Nonphysical results

Thus, the author considered forcing the solution to lie within the physical bounds. Besides eliminating the objectionable nonphysical result, this approach, it was hoped, would also improve the accuracy of the solution within the physical bounds. Because of the limited performance evidenced in previous literature available on deconvolution, the author was unprepared for the magnitude of the improvement that resulted. 

We focus only on the plus solution, the minus sign giving nonphysical results. Using Eq. (8), the kinetic energy of B relative to C is given as... 

Avoidance of Nonphysical Results in BOVB Calculations Of course, it would be tempting to further lower the BOVB energies by allowing the active orbitals to delocalize freely over the whole molecule. This, however, should never be done, as the procedure would end with severely biased results. The following cases are important to keep in mind ... 

When a function is defined by an infinite power series in terms of a parameter p, the traditional approach is to truncate the power series, retaining terms up to pq. However, if the power series fails to converge (i.e. outside the region of convergence of the local equation), including higher order terms does not save the truncated series from failure, and the truncated series may lead to nonphysical results in the limit of p —> oo. 

Several details with respect to implementation of Equations [22] and [23] deserve further discussion. Whereas the approximation of the solute residing in a spherical cavity is clearly of limited utility, since most molecules are not approximately spherical in shape, there is also the issue of the choice of the cavity radius, a. Obvious approaches include (1) recognizing that the spherical cavity approximation is arbitrary and thus treating a as a free parameter to be chosen by empirical rules, and (2) choosing a so that the cavity encompasses either the solvent-accessible van der Waals surface of the solute or the same volume. Wong et al. have advocated a quantum mechanical approach like the last method wherein the van der Waals surface is replaced by an isodensity surface. Because g depends on the third power of a, the calculations are quite sensitive to the radius choice, and some nonphysical results have been reported in the literature when insufficient care was taken in assigning a value to a. Implementations that replace the cavity sphere with an ellipsoid have also appeared. 

This VLLE algorithm is prone to the same kinds of problems discussed in 11.1.5 for the two-phase Rachford-Rice procedure the algorithm is sensitive to the initial guesses made for the Cs and Ks, and nonphysical results for L and V may be false roots, or they may indicate that three phases do not form at the given conditions. The latter interpretation may hinge on the models chosen for the equation of state and for the activity coefficients. In addition, the absence of three phases can cause the coefficient matrix in (11.1.37) to become singular. 

The model gives reasonable results for each case in which both material and thermal flow parameters are changed. Influences of many physical parameters of the SOFC are extracted from the current-voltage curve and can be investigated separately. The model is based on a combination of electric laws, gas flow relationships, solid material properties and electrochemistry correlations and is characterized by as low a number of requisite factors as possible. During calculations, the advanced model is very stable and can be used for both simulations and optimization procedures. In contrast, the classic model is very sensitive to input parameters and very often generates nonphysical results (e.g. for i = OA/cm ). 

Inputs which are very far away from experimental data can generate nonphysical results ... 

The expression above provides two values of x, one positive and one negative. The negative value can be ignored because it yields a nonphysical result—a negative value for [OH ]. 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Problems that once were thought to have been caused by evil spirits, curses, had luck, or other vague and nonphysical factors have now been shown to result from chemical changes that can he studied, identified, and understood. Every time that kind of progress occurs, a specific method of dealing with a disorder using chemical compounds becomes possible. 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

With the present definition of r, however, an overcorrection that would normally disappear gradually through ensuing iterations results in a value of d(k)(x) that vanishes for all subsequent iterations. This behavior occurs because further corrections to that value are prohibited. To use the method, the investigator is compelled to take small values for r0. Even in this case, erroneously nonphysical values of o(k) that have been forced to zero are never allowed to return to the finite range that might better represent the true spectrum o(x). This form of the method therefore demands excessive computation and yields a solution that, although physically realizable, is not the best achievable estimate. 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

The first line, /j(x), has a nonphysical behavior, if correlation of orientations is lacking (g = 1), since the susceptibility Xt(x) does not have an imaginary component in this case moreover, Xr(x) diverges at the rigorous resonance condition (x = 1), when %t = G(1 — x2) 1. These results mean that the correlation factor g should actually be nonzero and/or that the parabolic potential has a restricted applicability. 

Another aspect of the ab intio computation of molecular complexes is the basis set superposition error (BSSE). This is due to the use of an incomplete basis set for the description of molecular complexes. As a result, the basis set of one molecule provides a framework for artificial lowering the energy of its partner and vice versa. Consequently, the BSSE introduces a nonphysical attraction between two (or more) subunits which form a molecular complex. In another words, the BSSE decreases to zero as the atomic basis set used for the description of the molecular system becomes complete. 

The last ratio is independent of the parameters, a and 3 and may regarded as a purely group theoretical (kinematic) result. The same result is obtained dynamically without group theory through the Slater-Condon parameters which are integrals of e2/rjj over analytic hydrogenic orbitals. The chain has been interpreted as the sum of a p orbital-p orbital and a spin orbital-spin orbital magnetic dipole interaction. This inteipretation is clearly nonphysical. 

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