Asymptotic series, defined

Another common example arises in the expansion of the interaction energy of two atoms as a function of the inverse interatomic distance 1/Rab- This turns out to be a Taylor series which differ from the correct energy by a real contribution with zero Taylor expansion. The difference may be attributed to so-called exchange repulsion. In this case, it is nowadays known that die Taylor series has a zero convergence radius, so that the energy expression constitutes an example of an asymptotic series (to be defined in a moment) which is non-convergent for all Rab-... 

Sometimes, one is not so much concerned with the pointwise convergence of a series one merely wants each partial sum to be asymptotically better than the last Such an asymptotic series is almost always an inverse power series, and it is then defined as follows ... 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic series . Still, for the long range the multipole approximation to AE can be quite accurate, if properly truncated (for instance, after the smallest term). For shorter distances, the penetration between the molecular charge clouds becomes significant, the screening of the nuclei by the electrons becomes incomplete even for neutral molecules, and the power law for AE 5 is modified by contributions which increase exponentially with decreasing R. These penetration contributions we define as ... 

Ford, W. B., Studies on Divergent Series and Summability and The Asymptotic Developments of Functions Defined by MacLaurin Series, London Chelsea Publishing, 1960. 

The sum begins with the lowest resonance, and must be taken to infinity at the series limit. In MQDT, the bottom of the channel is not defined, but a sum to infinity does not arise either. The quantity Tra (the linewidth) scales as (n — /x)-3 for an asymptotically Coulombic potential. A zero in the cross section occurs between each resonance. 

If we know both series with coefficients Ap and Bp, we can in principle determine the exponents y and v which define the asymptotic behaviour of these series. In fact, if ac is the critical value of a (a value which depends on the cut-oif K0), we have... 

The electrostatic model proposed by JW was later thoroughly analyzed in a series of works performed by the Swedish-Australian group (17-19), where exact formal treatment and approximate methods were used to solve the problem. The authors of these papers considered two electrostatic models. In the first model they investigated the interaction between two planar surfaces (separated by a distance h) with mobile ions adsorbed onto them (the net surface charge was zero). The surfaces were immersed in the dielectric continuum with the dielectric constant eP Behind each surface a different dielectric medium (with the dielectric constant e2) was placed. In the second model the mobile ions were replaced by mobile dipoles that were oriented perpendicular to the surfaces. In both models the motion of the particles was restricted to the well-defined plane. From the analytical treatment, which included images and correlations, the following asymptotic results for the pressure were obtained for the first model ... 

Critical exponents are properly defined only in the asymptotic limit T thus, at non-zero values of t, there can be higher terms in a series which yield (e.g. from In Y against In t plots) an average value of an exponent for a particular range of t which differs from the limiting value. Such range effects have been looked for but until very recently not found. Two new experimental studies of fluids suggest that they are real. 

The dispersion energy is defined as E isp = T nA TIub ( A.o- A. )+( a.o- . ) of the electronic correlation. After applying the multipole expansion, the effect can be described as a series of instantaneous multipole instantaneous multipole interactions, with the individual terms decaying asymptotically as /f 2(t+/+l) -pjjg jjjQgj important contribution is the dipole rlipole (k = l — 1), which vanishes as R. ... 

Some models assume that a system reaches a steady state rather than equilibrium. Equilibrium is defined by the principle of detailed balance, which requires that the forward and reverse rates are equal and that each step along the reaction path is reversible. The forward and reverse rates of steady-state processes are equal but the process steps that produce the forward rate are different from those that produce the reverse rate. At steady state, the state variables of an open system remain constant even though there is mass and/or energy flow through the system. The steady-state assumption is especially useful for processes that occur in a series, because the concentrations of intermediates that are formed and subsequently destroyed are constant. Perturbation of a steady-state system produces a transient state where the state variables evolve over time and approach a new steady state asymptotically. 

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