Extrapolation Formulas

It can be noticed in formula (21) that increases with temperature. This variation can be represented in the following form [YUAN S.C., MOCK Y.I. - Hydrocarbon Processing, 47, 133 (1968)]  

The error with respect to the tabulated value (Ex. 22) is of the order of 2 %. 

Formula (48) allows the values of to be calculated at several temperatures between 1000 and 5000 K for example, and the corresponding NASA polynomials to be deduced. 

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Table 1.6 CCSD valence correlation energies calculated with the cc-pVXZ basis sets and using the extrapolation formula Eq. (5.14) are compared with the R12 values (mEh). The last column contains the mean absolute deviations from the R12 energies. All calculations have been carried out at the optimized all-electron CCSD(T)/cc-pCVQZ geometries [25].

This convergence is significantly accelerated by applying the extrapolation formula (5.14), the errors being reduced to 1 - 3 kJ/mol for all extrapolations except for the cc-pV(DT)Z energies. (Here and elsewhere we shall use the notation cc-pV(X - 1,X)Z for the energy obtained by extrapolation from the cc-pV(X - 1)Z and cc-pVXZ correlation energies.)... 

A short time later, using a continued fraction method, Aharoni constructed [83] for Aj a fairly long power series in ct and also showed numerically that Brown s expression (4.132) resembles the exact one with the accuracy of several percent for ct > 3. In the 1990s the eigenvalue A became a subject of extensive studies. Efficient numerical procedures were developed [84], and a number of extrapolation formulas with a good overall accuracy were proposed [71,85-87],... 

An equilibrium state is defined for generalized coordinates such that the total energy E( q ) is minimized. The energy gradients p = are forces with reversed sign. Coordinate and gradient displacements from m successive iterations are saved as m x n column matrices, Aq and Ap, respectively. The Hessian matrix is Ftj =, , such that Newton s extrapolation formula is Aq = —Gp°, where G = F 1. 

Fig. 1.13. Top Variation of defect formation enthalpies with Fermi level under zinc- (left) and oxygen-rich (right) conditions as obtained from GGA+U calculations. The gray shaded area indicates the difference between the calculated and the experimental band gap. The numbers in the plot indicate the defect charge state parallel lines imply equal charge states Bottom Transition levels in the band gap calculated within GGA (a), GGA+U (b) and using an extrapolation formula described in [115]. The dark gray shaded areas indicate error bars. Copyright (2006) by the American Physical Society...

The extrapolation formula was AE[]/ex = (AEis(Zmax) + AEis(Zmax- -l)) for different max All the values in eV... 

The extrapolation formula was the same as for the the first order (see Table 1). The results are... 

In many ways it would have been desirable to have obtained a continuous record during the reaction of the intrinsic viscosity, [rf], which can be found by extrapolating to zero concentration of the function rj/c. However, no convenient experimental procedure was devised for doing this, and no extrapolation formula could be used since the Huggins constant for the polymer changed during the reaction. 

Another well-known extrapolation formula was proposed by Kraemer (1938) ... 

If the left extreme test fails then a is too small. A trial °new s comPuted from the extrapolation formula... 

An independently measured value for Z), was obtained from a direct measurement of the effective diffusion coefficient Du using hydrogen at room temperature. The technique will be described in Section IV.5. Correcting this value for diffusion of cumene at reactor temperature, using the extrapolation formula applicable to Knudsen diffusion ... 

When the pore channel dimensions are small compared to the gaseous mean-free-path, however, diffusion is of the Knudsen type. The effective diffusion coefficient is then independent of gas pressure (as long as the gas pressure does not rise to values where the mean-free-path is no longer larger than the pore dimensions) and the appropriate extrapolation formulae are... 

The upper limit does not come from a technical limit, and if an exact and explicit extrapolation formula could be written, it would be possible to use measured values in a range as large as one would like. In fact, there is no such formula, and interpolation requires the use of limited expansions of type (5.1.107). 

Fig. 17. Distribution function P M) of the order parameter M plotted over a range of temperatures for N = 128, L = 80, J> = 0.5, using the bond fluctuation model of symmetric polymer mixtures and using the extrapolation formula Eq. (145) and data from a single temperature run at kBT/e = 266.4.The number of (statistically independent) samples was N — 16800. From Deutsch and Binder [92]...

The first three terms are the kinetic, exchange and correlation energies respectively, for the free-electron gas. The first two terms are exact for a free-electron gas. The correlation energy is an extrapolated formula developed by Wigner l. The last term gives the first-order correction to the kinetic energy if inhomogeneities have to be... 

As mentioned above, Edwards has used a self-consistent mean field theory to derive similar (though not identical) results to those obtained from scaling theory (Edwards, 1966 Edwards and Jeffers, 1979). One virtue of this approach is that the relationships were obtained from a single extrapolation formula which encompasses all concentrations from an infinitely dilute solution to bulk polymer. This means that a somewhat more intuitively acceptable picture of polymer solutions emerges, one where there is a gradual change from one concentration regime to another. 

The procedure can be iterated to calculate the next terms in the C/i series and so on and, therefore, to further improve the solution. By doing so, the Richardson extrapolation formulae are obtained and they are referred to as the Romberg method in the special case of definite integrals. 

In these derivations, it was implicitly assumed that s and D are independent of the concentration. Equation (9-97), therefore, only applies at infinite dilution. However, sedimentation and diffusion coefficients are measured at finite concentrations, and so must be extrapolated to c 0. The extrapolation formula for sedimentation coefficients is obtained from the reasoning that, according to Equation (9-96), the 5 values, and so also the 5c values, are inversely proportional to the frictional coefficients fs. Since fs is proportional to the viscosity rj, and this in turn is proportional to the concentration, 1 /sc must be directly proportional to the concentration. This dependence is usually formulated as... 

All the extrapolation formulas introduced to date are empirical. The much used expressions of Schulz and Blaschke, Huggins, and Kraemer start from the relationship... 

The extrapolation formula that is obtained by transforming Equation... 

Equation 16 forms the basis of the Richardson extrapolation formula [6] commonly used in numerical analysis, for improving the accuracy of the numerical solutions by accommodating for the truncation error estimates. 

In W2-F12, the Hartree-Fock component is calculated with the VQZ-F12 basis set. The valence CCSD-F12 correlation energy is extrapolated from the VTZ-F12 and VQZ-F12 basis sets, using the above two-point extrapolation formula with a = 5.94. The quasiperturbative triples, (T), corrections are obtained from standard CCSD(T)/VTZ-F12 calculations (i.e., without inclusion of F12 terms) and scaled by the factor f = 0.987 x mp2-fi2/-Emp2- This approach has been shown to accelerate the basis set convergence [31, 49]. The CCSD inner-shell contribution is calculated with the core-valence weighted correlation-consistent A PWCVTZ basis set, while the (T) inner-shell contribution is calculated with the PWCVTZ(no Q basis set. The scalar relativistic, spin-orbit coupling, DBOC, and ZPVE corrections are obtained in the same way as in W1-F12 theory. 

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