Asymptotic expansion coefficient

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Ab initio asymptotic-expansion coefficients for pair energies in MP2 perturbation theory for atoms " ... 

Single index labels (Mabels) for PW/m increments 2.5 Formulae for the PW/m asymptotic expansion coefficients of 161... 

The asymptotic behavior of the second-order energy of the M0ller-Plesset perturbation theory, especially adapted to take advantage of the closed-shell atomic structure (MP2/CA), is studied. Special attention is paid to problems related to the derivation of formulae for the asymptotic expansion coefficients (AECs) for two-particle partial-wave expansions in powers... 

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Refs. 32, 170, and 180. 

Secondly, due to the smallness of the rotational temperature for the majority of molecules (only hydrogen and some of its derivatives being out of consideration), under temperatures higher than, say, 100 K, we replace further on the corresponding summation over rotational quantum numbers by an integration. We also exploit the asymptotic expansion for the Clebsch-Gordan coefficients and 6j symbol [23] (JJ1J2, L > v,<0... 

Matched-Asymptotic Expansions Sometimes the coefficient in front of the highest derivative is a small number. Special perturbation techniques can then be used, provided the proper scaling laws are found. See Kevorkian, J., and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York (1981) and Lager-strom, P. A., Matched Asymptotic Expansions Ideas and Techniques, Springer-Verlag, New York (1988). 

These fits incur average errors of only 0.3, 0.3, 0.2, and 0.8%, respectively. It is possible [263] to fit the (p ) and (p ) moments by using measures of the numbers of electrons in the various subshells, but the fits are not as accurate, particularly for (p ), because these moments exhibit strongly periodic behavior. The Hartree-Fock values of Ag, the coefficient of the leading term in the large-p asymptotic expansion Eq. (5.42) for the 103 neutral atoms, can be fit with an average error of 0.1% to the simple expression [187,232]... 

Thus the power expansion (16. 9) seems to be valid practically only in the case of bounded perturbation. So we are obliged to regard it as asymptotic expansion even in the case of regular perturbation. In this section we shall assume (17.1) and examine under what conditions the first several coefficients of (16. 9) are significant, and then, discuss the validity of the asymptotic expansion. 

Validity of asymptotic expansion. We have shown that the coefficient w(0 exists if <, 3)(i o) Next we must examine whether the expansion (16. 9) is actually valid as an asymptotic expansion up to the second order. First we note that the 0-approximation (16. 5 ) is true a fortiori in our case. 

This is the asymptotic expansion. Note that the coefficients of the successive terms can be found without solving the M-equation but merely by applying the operator W a finite number of times. The expressions obtained for them are called sum rules . 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

Zener gives numerical solutions for the growth coefficients for one- and three-dimensional growth, and also asymptotic expansions for large and small values of the growth coefficients ... 

To up-scale the previous model to the macroscale we make use of a formal homogenization procedure based on asymptotic expansions in terms of a perturbation parameter e which quantify the ratio between the meso and macroscales. To describe the physics properly, the coefficients must be scaled. Further, denoting vref and T>ref reference velocity and diffusion coefficint, and... 

Since the coefficients C found from Eq. (4.218) are functions of cr, one has to perform asymptotic expansion in Eq. (4.219). This gives finally... 

As it should be, at (3 = 0 this formula reduces to Eq. (4.174), which was obtained for a one-dimensional case. We remark, however, that in a tilted situation ((3 / 0) the coefficient D2 acquires a contribution independent on a that assumes the leading role. This effect is clearly due to admixing of transverse modes to the set of eigenfunctions of the system, and it is just it that causes so a significant discrepancy between the zero-derivative approximation and the correct asymptotic expansion for x(3 ) curves in Figure 4.12. Evaluation of the coefficient L>4 is done according to the same scheme and requires taking into account a number of the perturbation terms that makes it rather cumbersome. 

Numerical values of c(K), v(K), for K < 4 are given in Table D.3. For expansion coefficients we use, following [249, 251], the same notation Aq as for the expansion coefficients of the quantum mechanical density matrix over real Tq . Such a choice of notation may be justified by the fact that classical Aq constitute an asymptotic limit of quantum mechanical... 

If an asymptotic expansion is carried out in the limit e — 0, then one obtains the preferential interaction coefficient at high salt (HS) [77, 78]... 

Its Hankel transform has no singularity at p -> 0, and so the expansion of the DCF at p = 0 keeps the analytic form (44). Accordingly, the total correlation function keeps the asymptotics (43). However, the matrices of the expansion coefficients Co, C2, C4,... in (44) have other, modified values. Through Equation (40) this, in general, changes the profile p and hence results in a modified inverse decay length appearing in the asymptotics (42), (43) and (46). 

The formulation of Section 9.5.1 has served to remove the chemistry from the field equations, replacing it by suitable jump conditions across the reaction sheet. The expansion for small S/l, subsequently serves to separate the problem further into near-field and far-field problems. The domains of the near-field problems extend over a characteristic distance of order S on each side of the reaction sheet. The domains of the far-field problems extend upstream and downstream from those of the near-field problems over characteristic distances of orders from to /. Thus the near-field problems pertain to the entire wrinkled flame, and the far-field problems pertain to the regions of hydrodynamic adjustment on each side of the flame in essentially constant-density turbulent flow. Either matched asymptotic expansions or multiple-scale techniques are employed to connect the near-field and far-field problems. The near-field analysis has been completed for a one-reactant system with allowance made for a constant Lewis number differing from unity (by an amount of order l/P) for ideal gases with constant specific heats and constant thermal conductivities and coefficients of viscosity [122], [124], [125] the results have been extended to ideal gases with constant specific heats and constant Lewis and Prandtl numbers but thermal conductivities that vary with temperature [126]. The far-field analysis has been... 

Figure 1. Spin-dependent terms Cq, —7, in Legendre expansion coefficients as functions of the interfragment distance R dashed curves from Asymptotic Theory (AT) solid curves from AT corrected to fit ah initio data.

---