Periodic boundaries limitations

Periodic boundary conditions force k to be a discrete variable with allowed values occurring at intervals of lull. For very large systems, one can describe the system as continuous in the limit of i qo. Electron states can be defined by a density of states defmed as follows ... 

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. 

Figure 3 Periodic boundary conditions realized as the limit of finite clusters of replicated simulation cells. The limit depends in general on the asymptotic shape of the clusters here it is spherical. Cations are presented as shaded circles anions as open circles.

Since plane waves are delocalised and of infinite spatial extent, it is natural to perform these calculations in a periodic environment and periodic boundary conditions can be used to enforce this periodicity. Periodic boundary conditions for an isolated molecule are shown schematically in Fig. 8. The molecular problem then becomes formally equivalent to an electronic structure calculation for a periodic solid consisting of one molecule per unit cell. In the limit of large separation between molecules, the molecular electronic structure of the isolated gas phase molecule is obtained accurately. 

Dixon and coworkers [25] have performed several CFD simulations of fixed beds with catalyst particles of different geometries (Figure 15.9). The vast number of surfaces and the problems with meshing the void fraction in a packed bed have made it necessary to limit the number of particles and use periodic boundary conditions to obtain a representative flow pattern. Hollow cylinders have a much higher contact area between the fluid and particles at the same pressure drop. However, with a random packing of the particles, there wiU be a large variation... 

As in classical simulations of biomolecules, there are two general frameworks for setting up QM/MM simulations for a biological system periodic boundary condition (PBC) and finite-size boundary condition (FBC). When the system of interest is small ( 200-300 amino acids), PBC is well suited because the entire system can be completely solvated and therefore structural fluctuations ranging from the residue level to domain scale can potentially be treated at equal footing, within the limit... 

There are several commercial packages that realise the above strategy for molecularly realistic systems. It is useful to discuss some of the limitations. Ideally, one would like to do simulations on macroscopic systems. However, it is impossible to use a computer to deal with numbers of degrees of freedom on the order of /Vav. In lipid systems, where the computations of all the interactions in the system are expensive, a typical system can contain of the order of tens of thousands of particles. Recently, massive systems with up to a million particles have been considered [33], Even for these large simulations, this still means that the system size is limited to the order of 10 nm. Because of this small size, one refers to this volume as a box, although the system boundaries are typically not box-like. Usually the box has periodic boundary conditions. This implies that molecules that move out of the box on one side will enter the box on the opposite side. In such a way, finite size effects are minimised. In sophisticated simulations, i.e. (N, p, y, Tj-ensembles, there are rules defined which allow the box size and shape to vary in such a way that the intensive parameters (p, y) can assume a preset value. 

Therefore, whenever we introduce symmetries into our systems, we risk observing behavior that is inconsistent with that observed when these symmetries are absent. Because opposing surfaces are almost always incommensurate unless they are prepared specifically, it will be important to avoid symmetries in simulations as much as possible. Unfortunately, it can be difficult to make two surfaces incommensurate in simulations, particularly when the interface is composed of two identical crystalline surfaces. These difficulties arise from the fact that only a limited number of geometries conform to the periodic boundary conditions in the lateral direction. Each geometry needs to be analyzed separately... 

Flat single-layer graphene is a zero band-gap semiconductor [50], in which every direction for electron transport is possible. However, when the graphene sheet is rolled up to form a SWCNT, the number of allowed states is limited by quantum confinement in the radial direction [17], i.e. the movement of electrons is confined by the periodic boundary condition [51] ... 

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

Two different boundary conditions are usually used for simulation processes. One is an isolated system and the other is a bulk system in which a periodical boundary condition is employed. The Ewald summation (17) is often introduced in the calculation of Coulombic interactions. For liquids and solutions the latter system has been used mostly, but the former has been examined in studying the dynamic behavior of a single molecule interacting with a limited number of particles. 

The main advantage of periodic boundary conditions is the elimination of edge effects or terminal atom problems that occur in the finite cluster approach. The Hamiltonian in slab calculations is generally limited to density functional theory11, an approach that is not always an appropriate choice. 

The models used in this study were limited under the periodic boundary condition so that the models should be applied to single-phase materials, although the formation of the p phase was experimentally observed to coexist with the a phase at temperatures below... 

Using periodic boundary conditions (BCs), our simulations are based on a lattice construction set up earlier by Berg [135]. We have performed multicanonical MC simulations for the two models with the lattice sizes that correspond to the number of water molecules N = 128, 360, 576, 896, and 1,600. Combining the two fit results in the thermodynamic limit (N —> oo) leads to our final estimate... 

The second approach was to employ periodic boundary conditions and molecular mechanics (COMPASS) to model the solvated SFA.55 73 These simulations were performed with Cerius2 4.2 (Accelrys, Inc.). Periodic boundary conditions create a bulk system with no surface effects and hence, this situation is more realistic compared to the experimental system of SFA dissolved in water. H20 molecules, however, must diffuse to allow motion of the SFA model, so that the SFA model conformations may be restricted due to this limited motion of the surrounding H20 molecules. Note also that periodic simulations must be charge neutral within the... 

---