Robust and Variational Fitting

Note that the added term, which involves the fit twice, is in that sense less accurate than Sambe and Felton s original term, which involves the fit only once. The point is, however, that if the fitted density is expressed as the exact density minus an error term, then the error in Eq. (5) is the Coulomb repulsion of the error with itself, which is always second order and positive [12]. This approach is readily extended to four-center Coulomb integrals [13], 

In this case, and perhaps for all robust fits, if the fit is robust then its LCAO coefficients can be determined by variation of the energy. In that case the fit is said to be variational. Quantum chemists are beginning to use variational fits, but they do not yet include robust energies, in a method that they call resolution of the identity [14,15]. Equation (6), with pab replaced by PlM where L and M are the usual multipole-moment quantum numbers, can also be used to remove the first order error from fast-multipole methods [16]. 

To evaluate Eq. (1) in a way that is analytic, robust and variational, it is first necessary to divide the density among the atoms. That is easy in any LCAO approach, where the only problem is to how to assign centers to the cross (two-center) terms of the density. The computationally most efficient way is to multiply each atomic orbital by a /8, where a is the appropriate Xa scaling factor for that atom. Then the scaled, and thus partitioned, density may be written 

The 4/3-power functional is not analytic about the origin there tire three cube roots of one. Nevertheless, the 4/3-power functional can be treated analytically using robust and variational fitting [17]. To do so it is convenient to define two new LCGTO functionals. The first functional approximates the cube root of the partitioned density, 

Using X, Y and n a unique robust approximation to the 4/3 power of the partitioned density can be constructed. 

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The Xa method has been made analytic and variational via single-center, LCGTO, robust, and variational fitting. It is quite promising because, unlike empirical and semiempirical methods, it is applicable to all elements. In fact, it has an independent parameter for each element that can be optimized to give... 

Dunlap, B. I. (2000). Robust and variational fitting. Physical Chemistry Chemical Physics, 2, 2113-2116. 

Resultant dimensions Fit and function Design limits Performance Sensitivity Robust to variation... 

Resultant Dimensions Fit and Function Design Limits Performance Sensitivity Robust to Variation... 

One of the striking conclusions to be drawn is that the quoted models are rather robust toward variation in particular input parameters. It turns out that under technical synthesis conditions, the surface vdll be largely covered by atomic nitrogen and that because of operation of the principle of detailed balance, a satisfactory description of the synthesis rate will be obtained if a good fit to the experimental thermal desorption data for N2 is used (32). This makes, of course, microkinetic modeling somewhat ambiguous and hence now the details of the rate-limiting step will have to be considered. 

This term refers to how resistant the precision and accuracy of an assay is to small variations in the method, e.g. changes of instrumentation, slight variations in extraction procedures, sensitivity to minor impurities in reagents, etc. Robust assays may not be capable of the highest precision or specificity but they are regarded as fit for the purpose for which they are designed. 

Their natural robustness has another consequence. The authors of a recent paper comment A protein s function is due to a comparatively small munber of residues, suitably interspersed throughout the sequence. This process of imbedding functional resides in a robust framework constitutes a versatile mechanism to confer multiple functions upon a given fold (Przytycka et al., 1999). The folds are thus able to maintain their core architectures in the face of considerable amino acid sequence variation, and this contributes another important element of fitness it makes possible adaptive substitutions that do not disrupt the underlying fold architecture, and this facilitates functional molecular evolution. It is the generic robustness of the basic fold frameworks that permits such sequential tampering and consequent functional variation. 

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