Computational methods periodic boundary conditions

The use of periodic boundary conditions also allows for an efficient evaluation of the ion-ion interaction. Ewald developed a method to compute the Coulomb energy associated with long range ion-ion interactions in solids. The Coulomb energy due to interactions between an ion at position R2 and an array of ions positioned at Rj+i is given by... 

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

One example of non-IRC trajectory was reported for the photoisomerization of cA-stilbene.36,37 In this study trajectory calculations were started at stilbene in its first excited state. The initial stilbene structure was obtained at CIS/6-31G, and 2744 argon atoms were used as a model solvent with periodic boundary conditions. In order to save computational time, finite element interpolation method was used, in which all degrees of freedom were frozen except the central ethylenic torsional angle and the two adjacent phenyl torsional angles. The solvent was equilibrated around a fully rigid m-stilbene for 20 ps, and initial configurations were taken every 1 ps intervals from subsequent equilibration. The results of 800 trajectories revealed that, because of the excessive internal potential energy, the trajectories did not cross the barrier at the saddle point. Thus, the prerequisites for common concepts of reaction dynamics such TST or RRKM theory were not satisfied. 

The structures of the surfaces, the surface adsorption and the alkali-doped crystal and the atom diffusion path (cf. Section 4) were investigated by different quantum-chemical methods. We used foremost ab initio methodologies. The main computational tool utilized was the program CRYSTAL [54]. This program makes it possible to treat molecules and in particular crystalline solids and surfaces at an ab initio level of theory for surfaces and solids the periodic boundary conditions are applied in 2 or 3 dimensions [55]. The familiar Gaussian basis sets can be used for systems ranging from crystals to isolated molecules, which enables systematic comparative studies of chemical properties in different forms of matter. In our studies, split-valence basis sets were used [56]. 

This general technique was employed to simulate the motion of a mono-layer of identical spherical particles subjected to a simple shear. Confinement to a monolayer represents tremendous economies of computer time compared with a three-dimensional simulation. Hopefully, these highly specialized monolayer results will provide comparable insights into the physics of three-dimensional suspensions. Periodic boundary conditions were used in the simulation, and the method of Evans (1979) was incorporated to reproduce the imposed shear. 

Fock molecular orbital (HF-MO), Generalized Valence Bond (GVB) [49,50] and the Complete Active Space Self-consistent Filed (CASSCF) [50,51], and full Cl methods. [51] Density Functional Theory (DFT) calculations [52-54] are also incorporated into AIMD. One way to perform liquid-state AIMD simulations, is presented in the paper by Hedman and Laaksonen, [55], who simulated liquid water using a parallel computer. Each molecule and its neighbors, kept in the Verlet neighborlists, were treated as clusters and calculated simultaneously on different processors by invoking the standard periodic boundary conditions and minimum image convention. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

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