Ewald lattice sums

Other formulations of the Ewald lattice sum problem based on the particle-particle particle-mesh (PPPM) method of Hockney and Eastwood have appeared. These methods also involve the implementation of Fast Fourier Transforms and are thus of order N n N). It is not clear at this time whether efficient adaptation of the PPPM to nonorthogonal unit cells can be implemented. 

U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen. The smooth particle mesh ewald method. J. Chem. Phys., 103 8577, 1995. Brock A. Luty, Ilario G. Tironi, and Wilfried F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

Ewald summation was invented in 1921 [7] to permit the efl5.cient computation of lattice sums arising in solid state physics. PBCs applied to the unit cell of a crystal yield an infinite crystal of the appropriate. symmetry performing... 

Ewald s formalism reduces the infinite lattice sum to a serial complexity of in the number of particles n, which has been reduced to n logn in more recent formulations. A review of variants on Ewald summation methods which includes a more complete derivation of the basic method is in [3]. 

Ewald Method. The best way to compute the lattice sum in Eq. (8.10.6) is the Ewald fast-convergence method [62], which uses an integral transform ... 

The alternative viewpoints here emphasize that the uniform neutralizing background for the individual contributions just permits the normal electric field to be zero on the boundary. These viewpoints avoid traditional (Valleau and Torrie, 1977) but inconclusive discussions of what periodic images might be doing when lattice sums are conceived with Ewald potentials. 

The Ewald potential is traditionally implemented as a lattice sum (Ziman, 1972 Leeuw et al, 1980). We just outlined a conceptualization of electrostatic interactions in periodic boundary conditions that involved adding a uniform neutralizing background for each charge, and the subsequent solution of the Poisson equation in periodic boundary conditions. Here we discuss the interconnections between that conceptualization and the traditional lattice sums, as presented in many sources, e.g. (Allen and Tildesley, 1987 Frenkel and Smit, 2002 Leeuw et al, 1980). 

For each point charge qi involved in Eq. (6.4). the resulting electrastatic potential decays in proportion to the inverse distance [see Eq. (6.2)], such that the lattice sum buried in the expression for the total energy Uc [see Eq. (6.1)] converges rather slowly. In view of this dilemma, the central idea of the Ewald summation techniques is to rewrite the d -like charge density in Eq. (6.4) as a sum of three contributions, pf (r ), and p] (r ). Each... 

Generalized Ewald Method for Coulombic Lattice Sums... 

The summation in eq. (62) is extended over ligands from the sublattice X. An explicit form of all parameters of the operator nonlinear in lattice variables (eq. 20) may be found in a similar way. If p = 2, conditionally converging lattice sums in eq. (59) should be calculated by use of the Ewald method ... 

Ewald summation is one of the procedures developed to solve the problems just mentioned. While VDW has rapid potential drop across certain interatomic distances due to its 6-12 exponential function, the electrostatic interaction s convergence over the interatomic distance variation is very slow due to its 1/r dependency. The use of a two-step summation (one in real space and one in reciprocal space) for the periodic system will give a more accurate value for the electrostatic interactions (63). One summation is carried out in reciprocal space the other is carried out in real space. Based on Ewald s formulation, the simple lattice sum can be reformulated to give absolutely convergent summations that define the principal value of the electrostatic potential, called the intrinsic potential. Given the periodicity present in both crystal calculations and in dynamics simulations using periodic boundary conditions, the Ewald formulation becomes well suited for the calculation of electrostatic energy and force. 

The atomic multipole expansion of the BI electrostatic potential is extremely useful, when the long-range, purely point-multipolar part of the potential yields an important contribution. This is the case in crystals, where the multipolar sums (up to the quadrupolar potential) are conditionally convergent lattice sums. Special techniques, like Ewald summation method [130, 131] and its generalizations [132] are needed to handle properly these infinite sums. Recently we applied the multipolar BI method, coupled with Ewald summation for the evaluation of electrostatic potentials and fields in zeolite cavities [133] and for the prediction of the IR frequency sequence of the different acidic sites in H-faujasite [134]. 

The appearance of the energy-wave number characteristic is not essential for the derivation, but is quite natural from perturbation theory with pseudopotentials. In the more elaborate treatments, as e.g. the dielectric formulation, the electronic contribution is more complicated, as is outlined in the related section. This contribution has then to be treated in the appropriate way. The main purpose of the Ewald-Fuchs method is to handle the ionic contribution, which is written as in the previous appendix, where q is an arbitrary constant, chosen to obtain optimum convergence for the direct and the reciprocal lattice sum. Denoting the equilibrium positions of the ions by... 

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... 

The first term is the lattice sum effected by the Ewald formula (28), the second is a surface dipole term which depends on the symmetry of the infinite lattice and e. The last term in (30) can be set equal to zero if z ... 

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