Periodic Boundary Conditions

Contents

Introduction

Periodic boundary conditions refers to the simulation of molecular systems in a periodic lattice of identical subunits. By applying periodic boundaries to simulations, the influence of bulk solvent or crystalline environments can be included, thereby improving the rigor and realism of a molecular model. This section describes the concept of periodic boundary conditions as implemented in the Discover program and the application of techniques.

The Discover program accepts only Cartesian coordinates. This is important when using asymmetric space groups, since the symmetry operators assume that the input coordinates correspond to the standard asymmetric unit as defined in the International Tables for Crystallography (Reidl 1983).

It is assumed that the x Cartesian axis corresponds to the a crystal axis and that the b axis lies in the x,y plane (Figure 2-2).

All the molecules in an asymmetric unit or unit cell must be complete molecules for calculations with Discover version 2.9.5, since it cannot handle bonds between symmetrically related objects. However, Discover version 95.0 can handle bonds across cell boundaries.

Minimum-Image Model

The minimum-image model is available in the Discover 2.9.5 program only.

The left side of Figure 2-3 shows a solute molecule surrounded by enough solvent to occupy the volume (and shape) of a cube. A simulation carried out on this isolated cubic system is a poor approximation of what would happen in a true bulk solvent environment. For example, the solute can diffuse toward a surface or solvent molecules can evaporate. To remedy this, on the right of Figure 2-3 the cube is replicated in three dimensions to form a 3 x 3 x 3 lattice of identical cubes. This is a much better representation of bulk solvent for the interior cube, because molecules near the surfaces now interact with solvent in adjacent cubes. The imaged atoms are used to calculate energies and forces on the real atoms in the interior cube. The energies and forces on the imaged atoms themselves are not calculated because their motions are computed as symmetry operations on the real atoms, for example, by translations along the cubic axes.

Consider the implications of this model for a specific case. In Figure 2-4, molecule A1 is located near an edge of the square. (For simplicity, this discussion focuses on a two-dimensional lattice.) In addition, eight images of A1 (A2-A9) are present in the adjacent symmetrically related squares. Consider the interactions of molecules A with molecules B. The closest image of B to A1 is actually not B1, but rather B5. If molecules in the interior cell are allowed to interact only with the molecule or molecular image closest to it, this is called a minimum-image model. Each molecule interacts only with those molecules and images within a distance of half the cell size. The advantage of this approach is its simplicity. It is straightforward to compute as required between a given pair of molecules without explicitly keeping track of the images in neighboring cells. All periodic boundary algorithms imply a cutoff criterion, but the minimum-image convention implies a maximum distance for this cutoff of no more than half the cell dimensions.

Explicit-Image Model

The Discover program can also use a more general approach by generating explicit periodic images of the interior molecules, called ghost molecules, which interact with the interior molecules. These ghost molecules are replicated to as great a distance as necessary (but no farther than necessary) to satisfy the desired potential energy cutoff criteria. In fact, since the nonbond potential is cut off on the basis of atom groups (i.e., only if two groups are within the cutoff is the interaction calculated), only those groups of molecular ghosts that are within the cutoff of a real group are included in the ghost atom list.

Ghost molecules shadow their symmetrically related counterparts. For example, the left side of Figure 2-5 shows molecule A1 interacting with several images of B (B1, B2, B3, B5) within the specified cutoff radius (shown as a magenta circle centered on A1). Note, in fact, that A1 interacts with several of its own images as well (A3, A5, A6, A8). However, when it comes time to move the molecules (in a dynamics step or minimization iteration), only the real molecules (A1 and B1) are moved according to the accumulated forces each molecule has felt. The ghost molecule positions are simply regenerated by applying the defined symmetry relations to the new positions of the molecules.

The right side of Figure 2-5 shows which molecules in the adjacent unit cells become explicit ghost molecules for a given cutoff distance. Note that not every molecule in an adjacent cell becomes a ghost. However, if a cutoff distance that is longer than the cell length is used, ghosts from unit cells beyond the nearest neighbor cells may be included. As molecules move in and out of the boundaries, the molecules that are ghosts can change. Therefore, the ghost list is regenerated periodically. Note that nonbond interactions do not have to be calculated between ghost atoms. This helps to significantly reduce computational time.

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Describe-System access Empirical Fit of the Surface Nonbond Cutoffs

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