Rational function expansions

Alternative GGA exchange functionals have been developed based on rational function expansions of the reduced gradient. These functionals, which contain no empirically optimized parameters, include B86, LG, P, PBE, and znPBE. 

Since the machine performs only arithmetic operations (and these only approximately), iff is anything but a rational function it must be approximated by a rational function, e.g., by a finite number of terms in a Taylor expansion. If this rational approximation is denoted by fat this gives rise to an error fix ) — fa(x ), generally called the truncation error. Finally, since even the arithmetic operations are carried out only approximately in the machine, not even fjx ) can usually be found exactly, and still a third type of error results, fa(x ) — / ( ) called generated error, where / ( ) is the number actually produced by the machine. Thus, the total error is the sum of these... 

The Redlich/Kister expansion, the Margules equations, and the van Laar equations are all special cases of a general treatment based on rational functions, i.e., on equations for G /x X2RT given by ratios of polynomials. They provide great flexibility in the fitting of VLE data for binary systems. However, they have scant theoretical foundation, and therefore fail to admit a rational basis for extension to multicomponent systems. Moreover, they do not incorporate an explicit temperature dependence for the parameters, though this can be supplied on an ad hoc basis. 

Derivation of the Basic Equations. The basic equations are obtained by forming the Taylor series expansion of the rational function In a logarithmic space and then retaining only the constant and linear terms (2,5,, 3.) This procedure yields a product of power functions In the corresponding Cartesian space. 

Often the enhancement factor is written as a rational function this is motivated by the observation that the gradient expansion diverges and by the knowledge that Pade approximants are often very effective approaches for resummation of divergent series. A representative functional of this form is the Depristo-Kress functional ... 

The rational function L(s)/D(s) is the approximation expression of function H(s) if the expansion in the power series with respect to s is identical with the expansion in the power series of function to the degree m +N. As the criterion of fitting, we can assume... 

So far, we have presented variational calculations with straightforward optimization by explicitly varying the parameters and increasing the basis. One may now try to elaborate a little on the convergence pattern. For instance, one may study the variational energy E N) as a function of the maximum number of states introduced into the h.o. expansion, and try to guess what E(p°) should be. To this end, E(N) can be written as a rational function... 

Sect. 1.10 if we restrict the discussion to the subsonic case. For subsonic velocities, /(co) can be expanded as a Taylor expansion in the velocity, where each term is a rational function of the moduli, as required by the argument in Sect. 1.10. We will see in the next section that the situation becomes more complex above the subsonic region. Taking the inverse Fourier transform of (7.2.3 a) gives... 

The exponential form of Eq. 6-28 is a nonrational transfer function that cannot be expressed as a rational function, a ratio of two polynominals in s. Consequently, (6-28) cannot be factored into poles and zeros, a convenient form for analysis, as discussed in Section 6.1. However, it is possible to approximate by polynomials using either a Taylor series expansion or a Fade approximation. 

As described above, simple mutation, regardless of rational or random, sometimes changes the function of enzymes in a drastic manner. Especially, in the case of enzymes belonging to enolase superfamily, including decarboxylases, consideration of the reaction mechanism is important because the apparently different transformations proceed via a similar key intermediate. Thus, the well-designed mutation and structure of the substrates will lead to a successful expansion of the application of enzymes in organic synthesis. 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

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