Periodic boundary condition method

Combined Use of Periodic Boundary Condition Methods and Embedded Cluster Methods... 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

There are three different algorithms for the calculation of the electrostatic forces in systems with periodic boundary conditions (a) the (optimized) Ewald method, which scales like (b) the Particle Mesh... 

The tests in the two previous paragraphs are often used because they are easy to perform. They are, however, limited due to their neglect of intermolecular interactions. Testing the effect of intennolecular interactions requires much more intensive simulations. These would be simulations of the bulk materials, which include many polymer strands and often periodic boundary conditions. Such a bulk system can then be simulated with molecular dynamics, Monte Carlo, or simulated annealing methods to examine the tendency to form crystalline phases. 

HyperChem supplements the standard MM2 force field (see References on page 106) by providing additional parameters (force constants) using two alternative schemes (see the second part of this book. Theory and Methods). This extends the range of chemical compounds that MM-t can accommodate. MM-t also provides cutoffs for calculating nonbonded interactions and periodic boundary conditions. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

The situation is envisaged in which the total energy of a box of atoms can be calculated, and one wants to obtain the excess energy of an interface which has been constructed within the box. The simplest situation is if a static calculation has been made and the atomic positions are relaxed to the structure of minimum energy. However, free energy calculations are also feasible. Periodic boundary conditions parallel to the interface are employed, and perhaps also three dimensional periodicity, which implies that two boundaries per box are necessary. These technicalities as well as the method for calculating energies will not be discussed further here. 

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... 

The efficiency of the methods outlined above has been tested by calculating the intermolecular Coulomb energies and forces for a series of water boxes (64,128,256, 512 and 1024) under periodic boundary conditions [15, 62], The electron density of each monomer is expanded on five sites (atomic positions and bond mid-points) using two standard ABSs, A2 and PI.These sets were used to fit QM density of a single water molecule obtained at the B3LYP/6-31G level. We have previously shown that the A1 fitted density has an 8% RMS force error with respect to the corresponding ab initio results. In the case of PI, this error is reduced to around 2% [15, 16], Table 6-1 shows the results for the 5 water boxes using both ABSs (Table 6-7). 

Based on the same underlying principles as the molecular-based quantum methods, solid-state DFT represents the bulk material using periodic boundary conditions. The imposition of these boundary conditions means that it becomes more efficient to expand the electron density in periodic functions such as plane waves, rather than atom-based functions as in the molecular case. The efficiency of the calculations is further enhanced by the use of pseudo-potentials to represent the core electrons and to make the changes in the electron density... 

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