Minimum-image convention

The Finite Range Problem. The minimum image convention requires (1) the use of interaction energy functions that are of finite range, i.e. that are non-zero only for distances below a certain limit, R. As a consequence, only a fraction of all minimum image pairs actually interact with non-zero energy this fraction must be less than ir/2d, i.e., in two dimensions maximally 0.79, in three dimensions maximally 0.52 (and actually often less than 0.3). It is desirable to efficiently eliminate from consideration all noninteracting atom pairs. 

Both a uniform bulk fluid and an inhomogeneous fluid were simulated. The latter was in the form of a slit pore, terminated in the -direction by uniform Lennard-Jones walls. The distance between the walls for a given number of atoms was chosen so that the uniform density in the center of the cell was equal to the nominal bulk density. The effective width of the slit pore used to calculate the volume of the subsystem was taken as the region where the density was nonzero. For the bulk fluid in all directions, and for the slit pore in the lateral directions, periodic boundary conditions and the minimum image convention were used. 

This holds whether or not the singlet potential is present. In the case of periodic boundary conditions, it is quite important to use the minimum image convention for all the separations that appear in this expression. This may be rewritten in the convenient form [4]... 

Fig. 2. (a) Periodic images surrounding the simulation box. Interactions are computed with respect to the nearest image which is indicated by the circle, (b) Violation of the minimum image convention resulting from the interaction of QM particle with point charge 1. 

In the simulations, the traditional algorithm was applied to an off-lattice at each step, and periodic boundary conditions were employed. The interactions were truncated using the minimum image convention, and the information was recorded every 50 cycles, after the system reached equilibrium 1000 samples were averaged in each simulation. 

Fock molecular orbital (HF-MO), Generalized Valence Bond (GVB) [49,50] and the Complete Active Space Self-consistent Filed (CASSCF) [50,51], and full Cl methods. [51] Density Functional Theory (DFT) calculations [52-54] are also incorporated into AIMD. One way to perform liquid-state AIMD simulations, is presented in the paper by Hedman and Laaksonen, [55], who simulated liquid water using a parallel computer. Each molecule and its neighbors, kept in the Verlet neighborlists, were treated as clusters and calculated simultaneously on different processors by invoking the standard periodic boundary conditions and minimum image convention. 

The inter-particle distance used in the simulation is calculated using the minimum image convention. It dictates that the distance between two particles m and k is the smallest of all the possible distances between particle m and k including all the replica images of particle k. 

The use of the periodic boundary conditions in the two directions perpendicular to the interface normal (X and Y) implies that the system has infinite extent in these directions. To make the computational cost reasonable, one must truncate the number of interactions that each molecule experiences. The simplest possible technique is to include, for each molecule i, the interaction with all the other molecules that are within a sphere of radius which is smaller than half the shortest box axis. One selects, from among the infinite possible images of each molecule, the one that is the closest to the molecule i under consideration. This is called the minimum image convention, and more details about its implementation can be found elsewhere [2]. To arrive at the correct bulk properties, any ensemble average calculated by this technique must be corrected for the contribution of the interactions beyond the cutoff distance. The fixed analytical corrections are calculated by assuming some simple form of the statistical mechanics distribution function for distances greater then R. ... 

However, because of periodic boundary conditions, one needs to make sure that as far as short-range interaction potentials are concerned, a molecule in the simulation cell interacts only with another peirticle in the simulation cell or one of its periodic images depending on which is closest. Tliis so-called minimum image convention can easily be implemented through the equations... 

Molecular dynamics simulations can be done on molecules in the gas phase (in vacuo), in the liquid phase as a pure liquid or dilute solution, and in the solid phase. In the simulation of molecules in the liquid and solid phase, periodic boundary conditions are used to reduce the surface effects because of the limited number of molecules that can reasonably be studied. The main principle is that as an atom or molecule leaves the main box, its image from one of the adjacent boxes enters. A natural consequence of periodic boundary conditions is the concept of minimum image convention. That is, a molecule will interact with all the N-1 molecules whose centers lie within a region of the same size and shape as that of the original box (see Figure 4). ... 

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