Generalized Ewald Method

Generalized Ewald Method for Coulombic Lattice Sums... 

Variational electrostatic projection method. In some instances, the calculation of PMF profiles in multiple dimensions for complex chemical reactions might not be feasible using full periodic simulation with explicit waters and ions even with the linear-scaling QM/MM-Ewald method [67], To remedy this, we have developed a variational electrostatic projection (VEP) method [75] to use as a generalized solvent boundary potential in QM/MM simulations with stochastic boundaries. The method is similar in spirit to that of Roux and co-workers [76-78], which has been recently... 

An analogous implementation for the standard Ewald method has been presented [44]. Conversely, direct use of the Ewald sum [45] or approximations to it [46 18], which are pairwise decomposable and hence suitable for MC simulations, have generally proven to be too inefficient for most modern applications [49]. Additionally, it should be pointed out that Ewald sums — independent of implementation — are incompatible with implicit solvent models that model a spatially varying dielectric with anything more than trivial functional dependencies [45]. 

Kuwajima, S. and Warshel, A. (1988) The extended Ewald method a general treatment of long-range electrostatic interactions in microscopic simulations, J. Chem. Phys. 89, 3751-3759. 

The coefficients Qjj> ( ) for given k only depend on the lattice structure (their explicit form, which is not needed here, can be found in the book by Born and Huang (16), eqn (30.31) a detailed description of the Ewald method of separation of the field is also given there). These coefficients for k —> 0 tend to values which do not depend on the direction s = k/k. It follows from formulas (2.40) and (2.42) that the quantity LQ/j(k) and, in consequence, the eigenvalues of the matrix .np (see (2.19)]) are, in general, nonanalytic functions of k. 

Here we show the dependence of the rms error in the force in Eq. 26 on the number of charged particles and their valence. Since the assumptions and arguments involved are of a rather general nature, the result is not specific to a certain type of Ewald method. 

Polarization effects, which induce fluctuating dipoles in addition to permanent dipoles on polarizable atoms or molecules, do not produce essential modifications to the energy expressions of pure permanent dipolar systems when written in the context of the Ewald method. Generalization of the Ewald expressions for mixtures of molecules possessing charges, dipoles, quadrupoles and polarizabilities can be found in [49,50] and [51]. 

The sum (6.50) can be calculated for k kj, for example, by the Ewald method. However, for k = kj the series (6.50) appears to be divergent [95]. This divergence is the result of the general asymptotic properties of the approximate density matrix calculated by the summation over the special poits of BZ (see Sect. 4.3.3). The difficulties connected with the divergence of lattice sums in the exchange part have been resolved in CNDO calculations of solids by introduction of an interaction radius... 

In this section we describe the methods to extend Ewald sum methodologies to accelerate the calculation of the intermolecular interactions using PBC. For simplicity, we begin with a generalization of Ewald sums to interacting spherical Hermite Gaussians (e.g. GEM-0 [14]). This is followed by the extension to arbitrary angular momentum. Finally, we describe the implementation of methods to speed up both the direct an reciprocal terms in the Ewald sum [62],... 

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]. 

Ewald 37) developed a method for evaluating the Madelung constants and other sums for crystal lattices which depends on the use of a 5-function transformation. This is probably the method of most general application though it suffers from the disadvantage that the convergence of the series obtained is less rapid than that obtained by the Emersleben method. 

The well-known correction for the speed of the transition of a reflection through the Ewald membrane is attributed to a lecture given by Lorentz. In its form applicable to a perfect single crystal it normalizes the intensity of a single reflection to the shortest traversal of the Ewald sphere. This motion is brought about by the rotation of the crystal in direct space. A consequence is that the correction is not only dependent upon the rotation vector of the crystal but also on the detection method. The general formulation takes the form ... 

The cell multipole method (also called the fast multipole method) is an algorithm that enables all N N — 1) pairwise non-bonded interactions to be enumerated in a time that scales linearly with N, rather than N, as in the standard Ewald approach [Greengard and Roklin 1987 Ding et al. 1992a, b Greengard 1994]. The cell multipole method can be used to evaluate interactions that can be expressed in the following general form ... 

Raw computational speed has been considered one of the key advantages of the GB model. However, note that the cost of a calculation based directly on Eq. (3) is generally O(N ) for a system of N atoms, while the scaling is more favorable, N log(N), for Ewald-based methods used in explicit solvent simulations. For large systems, e.g. the nucleosome (25,000 atoms), the number of nanoseconds of MD per GPU hour may actually be less in a GB-based simulation (without additional approximations such as cut-offs) than in a comparable explicit solvent run [54], although the conformational search is still much faster in the implicit solvent. 

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