Asymptotic expressions

The asymptotic expressions for the corresponding Clebsch-Gordan coefficients take the form... 

From Eq. (13.3) derive an asymptotic expression for the current density which is valid in the region At1/2 3> 1. 

The remaining terms in the expression for the activity coefficient all involve the function mu defined in Eq. (140). Using the asymptotic expression for the Fourier transform in Eq. (155) it is found that... 

Substituting these asymptotic expressions in equations (30.18) and (30.10) we find that, as - -co,... 

Results of similar accuracy as relativistic TFDW are found with a simple procedure based on near-nuclear correction which leave space for further improvements. For the reasons mentioned at the end of previous section the direct way to improve the present approach seems to be the refinement of the near nuclear corrections, a problem that we have just tackled with success in the non-relativistic framework [31,32]. The aim was to describe the near-nuclear region accurately by means of using the quantum mechanical exact asymptotic expression up to of the different ns eigenstates of Schodinger equation with a fit of the semiclassical potential at short distancies to the exact asymptotic behaviour (with four terms) of the potential near the nucleus. The result is that the density below Tq becomes very close to Hartree-Fock values and the improvement of the energy values is large (as an example, the energy of Cs+ is improved from the Ashby-Holzman result of-189.5 keV up to -205.6, very close to the HF value of -204.6 keV). This result makes us expect that a similar procedure in the relativistic framework may provide results comparable to Dirac-Fock ones. 

In the limit of infinite bond distance the width of the peak of the potential Vki approaches zero and the height of the peak approaches a value equal to the ionization energy of the molecule [43,78], The asymptotic expression for T, = T — T. can be obtained with T calculated with the Heitler-London... 

Equations A2-8a and A2-8b always converge but for large absolute values of z (e.g., >5) the convergence is slow and truncation errors may dominate. Hence, in practice. Equation A2-8a is often applied for z < 1, and Equation A2-8b is often applied for l< z < 4.5. Equation A2-8c is an asymptotic expression and must be truncated at or before the absolute value of the term in the series reaches a minimum. For large z (z> 10), ze erfc(z) approaches 1/ /n. 

The coefficient x is expected to be slightly above one, and X in the interval between zero and one (intrinsically different from the homonuclear case where both and X are identically 1/Smx)- Equation (1) can be re-adapted if Hm and Hx are no longer the diagonal elements of energy, but rather related to the asymptotic expressions for the local contributions to the kinetic energy for large distances r from the nucleus, viz. in atomic units [42)... 

The first of these asymptotic expressions corresponds to an outgoing spherical wave the second corresponds to an incoming spherical wave. If, on physical grounds, the scattered field is to be an outgoing wave at large distances from the particle, then only should be used in the generating functions. When we consider the scattered field at large distances we shall also need the asymptotic expression for the derivative of h it follows from the identity... 

We assume that the series expansion (4.45) of the scattered field is uniformly convergent. Therefore, we can terminate the series after nc terms and the resulting error will be arbitrarily small for all kr if nc is sufficiently large. If, in addition, kr n, we may substitute the asymptotic expressions (4.42) and (4.44) in the truncated series the resulting transverse components of the scattered electric field are... 

Asymptotic expressions for extinction and absorption efficiencies of spheres averaged over a size parameter interval Ax it (i.e., with no ripple structure) have been derived by Nussenzveig and Wiscombe (1980). 

In Fig. 1, the experimental results of Gryzagoridis [25] are compared with the algebraic equation (72) and with the series solution of Szewczyk [26], which is accurate only for Bx < 0.5. It is of interest to note that while the interpolation equation (72) is expected to be valid at large Prandtl numbers only, it remains valid even for Pr a 1 if the appropriate expressions are employed for the two asymptotic expressions. 

The proofs of Theorems 3-6, which will be published elsewhere,17 are fairly elementary, although a little lengthy. A principal step in the argument involves the construction of explicit asymptotic expressions for the elements of the inverse matrix XN = CN 1 in terms of the coefficients p , q , / , and m . These formulas may themselves be useful in certain applications. 

Molecular gases. In the case of molecular gases, or of mixtures involving a molecular gas, one must in general account for several induced dipole components, as the asymptotic expressions of Section 4.7 suggest. Besides these multipole-induced terms, one or more overlap-induced terms are usually necessary, especially if induction in species of low polarizability is considered (He atoms). While for certain systems just one or two dipole components need to be accounted for, for other systems elaborate sets had to be assumed for a satisfactory fit of the spectra. Moreover, for the molecular systems, we have to consider a number of different dipole models for the different bands (rototranslational, rotovibrational and overtones). It is, therefore, impractical to repeat here even the most important measurements. Table 4.2 quotes some sources where such information may be found. Examples of what is expected for a few representative systems are given below. These are based on first principles. 

As noted above, the asymptotic formulae given here, Eqs. 4.47 through 4.86, are valid for two interacting diatomic molecules, i = 1 and 2. For symmetric molecules like H2, only even 2, occur. In that case, for example, no octopoles exist. If one of the interacting partners is an atom, the associated 2, can assume the value 0 only this reduces the amount of computations needed significantly. Empirical dipole model components have been proposed in the past that were consistent with the asymptotic expressions above, sometimes with exponential overlap terms of the form of Eq. 4.1 added. 

Exercise. From the known properties of In follows the asymptotic expression for large t (more precisely t- oo, n- oo with n2/t fixed),... 

Considerable progress has recently been made in developing the theoretical background necessary for the application of the above method of transient kinetic analysis. An important step in this direction was the use of WKB asymptotics to derive approximate analytical expressions for short- and long-time transient sorption and permeation in membranes characterized by concentration-independent continuous S(X) and Dt(X) functions 150-154). The earlier papers dealing with this subject152 154) are referred to in a recent review 9). The more recent articles 1S0 1S1) provide the correct asymptotic expressions applicable to all kinetic regimes listed above the usefulness... 

The case of small pressure differentials along the slit channel. In this case parameter i in Eq. (4.41) is close to unity, more explicitly E, = 1 — e, where 0 < e 1. Then Eq. (4.41) is reduced to the following asymptotic expression ... 

As shown elsewhere 36,54), the analysis leads to the following asymptotic expression ... 

Several groups (84-86) have extended the similarity analysis of Burton et al. (73) to the case in which an axial magnetic field is imposed on the melt with sufficient strength such that Ha >> 1 and N 1. With these limits, a closed-form asymptotic expression describes the variation in the flow field across the thin 0(Ha 1/2) Hartmann layer adjacent to the disk. Axial solute segregation across this layer was analyzed by assuming that the melt outside of the Hartmann layers is well mixed. The effective segregation coefficient approaches 1 when the field strength is increased, as expected for any mechanism that damps convection near the crystal. 

Comparing this again with the asymptotic expressions, one obtains the... 

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