Comparison Equation Method

The Gamov formula derived above is a deep tunneling approximation at Emax. where is the potential barrier top. Actually, Equation (2.55) gives 1.0 at = Vmax, but the correct transmission probability at A Enax is 1 /2. This error is obvious, since the linear potential approximation does not hold in the vicinity of the potential barrier top. It is better to use a quadratic potential approximation. Instead 

We try to solve a more general equation that has the same analytical structure, 

The approximate function fappix) can be shown to satisfy the equation 

fappix) can be a good approximation, when the following condition is satisfied  

The important question about this comparison equation method is what the same analytical structure means. In order for the function yix) given by Equation (2.64) not to diverge, the order and the number of zeros of Kix) should be the same as those of /(t). That is to say, if the conditions 

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In the case of a general potential that has the same analytical structure as that of the parabolic potential barrier, we can use the comparison equation method explained above. Equation (2.67) requires... 

Unfortunately, this method cannot be used in the case of an Eckart potential barrier, discussed in Section 2.1.3. The corresponding appropriate comparison equation method has not yet been developed in this case. 

Here we have used the comparison equation method for the case of quadratic potential barrier. This transfer matrix M is the same as Equation (2.24), and ij) and e are given by Equations (2.25) and (2.69). 

A model problem. Comparison of methods. Further comparison of various iterative methods will be conducted by having recourse to the Dirichlet problem associated with Poisson s equation in the square 0 < < 1,... 

A model problem. Comparison of methods. Further comparison of various iterative methods will be conducted by having recourse to the Dirichlet problem associated with Poisson s equation in the square 0 < x1 < 1, 0 < x2 S 1 °f the unit sides /j = l2 = 1 and posed on a square grid u>h with steps h1 = h2 = h. As a special case of problem (2) in Section 2, the problem of interest is characterized by the grid equations... 

In the single site model with two (spin-up and spin-down) levels it is possible to make the direct comparison between our Ansatz and the master equation methods. For the latter, we used the well known master equations for quantum dots [180,181],... 

So far as Eq. (5.197) can deliver them, quantitatively accurate reaction rates, say to within a factor of 2, require activation energies accurate to within about 2 kJ mol-1. Nevertheless, the equation does provide a simple way of obtaining serviceably good rate constants. The (admittedly small) selection of reactions here shows no bias toward low or high calculated barriers for any of the four methods, and for a particular kind of reaction it is advisable to choose a method based on a comparison of methods with experiment results where this information is available. 

Yang, J. Griffiths, P.R. Goodwin, A.R.H. Comparison of methods for calculating thermodynamic properties of binary mixtures in the sub and super critical state Lee-Kesler and cubic equations of state for binary mixtures containing either CO2 or H2S. J. Chem. Thermo. 2003, 35, 1521-1539. 

FFF should yield a diameter [the Stokes diameter d, see Equation (41.8)] more nearly representative of the true hydrodynamic diameter of the particles and thus a diameter larger than that measured by sedimentation FFF. This is indeed the case, as shown by the results listed in Table 41.5. The ratio of the diameters measured by these alternative methods to the SdFFF diameter should provide a measure of the bulkiness (or fractal dimension) of the chainlike structures. More work is obviously needed to exploit the comparison of methods. 

It is immediate to see from equation (A2) that whenever elements of P are small, since /2m is a small parameter, equations adiabatically decouple into one-dimensional problems for the effective potentialse (p). In turn, these problems can be analyzed by the Liouville-Green WKB technique, which requires special care whenever e = E (turning points) but this problem is to be considered as effectively solved by the method of comparison equations. It is important to realize that proper coordinate choices may lead to wide regions of p space where this decoupling is very effective in such a case, approximate quantum numbers can be assigned, and it is possible to compute semiclassically bound or resonance states and scattering properties. 

Corticosteroids Blue Tetrazolium Use of the Kalman filter algorithm to resolve mixtures of cortisone and hydrocortisone with a pseudo-first order rate constant ratio as low as 1.8 Comparison with logarithmic-extrapolation and proportional-equations methods... 

See for example, P. Saxe, H. F. Schaefer 111 and N. Handy, Chem. Phys. Lett., 79, 202 (1981). Exact Solution (within a Double Zeta Basis Set) of the Schrodinger Elearonic Equation for Water or C. W. Bauschlicher, Jr. and S. R. Langhoff,/. Chem. Phys., 86,5595 (1987). Full Cl Benchmark Calculations on Nj, NO and O2 A Comparison of Methods for Describing Multiple Bonds. 

The empirical data obtained by INL for HeXe gas tnixmre viscosity are provided for a temperature range of approximately 300 K to 800 K at various mole fractions. The correlation methods described above were all computed using the DIPPR pure component equations, with the exception of the Lucas Rules, which do not utilize pure component values. This allows a consistent comparison of methods. The methods used for the comparison in Figure 5 and labeled as such are as follows ... 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

Davis, M. E., McCammon, J. A. Solving the finite difference linearized Poisson-Boltzmann equation A comparison of relaxation and conjugate gradients methods.. J. Comp. Chem. 10 (1989) 386-394. 

The comparison between the finite element and analytical solutions for a relatively small value of a - 1 is shown in Figure 2.25. As can be seen the standard Galerkin method has yielded an accurate and stable solution for the differential Equation (2.80). The accuracy of this solution is expected to improve even further with mesh refinement. As Figmre 2.26 shows using a = 10 a stable result can still be obtained, however using the present mesh of 10 elements, for larger values of this coefficient the numerical solution produced by the standard... 

The results of a comparison between values of n estimated by the DRK and BET methods present a con. used picture. In a number of investigations linear DRK plots have been obtained over restricted ranges of the isotherm, and in some cases reasonable agreement has been reported between the DRK and BET values. Kiselev and his co-workers have pointed out, however, that since the DR and the DRK equations do not reduce to Henry s Law n = const x p) as n - 0, they are not readily susceptible of statistical-thermodynamic treatment. Moreover, it is not easy to see how exactly the same form of equation can apply to two quite diverse processes involving entirely diiferent mechanisms. We are obliged to conclude that the significance of the DRK plot is obscure, and its validity for surface area estimation very doubtful. 

Computer codes are used for the calculational procedures which provide highly detailed data, eg, the Ruby code (70). Rapid, short-form methods yielding very good first approximations, such as the Kamlet equations, are also available (71—74). Both modeling approaches show good agreement with experimental data obtained ia measures of performance. A comparison of calculated and experimental explosive detonation velocities is shown ia Table 5. 

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