Lattice summation methods

The past 20 years have seen a renewal of interest in lattice summation methods, catalyzed by the advances in high-performance computing and the ability thereby provided to approach molecular dynamics and condensed-phase structural problems that had previously seemed inaccessible. In this respect, an important development was the so-called fast multipole method (FMM) [5]. With its help, the electrostatic energies of arrays of charged particles can be evaluated in computing times that are nearly linear in the number of particles. One of the strengths of FMM is that the charge distribution need not be periodic, and methods of Ewald character can be combined with FMM concepts for studies of periodic systems [6]. 

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]. 

Table 6 lists the HF and MBPT(2) band gaps with three basis sets for polymethineimine. The structure of the system used in each calculation is the optimized geometry obtained with the same method and basis. The number of unit cells in the lattice summation is the same as that used in geometry optimization, namely 21. From the table, we can see that electron correlation... 

The electrostatic part of the potential is slowly convergent when lattice summations are carried out, and this problem is overcome by the use of the Ewald method, details of which are given in contrasting but complementary approaches by Tosi (1964) and by Jackson and Catlow (1988). A detailed discussion follows in Chapter 4. 

Solid state applications of the EEM formalism require the structural information (1/Re,p) in each of the equations to be generated by a Madelung-type summation [21]. This is achieved by performing the slowly converging lattice summations in reciprocal space, i.e. the Ewald method [22]. The structural information needed in the model is evaluated as the sensitivity of the external potential (at atom oc) for a charge shift on atom p (or another, symmetrically equivalent a ) ... 

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 direct lattice summations in the Fock matrix elements and the k dependence of the one-electron DM, energy levels and COs are the main difficulties of the HF LCAO method for periodic systems, compared with molecules. A special strategy must be specified for the treatment of the infinite Coloumb and exchange series as well as for the substitution of the integral that appears in DM with a weighted sum extended to a finite set of fc-points. The efficient solution of these problems has been implemented in the CRYSTAL code [23]. These problems are also valid for UHF and ROHF LCAO methods for periodic systems considered in the next subsection. 

The EHT method is noniterative so that the results of COM apphcation depend only on the overlap interaction radius. The more complicated situation takes place in iterative Mulliken-Riidenberg and self-consistent ZDO methods. In these methods for crystals, the atomic charges or the whole of the density matrix are calculated by summation over k points in the BZ and recalculated at each iteration step. The direct lattice summations have to be made in the surviving integrals calculation before the iteration procedure. However, when the nonlocal exchange is taken into account (as is done in the ZDO methods) the balance between direct lattice and BZ summations has to be ensured. This balance is automatically ensured in cychc-cluster calculations as was shown in Chap. 4. Therefore, in iterative MR and self-consistent ZDO methods the increase of the cyclic cluster ensures increasing accuracy in the direct lattice and BZ summation simultaneously. This advantage of COM is in many cases underestimated. 

Molecular dynamics was performed at constant temperature with AMBER 4.1 all-atom force field [121] and Particle Mesh Ewald method (PME) was used for the calculation of electrostatic interactions [122]. This is a fast implementation of the Ewald summation method for calculating the full electrostatic energy of a unit cell in a macroscopic lattice of repeating images. The PME grid spacing was 1.0A. It was interpolated on a cubic B-spline, with the direct set tolerance set to 0.000001. Periodic boundary conditions were imposed in all directions. All solute-solute non-bonded interactions were calculated without jmy cut-off distance, while a non-bonded residue based cutoff distance of 9A was used for the solvent-solvent and for the solute-solvent interactions. The non-bonded pair list was updated every 20 steps and the... 

Statistical mechanics methods such as Cluster Variation Method (CVM) designed for working with lattice statics are based on the assumption that atoms sit on lattice points. We extend the conventional CVM [1] and present a method of taking into account continuous displacement of atoms from their reference lattice points. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. Then the summation over the species in the conventional CVM changes into an integral over r. An example of the 1-D case was done successfully before [2]. The similar treatments have also been done for... 

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