Lekner method

For the Lekner method, upper error boimds can be given easily since the Bessel functions drop essentially exponentially fast, allowing for a simple approximation of the sum by an integral. These error estimates are much less sharp than the error estimates for the Ewald-type methods, but here the error bound only enters logarithmically into the computation time, so that excessive accuracy has only a small impact on the overall performance. 

Boundary conditions where only one dimension is periodic and the other two are open or finite appear in physical situations such as electronic structures of supported structures on metal surfaces, e.g., steps and atomic chains in one dimension, or in systems which can model a charged stiff polymer or DNA piece where, for avoiding end effects, the rod is made infinitely long [64-66]. It was only very recently that a ID Ewald method (EWID) for these systems was developed [67,68], although the Lekner method [11] de-... 

In general, the precision with which the energy is calculated by the Ewald or Lekner methods can be estimated without ambiguity [56,81,88]. Quantitative estimates of the performances obtained for the computation times are more uncertain, depending crucially on the type of computer (processor, cache-memory etc...), compiler and details of the program coding, though they are certainly correct in a quahtative way e.g. [44,64,77,91). 

Two standard methods are in common use in the MD community the reaction field method [79,80] and the Ewald summation technique [72,81-83]. There are also various hierarchical algorithms which are quite attractive in principle, but have proved to be difficult to implement efficiently in practice [67,84-87]. An alternative and potentially development interesting complement, is the summation formula developed by Lekner [88,89] which has been given an alternative and more general derivation by Sperb [90]. 

Although the above method of calculating the reflectivity can be extended to multilayer systems with any number of discrete layers, it becomes unwieldy as soon as the number of layers involved exceeds four or five. A computationally more efficient method is offered by the use of an optical transfer matrix. Here we simply summarize the method as described by Lekner.6 For any single layer j within the multilayer... 

In principle matrix methods analogous to the ones discussed in the section about neutron reflectivity can be applied to calculate ellipsometric angles for an arbitrary refractive index profile (Lekner 1987) and analytical approximations have also been developed (Charmet and de Gennes 1983). In practice the use of ellipsometry to obtain fine details of the structure of interfaces at the level of tens of angstrom units is likely to be difficult and to require extreme care. 

A completely different approach to the problem is due to Lekner [11] and Sperb [12,13]. Although the method in the original Lekner formulation has (N ) complexity, Strebel and Sperb have developed a factorization approach which yields an 0 NlogN) algorithm [14], called MMM The Strebel approach can also be adapted for partially periodic systems [15] however, the method becomes computationally more expensive. 

Starting from this convergence factor approach along the lines of the Lekner sum, R. Strebel and R. Sperb constructed a method of computational order O (N log N), MMM [54]. The favorable scaling is obtained, very much as in the Ewald case, by technical tricks in the calculation of the far formula. The far formula has a product decomposition and can be evaluated hierarchically, similar in spirit to the fast multipole methods. 

Upper error bounds can be foimd easily by approximating the sums with integrals (see [54]). As for the Lekner sum, the additional accuracy has to be paid for with only a small decrease of computational performance therefore, MMM is the method of choice if high accuracy is required. 

In the following we will give a summary of the energy expressions obtained by the Ewald and Lekner summation methods, for 3D and 2D point dipolar systems periodic in all spatial directions or periodic in only some directions and finite in the others. The energies obtained with the Lekner summation are often quite lengthy and will be omitted when available in the original papers. 

The Lekner expressions of the energy of systems of charges or dipoles periodic in one or two directions [39] contain only terms in r-space, and therefore seem particularly adequate for an efficient calculation. However, it is also supposed that use of the Lekner sum, as was mentioned for the 3D systems, overcomes the numerical instabilities arising from the divergence of the functions Ko and Ki for small values of their arguments [57]. An extension of the MMM method to the systems with a slab geometry is discussed and checked in [84]. 

At least in the case of liquid simple metals, a knowledge of the effective pan-potentials describing the interaction between the ions in the liquid metal can also be utilized to calculate g(r) and A K). The most common such method involves the assumption of a hard-sphere potential in the Percus-Yevick (PY) equation its solution provides the hard-sphere structure factor, /4hs( C). (See Ashcroft and Lekner 1966.) The two parameters that must be provided for a calculation of Ahs( ) are the hard-sphere diameter, a, and the packing fraction, x. It is found that j = 0.45 for most liquid metals at temperatures just above their melting points. A hard-sphere solution of the PY equation has also been obtained for binary liquid metal alloys, and provides estimates of the three partial structure factors describing the alloy structure (Ashcroft and Langreth 1967). To the extent that the hard-sphere approximation appears to be valid for the liquid R s, pair potentials should dominate these metals also, at least at short distances. 

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