Conditions periodic boundary

2 Periodic Boundary Conditions. - Computer simulations of interfacial systems have traditionally employed 2D periodicity and implemented a single layer embedded within a vacuum, or, with 3D periodicity, a lamellar structure. Wong and Pettitt devised a new boundary condition which contains only one interface, using an asymmetric unit of space group Pb. The lower half of the simulation cell is obtained by applying to the asymmetric unit a combination of reflection and translation operations across the intervening surface boundary plane.  

The problems already mentioned at the solvent/vacuum boundary, which always exists regardless of the size of the box of water molecules, led to the definition of so-called periodic boundaries. They can be compared with the unit cell definition of a crystalline system. The unit cell also forms an endless system without boundaries when repeated in the three directions of space. Unfortunately, when simulating hquids the situation is not as simple as for a regular crystal, because molecules can diffuse and are in principle able to leave the unit cell. 

A realistic model of a solution requires at least several hundred solvent molecules. To prevent the outer solvent molecules from boiling off into space and minimizing surface effects, periodic boundary conditions are normally employed. The solvent molecules 

If a solvent molecule leaves the central box through the right wall, its image will enter the box through the left wall from the neighbouring box. This means that the resulting solvent model becomes quasi-periodic, with a periodicity equal to the dimensions of the box. 

An alternative form for the central part that also has vanishing second derivatives at both limits is given in eq. (14.30). 

A variation of this is to use a shifting function, which corresponds to a switching function with Ri = 0. Such functions modify the potential for all r values less than R2 and an example is given in eq. (14.31). 

The use of both switching and shifting functions modifies the model, since the potential and forces are changed, and therefore affects the results of the simulation. Whether these changes are significant relative to the other approximations in the model depends on the specific system and properties. 

In the application of either Monte Carlo or molecular dynamics, the number of particles that can be treated is small (at most a few thousand) compared to Avogadro s number. In order that such small systems have properties similar to macroscopic systems, we impose periodic boundary conditions. It is then necessary that the volume V of the system have a shape that is d-dimensional space filling under appropriate translations. For notational convenience, we will suppose it to be a d-dimensional cube of edge L = although in practice in two dimensions V is usually a rectangle rather 

If particle i is in primary volume V at r,-, then image particles are located at positions r,- +vL for all d-vectors v having integer components, —oo i/, 00. Under periodic boundary conditions, each particle interacts with all other particles, whence the potential energy and virial become 

The properties of such periodic systems are discussed in some detail elsewhere and we state the results of interest, 

The system has d ideal-gas degrees of freedom corresponding to free translation of the center of mass. 

While the imposition of periodic boundary conditions is desirable, it should be mentioned particularly in connection with so-called nonequilibrium molecular dynamics that other boundary conditions have been used. This will be discussed in Section 6, but we note that a greater dependence on system size can be expected in such calculations. 

Perhaps the most important disadvantage of a finite size system with walls bounding the available volume is that a lot of particles will be located at or near the wall surface. Consequently, the properties obtained from the simulation will not be the bulk properties of the system. An ingenious solution to the challenges of simulating small system sizes is based on periodic boundary conditions, presented next. 

With periodic boundary conditions we simulate small systems in the bulk state (Fig. 14.3). The trick is to imagine surrounding the primary volume cell with images of itself in all directions. If a particle moves and leaves the central box, it gets replaced by one of its images in the 

Currently, most MD simulations involve systems containing 0(10 -10 ) atoms, although simulations using 0(10 -10 ) atoms have been performed (Omeltchenko et al. 2000) and will become more prevalent as both computer power increases and the need to study such large systems becomes more important. In the more common, smaller simulations, the relatively small system size means that a significant proportion of the 

In order to describe the previous paragraph mathematically, a brief fortran algorithm showing the PBC features for closest neighbor distance is given below in one dimension 

FIGURE 1.1 (a) Minimum-image structure and (b) explicit-image structure showing cutoff radius. 

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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 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... 

Figure B3.3.3. Periodic boundary conditions. As a particle moves out of the simulation box, an image particle moves in to replace it. In calculating particle interactions within the cutoff range, both real and image neighbours are included.

Felderhof B U 1980 Fluctuation theorems for dielectrics with periodic boundary conditions Physice A 101 275-82... 

Figure 3. Floquet band structure for a threefold cyclic barrier (a) in the plane wave case after using Eq. (A.l 1) to fold the band onto the interval —I < and (b) in the presence of a threefold potential barrier. Open circles in case (b) mark the eigenvalues at = 0, 1, consistent with periodic boundary conditions. Closed circles mark those at consistent with sign-changing...

If the simulated system uses periodic boundary conditions, the logical long-range interaction includes a lattice sum over all particles with all their images. Apart from some obvious and resolvable corrections for self-energy and for image interaction between excluded pairs, the question has been raised if one really wishes to enhance the effect of the artificial boundary conditions by including lattice sums. The effect of the periodic conditions should at least be evaluated by simulation with different box sizes or by continuum corrections, if applicable (see below). 

Wood, R.H. Continuum electrostatics in a computational universe with finite cut-off radii and periodic boundary conditions Correction to computed free energies of ionic solvation. J. Chem. Phys. 103 (1995) 6177-6187. 

Very recently, we have developed and incorporated into the CHARMM molecular mechanics program a version of LN that uses direct-force evaluation, rather than linearization, for the fast-force components [91]. The scheme can be used in combination with SHAKE (e.g., for freezing bond lengths) and with periodic boundary conditions. Results for solvated protein and nucleic-... 

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... 

Fig. 1. Periodic boundary conditions protect the inner simulation cell from disturbing effects of having all its particles close to the surface. With PBCs in force, as a particle moves out of the box on one side, one of its images will move back into the box on the opposite side.

A typical molecular dynamics simulation comprises an equflibration and a production phase. The former is necessary, as the name imphes, to ensure that the system is in equilibrium before data acquisition starts. It is useful to check the time evolution of several simulation parameters such as temperature (which is directly connected to the kinetic energy), potential energy, total energy, density (when periodic boundary conditions with constant pressure are apphed), and their root-mean-square deviations. Having these and other variables constant at the end of the equilibration phase is the prerequisite for the statistically meaningful sampling of data in the following production phase. 

Often yon need to add solvent molecules to a solute before running a molecular dynamics simiilatmn (see also Solvation and Periodic Boundary Conditions" on page 62). In HyperChem, choose Periodic Box on the Setup m en ii to enclose a soln te in a periodic box filled appropriately with TIP3P models of water inole-cii les. 

Note MM-i- is derived from the public domain code developed by Dr. Norm an Allinger, referred to as M.M2( 1977), and distributed by the Quantum Chemistry Program Exchange (QCPE). The code for MM-t is not derived from Dr. Allin ger s present version of code, which IS trademarked MM2 . Specifically. QCMPOlO was used as a starting point Ibr HyperChem MM-t code. The code was extensively modified and extended over several years to include molecular dynamics, switching functuins for cubic stretch terms, periodic boundary conditions, superimposed restraints, a default (additional) parameter scheme, and so on. 

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. 

Isolated gas ph ase molecules are th e sim plest to treat com pii tation -ally. Much, if not most, ch emistry lakes place in the liq iiid or solid state, however. To treat these condensed phases, you must simulate continnons, constant density, macroscopic conditions. The usual approach is to invoke periodic boundary conditions. These simulate a large system (order of 10" inoleeti les) as a contiruiotis replication in all direction s of a sm nII box, On ly th e m olceti Ics in the single small box are simulated and the other boxes arc just copies of the single box. 

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

For some simulations it is inappropriate to use standard periodic boundary conditions in all directions. For example, when studying the adsorption of molecules onto a surface, it is clearly inappropriate to use the usual periodic boundary conditions for motion perpendicular to the surface. Rather, the surface is modelled as a true boundary, for example by e, plicitly including the atoms in the surface. The opposite side of the box must still be treated when a molecule strays out of the top side of the box it is reflected back into the simulation cell, as indicated in Figure 6.6. Usual periodic boundary conditions apply to motion parallel to the surface. 

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