any molecular mechanics force field. The application of this method at the IMOMM(HF MM3) and IMOMM(MP2 MM3) levels to the oxidation addition reaction of H2 to Pt(PRa)2, where R = H, Me, r-Bu, and Ph [88], has shown a promise that more realistic models of elementary reactions and catalytic cycles may be studied in the near future with this method. 

Since full Cl (FCI) is impossible except for small molecules and small basis sets, one resorts to limited Cl, the most common form of which is CI-SD (Section 13.21). In addition, the frozen-core (FC) approximation is often used here, excitations out of the inner-shell (core) MOs of the molecule are not included. TTie contribution of such excitations is not small, but their contribution changes very little with change in environment. 

The valence electrons oscillate in the core region as is shown in Fig. A5, which is difficult to treat using plane wave basis functions. Since the core electrons are typically insensitive to the environment, they are replaced by a simpler smooth analytical function inside the core region. This core can also now include possible scalar relativistic effects. Both the frozen core and pseudopotential approximations can lead to significant reductions in the CPU requirements but one should always test the accuracy of such approximations. 

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