PadD© approximation

Thus, we have two expressions for E(v, J), Eqs. (II.7) and (2), as functions of v and J. The first, the Dunham expression, is valid for v near the bottom of the potential well. The second, the Le Roy-Bemstein-Stwalley expression, is applicable in the dissociation limit. Naturally, it would be expected to derive a formula for E(u, J) in the complete space, whose limiting cases coincide with Eqs. (II.7) and (2). Evidently, this formula, if it exists in general, may have a more complex expression. One of the possible ways to construct this formula is based on the application of the Padd approximant technique. This way was exploited by Beckel and co-workers (Hashemi-Attar et al., 1979 Hashemi-Attar and Beckel, 1979), who suggested... 

The table of the right-hand side ratios, Eq. (9), is called the C-ratio table. Equation (9) is the basic relation in the Padd theory, and emphasizes the existence of the correlation rule between the convergence properties of Padd approximants and the internal structure of the C-table or the C-ratio table. In fact, this statement was confirmed numerically by Gilewicz (1978), who called the characteristic internal structure of the C-table the valley structure. The valley structure is constructed by the following procedure ... 

The type II approximant is a new approach completely. It is evident that this class of Padd approximant involves as a particular case type I,... 

Finally we note that we have not been concerned with all applications of the Padd approximant approach to diatomic potential theory (Epstein, 1968 Brandas and Micha, 1972 Barnsley and Aguilar, 1978 Goscinski and Tapia, 1972 Sullivan, 1978 Thakkar, 1978), but we have discussed here only the aspects of the Padd approach that, in our opinion, will have fundamental value in diatomic potential theory and will elevate the theory of Padd to a level of distinction such as it deserves. 

Baker, G. A., Jr. (1975). The Essentials of Padd Approximant. Academic Press, New York. 

Gilewicz, J. (1981a). Abstracts of The First French-Polish Meeting on Padd Approximation and Convergence Acceleration Techniques, Warsaw, p. 22. 

Graves-Morris, P. R., ed. (1973a). Padd Approximants. Institute of Physics, London. Graves-Morris, P. R., ed. (1973b). Padd Approximants and Their Applications. Academic Press, New York. 

The apparent garbage produced by the perturbational series for R = 3.0 a.u. represented for the Padd approximants precise information that the absurd perturbational corrections pertain the energy of the... 2pcr state of the hydrogen atom in the electric field of the proton. Why does this happen Visibly low-order perturbational corrections, even if absolutely crazy, somehow carry information about the physics of the problem. The convergence properties of the Rayleigh-Schrddinger perturbation theory depend critically on the poles of the function approximated (see the discussion on p. 250). A pole cannot be described by any power series (as happens in perturbation theories), whereas the Pade approximants have poles built in the very essence of their construction (the denominator as a polynomial). This is why they may fit so well to the nature of the problems under study. 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

Table 13.4. Convergence of the MS-MA perturbational series for the hydrogen atom in the field of a proton (Ipou state) for internuclear distance R (a.u.). The Table reports the error (in %) for the sum of the original perturbational series and for the Padd [L + 1, L] approximant. The error is calculated with respect to the variational method (i.e., the best for the basis set used).

---