The Periodic Boundary Conditions

It is also possible to simulate liquid droplets by surrounding a solute by a finite number of water molecules and performing the simulation without a periodic box. The water, of course, eventually evaporates and moves away from the solute when periodic boundary conditions are not imposed. If the water is initially added via periodic boundary conditions, you must edit the resulting HIN file to remove the periodic boundary conditions, if a droplet approach is desired. 

Periodic boundary conditions can also be used to simulate solid state conditions although HyperChem has few specific tools to assist in setting up specific crystal symmetry space groups. The group operations Invert, Reflect, and Rotate can, however, be used to set up a unit cell manually, provided it is rectangular. 

HyperChem avoids the discontinuity and anisotropy problem of the implied cutoff by imposing a smoothed spherical cutoff within the implied cutoff. When a system is placed in a periodic box, a switched cutoff is automatically added. The default outer radius, where the interaction is completely turned off, is the smallest of 1/2 R, 1/2 R and 1/2 R, so that the cutoff avoids discontinuities and is isotropic. This cutoff may be turned off or modified in the Molecular Mechanics Options dialog box after solvation and before calculation. 

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Hor the periodic boundary conditions described below, the ctitoff distance is fixed by the nearest image approximation to be less than h alf th e sm allest box len gth. W ith a cutoff an y larger, more than nearest images would be included. 

Figure 2 Snapshot from an MD simulation of a multilamellar liquid crystalline phase DPPC bilayer. Water molecules are colored white, lipid polar groups gray, and lipid hydrocarbon chains black. The central simulation cell containing 64 DPPC and 1792 water molecules, outlined m the upper left portion of the figure, is shown along with seven replicas generated by the periodic boundary conditions. (From Ref. 55.)...

Consider a more general eigenvalue equation without imposing the periodic boundary condition,... 

Structurally, carbon nanotubes of small diameter are examples of a onedimensional periodic structure along the nanotube axis. In single wall carbon nanotubes, confinement of the stnreture in the radial direction is provided by the monolayer thickness of the nanotube in the radial direction. Circumferentially, the periodic boundary condition applies to the enlarged unit cell that is formed in real space. The application of this periodic boundary condition to the graphene electronic states leads to the prediction of a remarkable electronic structure for carbon nanotubes of small diameter. We first present... 

The orientational relationships between the martensite and austenite lattice which we observe are partially in accordance with experimental results In experiments a Nishiyama-Wasserman relationship is found for those systems which we have simulated. We think that the additional rotation of the (lll)f< c planes in the simulations is an effect of boundary conditions. Experimentally bcc and fee structure coexist and the plane of contact, the habit plane, is undistorted. In our simulations we have no coexistence of these structures. But the periodic boundary conditions play a similar role like the habit plane in the real crystals. Under these considerations the fact that we find the same invariant direction as it is observed experimentally shows, that our calculations simulate the same transition process as it takes place in experiments. The same is true for the inhomogeneous shear system which we see in our simulations. 

Here, ry is the separation between the molecules resolved along the helix axis and is the angle between an appropriate molecular axis in the two chiral molecules. For this system the C axis closest to the symmetry axes of the constituent Gay-Berne molecules is used. In the chiral nematic phase G2(r ) is periodic with a periodicity equal to half the pitch of the helix. For this system, like that with a point chiral centre, the pitch of the helix is approximately twice the dimensions of the simulation box. This clearly shows the influence of the periodic boundary conditions on the structure of the phase formed [74]. As we would expect simulations using the atropisomer with the opposite helicity simply reverses the sense of the helix. 

The heptane water and toluene water interfaces were simulated by the use of the DREIDING force field on the software of Cerius2 Dynamics and Minimizer modules (MSI, San Diego) [6]. The two-phase systems were constructed from 62 heptane molecules and 500 water molecules or 100 toluene molecules and 500 water molecules in a quadratic prism cell. Each bulk phase was optimized for 500 ps at 300 K under NET ensemble in advance. The periodic boundary conditions were applied along all three directions. The calculations of the two-phase system were run under NVT ensemble. The dimensions of the cells in the final calculations were 23.5 A x 22.6 Ax 52.4 A for the heptane-water system and 24.5 A x 24.3 A x 55.2 A for the toluene-water system. The timestep was 1 fs in all cases and the simulation almost reached equilibrium after 50 ps. The density vs. distance profile showed a clear interface with a thickness of ca. 10 A in both systems. The result in the heptane-water system is shown in Fig. 3. Interfacial adsorption of an extractant can be simulated by a similar procedure after the introduction of the extractant molecule at the position from where the dynamics will be started. 

A polymer coil does not only possess a structure on the atomistic scale of a few A, corresponding to the length of covalent bonds and interatomic distances characteristic of macromolecules are coils that more or less, obey Gaussian statistics and have a diameter of the order of hundreds of A (Fig. 1.2) [17]. Structures of intermediate length scales also occur e. g., characterized by the persistence length. For a simulation of a polymer melt, one should consider a box that contains many such chains that interpenetrate each other, i. e., a box with a linear dimension of several hundred A or more, in order to ensure that no artefacts occur attributable to the finite size of the simulation box or the periodic boundary conditions at the surfaces of the box. This ne-... 

It was necessary periodically to generate an adiabatic trajectory in order to obtain the odd work and the time correlation functions. In calculating E (t) on a trajectory, it is essential to integrate E)(t) over the trajectory rather than use the expression for E (T(f)) given earlier. This is because is insensitive to the periodic boundary conditions, whereas j depends on whether the coordinates of the atom are confined to the central cell, or whether the itinerant coordinate is used, and problems arise in both cases when the atom leaves the central cell on a trajectory. 

In all considered above models, the equilibrium morphology is chosen from the set of possible candidates, which makes these approaches unsuitable for discovery of new unknown structures. However, the SCFT equation can be solved in the real space without any assumptions about the phase symmetry [130], The box under the periodic boundary conditions in considered. The initial quest for uy(r) is produced by a random number generator. Equations (42)-(44) are used to produce density distributions T(r) and pressure field ,(r). The diffusion equations are numerically integrated to obtain q and for 0 < s < 1. The right-hand size of Eq. (47) is evaluated to obtain new density profiles. The volume fractions at the next iterations are obtained by a linear mixing of new and old solutions. The iterations are performed repeatedly until the free-energy change... 

Rao and Singh32 calculated relative solvation free energies for normal alkanes, tetra-alkylmethanes, amines and aromatic compounds using AMBER 3.1. Each system was solvated with 216 TIP3P water molecules. The atomic charges were uniformly scaled down by a factor of 0.87 to correct the overestimation of dipole moment by 6-31G basis set. During the perturbation runs, the periodic boundary conditions were applied only for solute-solvent and solvent-solvent interactions with a non-bonded interaction cutoff of 8.5 A. All solute-solute non-bonded interactions were included. Electrostatic decoupling was applied where electrostatic run was completed in 21 windows. Each window included 1 ps of equilibration and 1 ps of data... 

The periodic boundary conditions applied in the system have the consequence that one bilayer leaflet can interact in the normal direction with the other leaflet, not only through the contact region in the core of the bilayer, but also through the water phase. To minimise artifacts, one should systematically increase the size of the water phase. However, this is expensive, especially if the main interest is in the behaviour of the lipids. Another solution is to cut off the... 

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] ... 

The FeSa (100) surfaces are modeled using the supercell approximation. Surfaces are cleaved fi om a GGA optimized crystal structure of pyrite. A vacuum spacing of 1.5 nm is inserted in the z-direction to form a slab and mimic a 2D surface. This has been shown to be sufficient to eliminate the interactions between the mirror images in the z-direction due to the periodic boundary conditions. 

The Monte Carlo (MC) simulation is performed using standard procedures [33] for the Metropolis sampling technique in the isothermal-isobaiic ensemble, where the number of molecules N, the pressure P and the temperature T are fixed. As usual, we used the periodic boundary conditions and image method in a cubic box of size L. In our simulation, we use one F embedded in 1000 molecules of water in normal conditions (T—29S K and P= 1 atm). The F and the water molecules interact by the Lennard-Jones plus Coulomb potential with three parameters for each interacting site i (e, o, - and qi). 

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