Numerical fitting

As mentioned above, the numerical instabilities in the fit arise from the attempt to fit the nuclear cores. Thus, if the density at the cores is discarded, then the fit should become more stable. This can be achieved by using numerical grids to evaluate a given molecular property and discarding points at and near the core. This can be achieved by minimizing the following fitting function  

The minimization of Equation 8.5 leads to a linear system of equations that can be expressed as c — c = As was the 

In this subsection, we present the methodology to obtain Cartesian point multipoles from the Hermite coefficients obtained in the fitting procedure. In all our work we have purposefully employed ABSs with a maximum angular momentum of 2, which results in 

Briefly, we have expanded on the work by Challacombe et al., who have shown that Hermite Gaussians have a simple relation to elements of the Cartesian multipole tensor (Challacombe et al., 1996). Once the Hermite coefficients have been determined, they may be employed to calculate point multipoles centered at the expansion sites. Thus, if hctuv represents the coefficient of a Hermite Gaussian of order Atuv, then if this Hermite is normalized we have 

This guarantees that higher order multipole integrals will integrate to integer numbers, for example, for the dipole integral in the z direction, d  

---

Figure 35 shows how the equilibrium transition curves of [(Ala-Gly-Pro)n]3Lys-Lys shift to higher temperatures with increasing chain length. The continuous curves represent the numerical fit to the experimental transition data. 

The data can be analyzed by graphical or numerical fitting to Eq. (2-19). A plot of In ([B]r/tA],) versus time affords a straight line of slope kAo- The value of the rate constant depends inversely on the difference Ao. The closer the initial concentrations are to each other, the less accurately one knows A0. 

Cisneros GA, Elking D, Piquemal J-P, Darden TA (2007) Numerical fitting of molecular properties to Hermite Gaussians. J Phys Chem A 111 12049... 

In this form, which is the more common expression used in numerical fitting, the matrix A is not as simply related to the partial widths. For an INR,... 

Initial numerical simulations of population density dynamics incorporated experimental data of Figures 2 and 4 and Equations 16, 20, 23, and 27 into a Runge-Kutta-Gill integration algorithm (21). The constant k.. was manipulated to obtain an optimum fit, both with respect to sample time and to degree of polymerization. Further modifications were necessary to improve the numerical fit of the population density distribution surface. 

By numerically fitting the decay curves of [Ti(III)] with the simulation program Gepasi [52], it was established that the dimer opens the epoxide with a rate constant of k = 1.4 M 1 s, whereas the monomer reacts more slowly (k = 0.5 M 1 s 1). At the usual initial Cp2TiCl2 concentration of 10 mM, this means that 84% of 25 molecules are opened by the dimer. 

Figure 21.5 Fluorescence dynamics of Au(0) i, for excitation at 395 nm, and emission at 570 nm. The corresponding numerical fits to the data are indicated by the thin solid lines. The residuals of the fit are shown at the top of graph.

When we wanted to numerically fit experimental PFGE data of water diffusion in a water-in-oil emulsion, we found that for a beginner in this field the literature is quite confusing. First, all three expressions for diffusion in a sphere with reflecting walls are somewhat different and lead to very different fitting results, especially when the formulas are combined with a radius distribution function. Since the derivation of the published expressions needs some tedious algebra (which has not been published), it is not trivial to check the derivation in order to establish which expression is the correct one. Here we use a numerical approach to decide which expression is correct. 

Fig. 3. (a-c) Time resolved changes of the O-H stretching absorption of OH/OH dimers as measured with spectrally integrated probe pulses centered at Epr and corrected for rotational diffusion (symbols, pump pulses centered at Ep=2950 cm"1). The solid lines represent numerical fits based on exponential kinetic components with time constants of 200 fs, 1 ps and 15 ps. Inset of Fig. (c) Time evolution up to a 70 ps delay time, (d-f) Oscillatory component of the signals in Figs, (a-c) and Fourier transforms (insets). 

The analysis of the spectrum is a multiple-step process. The first step is the removal of the contribution to the spectrum due to the sampling apparatus. The next step is subtraction of the relatively large contribution due to N02. The remaining absorptions are due to H02, other organic peroxy radicals, and possibly other unknown radicals. A numerical fitting procedure... 

In recent years a compromise has been found which presently dominates polyatomic calculations. Each function fj is expanded as a linear combination of gaussian orbitals (f is then called a contracted gaussian function). Since this is basically a numerical fitting procedure, various choices have been suggested for the contraction scheme. The most popular choices are presently Pople s approximations (15) to Slater orbitals and Dunning s approximations (16) to free atom Hartree-Fock orbitals. 

Model Equations to Describe Component Balances. The design of PVD reacting systems requires a set of model equations describing the component balances for the reacting species and an overall mass balance within the control volume of the surface reaction zone. Constitutive equations that describe the rate processes can then be used to obtain solutions to the model equations. Material-specific parameters may be estimated or obtained from the literature, collateral experiments, or numerical fits to experimental data. In any event, design-oriented solutions to the model equations can be obtained without recourse to equipment-specific fitting parameters. Thus translation of scale from laboratory apparatus to production-scale equipment is possible. 

Figure 5.15 reveals that at E = °° the conversion-time profile bends sharply at a conversion of 50% and stays constant after that point. At lower E values, however, the discontinuity in the profile is not as discernible and the final degree of conversion is much higher than 50%. If only the racemate is available but none of the enantiomers, and thus e.e.s and/or e.e.P cannot be measured, the determination of the E value of the reaction can only be performed by numerical fitting of the conversion-time profile. 

The resonance parameters are then extracted by numerically fitting the formula to the results of the scattering calculation. Numerical instabilities that sometimes arise in fitting the pole structure, either using Eq. (7) for one resonance or Eq. (13) for two resonances, may be assuaged using more sophisticated approaches such as the Pade approximation [64]. 

The numerical fitting has its merits and disadvantages. Certainly, it is simple, but on the other hand it has empirical character and provides no insight into the mathematical nature of scaling relations. 

Figure 10 shows a typical measured homodyne waveform and the corresponding numerical fit (solid lines). The measured THz waveform exhibits both the fundamental ECDL difference frequency (Fig. 10(a)) and higher harmonics - predominantly the third harmonic (Fig. 10(b)). Multiple harmonic generation in THz photo-mixers has been previously reported [103], By fitting the observed waveform to a sum of harmonic sinusoidal functions, the amplitude and phase of the THz electric field can be determined separately for the fundamental and third harmonic. The solid line shows a numerical fit to the data. The fundamental extracted frequency, 0.535 THz, compares well to the expected frequency based on the frequency difference of the two ECDL. The extracted E field amplitudes and phases are 3.37 x 10 4 and 2.17 radians for 0.535 THz (Fig. 10(a)) and 5.61 x 10-5 and 3.94 radians for the 1.605 THz third harmonic, respectively (Fig. 10(b)). 

---