Extrapolation methods

In principle it is known how to compute the thermochemical properties of most molecules to very high accuracy (uncertainty of 0.5 kcal/mol). This can be achieved by accounting for electron correlation, such as can be obtained by means of coupled clustered theory with single, double, and triple excita- 

Gaussian-2 (G2) theory Ideally, a sucessful method for computation of thermochemical data has several features (1) it should be applicable to any molecular system in an unambiguous manner, (2) it must be computationally efficient so that it can be widely applied, (3) it should be able to reproduce known experimental data to a prescribed accuracy, and (4) it should give similar accuracy for species for which the data are not available or for which experimental uncertainties are large. The Gaussian-n methods were developed with these objectives in mind. Gaussian-1 (Gl) theory was the first in this series.27 28 We will not cover Gl theory in this chapter because it was replaced by G2 theory, which eliminated several deficiencies in Gl, and because G2 is currently the most widely used method of this series. 

An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68 69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 

The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies u, which are then scaled empirically by a factor of 0.8929 to take account of known deficiencies at this level.70 These frequencies give the zero-point energy, A (ZPE) 

The equilibrium geometry is refined at the MP2/6-31G(d) level [Moller-Plcsset perturbation theory to second order with the 6-31G(d) basis 

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The method proposed by Papoulis [7] to determine h(t) as a function of its Fourier transform within a band, is a non-linear adaptive modification of a extrapolation method.[8] It takes advantage of the finite width of impulse responses in both time and frequency. 

Reviews of Ganssian theory and other extrapolation methods are... 

Corrosion Rate by CBD Somewhat similarly to the Tafel extrapolation method, the corrosion rate is found by intersecting the extrapolation of the linear poi tion of the second cathodic curve with the equihbrium stable corrosion potential. The intersection corrosion current is converted to a corrosion rate (mils penetration per year [mpy], 0.001 in/y) by use of a conversion factor (based upon Faraday s law, the electrochemical equivalent of the metal, its valence and gram atomic weight). For 13 alloys, this conversion factor ranges from 0.42 for nickel to 0.67 for Hastelloy B or C. For a qmck determination, 0.5 is used for most Fe, Cr, Ni, Mo, and Co alloy studies. Generally, the accuracy of the corrosion rate calculation is dependent upon the degree of linearity of the second cathodic curve when it is less than... 

For strong electrolytes the molar conductivity increases as the dilution is increased, but it appears to approach a limiting value known as the molar conductivity at infinite dilution. The quantity A00 can be determined by graphical extrapolation for dilute solutions of strong electrolytes. For weak electrolytes the extrapolation method cannot be used for the determination of Ax but it may be calculated from the molar conductivities at infinite dilution of the respective ions, use being made of the Law of Independent Migration of Ions . At infinite dilution the ions are independent of each other, and each contributes its part of the total conductivity, thus ... 

It is simpler, though less exact, to apply the extrapolation method. The part of the residual current curve preceding the initial rise of the wave is extrapolated a line parallel to it is drawn through the diffusion current plateau as shown in Fig. 16.6(h). For succeeding waves, the diffusion current plateau of the preceding wave is used as a pseudo-residual current curve. 

A typical behavior of amplitude dependence of the components of dynamic modulus is shown in Fig. 14. Obviously, even for very small amplitudes A it is difficult to speak firmly about a limiting (for A -> 0) value of G, the more so that the behavior of the G (A) dependence and, respectively, extrapolation method to A = 0 are unknown. Moreover, in a nonlinear region (i.e. when a dynamic modulus depends on deformation amplitude) the concept itself on a dynamic modulus becomes in general not very clear and definite. 

While the arbitrary extrapolation methods used to evaluate v( and f° for a supercritical component are partly compensated by evaluating dt from data for binary mixtures, such compensation cannot apply generally to mixtures containing supercritical components i.e., for a supercritical component, 5, found from data for solutions of i in one solvent may be quite different from that found from data for the same component i in another solvent. 

The value was obtained using an extrapolation method with a numerical method = 4.1 cm s was obtained. 

The different organic modifiers used to derive the most suitable mobile phases lead to different parameters namely isocratic logfe and extrapolated logkw. The extrapolation method has no reality in terms of chromatographic behavior of solutes. However, mainly by correlation with log Pod (Eqs. 2 and 3) several studies have demonstrated the interest of these extrapolated retention factors as predictors of the lipophilicity of solutes. 

Pliego JR Jr, Riveros JM. 2000. On the calculation of the absolute solvation free energy of ionic species Application of the extrapolation method to the hydroxide ion in aqueous solution. J Phys Chem B 104 5155-5160. 

Experimentally there are two methods of determining the ] extracolumn band broadening of a chromatographic instrument. The linear extrapolation method, discussed above, is relatively straightforward to perform and interpret but rests on the validity.. of equation (5.1) and (5.3). The assu itlon that the individual contributions to the extracolumn variance are independent, may not be true in practice, and it may be necessary to couple some of the individual contributions to obtain the most accurate values for the extracolumn variance [20]. It is assumed in equation (5.3) ... 

This model tends to approach a zero probability rapidly at low doses (although it never reaches zero) and thus is compatible with the threshold hypothesis. Mantel and Bryan, in applying the model, recommend setting the slope parameter b equal to 1, since this appears to yield conservative results for most substances. Nevertheless, the slope of the fitted curve is extremely steep compared to other extrapolation methods, and it will generally yield lower risk estimates than any of the polynomial models as the dose approaches zero. 

Taylor expansion of (6.89) is not based on a physical model, and more-complicated nonlinear forms could be used. However, nonlinear extrapolation methods seem to offer little improvement over their linear counterparts [58]. Larger improvements appear to be achieved using the cumulative integral extrapolation method discussed in Sect6.7.3. 

Since the linear and related expansion formulas depend on fits to regions of the curve that are statistically less and less reliable, it makes sense to find a measure for extrapolation that depends on the relative accuracy of the relative free energy estimate for all points along the curve. The cumulative integral extrapolation method is one approach to this idea. 

Mordasini, T.Z. McCammon, J.A., Calculations of relative hydration free energies a comparative study using thermodynamic integration and an extrapolation method based on a single reference state, J. Phys. Chem. B 2000,104, 360-367... 

Chloride can also be estimated by potentiometric titration using standard silver nitrate [27]. The results are recorded directly and evaluated by means of a computer program based on the Gran extrapolation method. The determinations have a precision of 0.02% and since many samples can be titrated simultaneously, the time for a single determination can be reduced to less than 5 min. 

From the measurements published in the paper it cannot be inferred that the concentration-time curves can safely be extrapolated to zero time. The authors do not communicate details as to how well the beginning of the reaction could be defined. It should be pointed out that reaction kinetics derived by these means are subject to the uncertainties inherent in the extrapolation method. This holds particularly for a rapidly accelerated reaction. Furthermore, the possibility that the decomposition was initiated heterogeneously could not be excluded with certainty. These objections have to be considered when regarding the following kinetic results. 

Early determinations of RSE values employed unrestricted Hartree-Fock (UHF) theory in combination with 3-21G [9] or 4-31G [10] basis sets to evaluate the RSE according to Eq. 1. The appropriate consideration of correlation effects, the avoidance of spin contamination, and the treatment of thermochemical corrections have in detail been studied in the following, in particular by Bauschlicher [11], Coote [12-14], Morokuma [15-18], and Radom [19-25]. Highly accurate RSE and BDE results can be obtained with high level compound methods such as the G2 [26-30] and G3 [31-34] schemes (and variants thereof [11,15-18]), as well as extrapolation methods such as the CBS schemes [35,36], Wl, or W2 [37-39]. Generally, the accurate... 

S. R. Gunn. On the Calculation of the Corrected Temperature Rise in Isoperibol Calorimetry. Modifications of the Dickinson Extrapolation Method of Treatment of Thermistor-Thermometer Resistance Values. J. Chem. Thermodynamics 1971, 3, 19-34. 

Recently Hoover 29> compared various extrapolation methods for obtaining true solution resistances concentrated aqueous salt solutions were used for the comparisons. Two Jones-type cells were employed, one with untreated electrodes and the other with palladium-blacked electrodes. The data were fitted to three theoretical and four empirical extrapolation functions by means of computer programs. It was found that the empirical equations yielded extrapolated resistances for cells with untreated electrodes which were 0.02 to 0.15 % lower than those for palladium-blacked electrodes. Equations based on Grahame s model of a conductance cell 30-7> produced values which agreed to within 0.01 %. It was proposed that a simplified equation based on this model be used for extrapolations. Similar studies of this kind are needed for dilute nonaqueous solutions. 

S. E. Mason, I. Grinberg, and A. M. Rappe, First-principles extrapolation method for accurate CO adsorption energies on metal surfaces, Phys. Rev. B Rapid Commun. 69, 161401—161404 (2004). 

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