Asymptotic method, semi

When the transform solution P(p,to,A) is obtained by numerical procedures for values of p not too close to /x —i.e., such that the limiting formulas (11) are not obeyed—it has been found convenient to represent the numerical results analytically by a sum of terms of the form (11a) or (11b). The inverse transform 7(2 ,to, A) is then given by a sum of terms of the form (12a) or (12b) with suitable coefficients and suitable values of the parameters /xs, K, or 7] (see [5] and [1, 21]). This approach, which has been called a semi-asymptotic method can be extended by fitting P(p,to,A), in the range of p about the saddle point of the integrand in (9), with any suitable analytic expression whose inverse transform is known. For example, sums of terms of the form (a — pYj b — pY, whose inverse transforms are confluent hypergeometric functions, have been utilized. 

We can conclude that the present method of correcting TF calculations provides adequate estimations of expectation values for ground state atoms taking into account the simplicity of the model and it self-consistent nature, where no empirical parameters are used. It provides information about the asymptotic behaviour of quantities such as p(0) and (r 2) that cannot be evaluated with the standard semi classical approach and allow us to estimate momentum expectation values which are not directly related to the density in an exact way. 

Knopfinacher and Warlimont, following earlier work, use much more esoteric mathematical machinery (concerning semi-groups of polynomial-like structures) to develop methods to determine the asymptotic (large-A) form of isomer counts for A-atom alkanes. Polya " in fact developed asymptotic results, but Knopfinacher and Warlimont are interested in extensions, particularly to obtain asymptotics for the numbers of generalized isomers , viewed as sets of molecules such that different sets as a whole have the same elemental composition. Their main result is a theorem on the asymptotic number of ways in which a mixture of molecules may be realized when the total number of particular kind of atoms is fixed, and the asymptotic results for ordinary single-molecule isomers is already known. Examples of application to mixtures of alkanes, alkenes, substituted alkanes and achiral alkanes and alkanols are given. 

Schafer and Witten" have applied the RG to excluded volume, and established scaling laws , for example for the osmotic pressure. One of the objects of the RG method is to establish such scaling laws, and to demonstrate scale invariance . Then experimentally observable correlation functions can be shown to obey particular scaling behaviour, and the critical exponent calculated may be compared with that obtained by experiment. Critical exponents calculated by the RG will generally differ from that obtained by classical mean field e.g. SCF approaches - Mackenzie " in a recent review has pointed out that discrimination between the two lies with experiment. For example, Le Guillou and Zinn-Justin have calculated v in equation (7) to be 0.588 (c/. the SCF-fifth-power law value of 0.60). However, to discriminate between these values is beyond the capability of current experimental techniques. Moore has used the RG to explore the asymptotic limit, and recently demonstrated that when the ternary cluster integral vanishes, an expression for the osmotic pressure may be derived which holds for both poor and good solvents, in semi-dilute solutions. 

For polyatomic molecules the situation is more complex, and detailed knowledge of the potential hypersurface is available only for few systems. Semi-empirical methods are therefore introduced. As in the construction of the LEPS surface one should in the semi-empirical method, use expressions which behave correctly in the asymptotic limit. Thus for the molecule ABC the potential should approach Vab for bc oo Vbc for Rab oo, and Vac for Rab oo. The LEPS function discussed above has this property. However, it is advantageous to have a recipe which uses the extensive information on force-field potentials which is available from spectroscopic measurements. For a molecule as CO2, the... 

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