Coulomb interaction infinite range

It should be stressed at this point that, as we shall see, the in and out negaton-positon and electromagnetic fields given by Eqs. (11-56), (11-57), and (11-62) are ill-defined. For the matter field, the reason is that the Coulomb field has an infinite range, and, hence, charged particles, no matter how far apart, still interact with one... 

Note that, due to their infinite-range character, pure Coulombic potentials can actually lead to significant bond non-additivity for any proposed separation into bonded and nonbonded units. This reflects the fact that classical electrostatics is oblivious to any perceived separation into chemical units, because all Coulombic pairings (whether in the same or separate units) make long-range contributions to the total interaction energy. 

This simple result may be improved in various ways first, we may relax the "static approximation and keep the plasma assumption (315). In order to eliminate the divergences brought in by the long-range Coulomb interactions (114), it is then necessary to sum over an infinite class of diagrams, known as the ring... 

We turn now to theories of ionic criticality that encompass nonclassical phenomena. Mean-field-like criticality of ionic fluids was debated in 1972 [30] and according to a remark by Friedman in this discussion [69], this subject seems to have attracted attention in 1963. Arguments in favor of a mean-field criticality of ionic systems, at least in part, seem to go back to the work of Kac et al. [288], who showed in 1962 that in D = 1 classical van der Waals behavior is obtained for a potential of the form ionic fluids with attractive and repulsive Coulombic interactions have little in common with the simple Kac fluid. 

A method which would seem to have particular relevance to hydrogen-bonded systems in view of the Coulombic nature of the longer-range hydrogen-bond forces is one which evaluates long-range Coulombic interactions within the framework of the LCAO-MO method [314]. Hitherto this method has been applied to infinite polymers, where comparison with experimental structural data is not possible. 

If the interaction potential u is long range, all terms in the infinite sum should be taken into account. The brute force approach here is to truncate the summation at some large enough values of ny, rix. Efficient ways to do this for Coulombic interactions—the Ewald summation, fast multipole, and particle-mesh methods— will be described in the following chapters. 

Calculating the sum over all atoms is fairly straightforward for finite systems, but non-trivial for extended systems As the Coulomb interaction is very long ranged, the (infinite) sum in (6.13) is very slowly convergent. There is, however, a technique due to Ewald that allows us to circumvent this problem and evaluate (6.13) (see Sect. 6.2.3). 

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