Simulations minimum image convention

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. 

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. 

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). ... 

Smith, W., The minimum image convention in non-cubic MD cell, CCP5 Infotmation Quarterly for Computer Simulation of Condensed Phases, 30 35, Infomial Newsletter, Daresbury Laboratory, England, 1989. 

The minimum image convention was used in all these simulations, and no tail corrections to the thermodynamic properties were applied, since we were investigating tte relative performance of various algorithms and did not concentrate on the precise values of thermodynamic properties. 

Periodic boundary conditions refer to the simulation of structures consisting of a periodic lattice of identical subunits. Periodic boundaries help simulate bulk-material, solvent, and crystalline systems. Ideally, a periodic system infinitely replicates in all dimensions to form a periodic lattice. However, in practice, all periodic boundary algorithms imply a cutoff criterion for computational efficiency (Figure 1.1). In these cutoff schemes, each atom interacts with the nearest images of other N - I atoms (minimum-image convention) or only with the explicit images contained in a sphere... 

A fluid molecule in a channel receives not only the above fluid-solid interaction but also those from fluid molecules within the cutoff distance, some of which may exist in other compartments beyond the ultrathin walls. This is the reason the unit cell contains N-hy-N array of channels. The number N was at least three or more, determined so as to satisfy the usual condition of (Unit cell length)/2 > (Cutoff distance). Though the molecules themselves never go beyond the unit cell along the confining direction, the periodic boundary conditions and the minimum image convention for all the three directions were set in the simulations to take the above explained interactions into account. 

For the computation of the free energy Ap in Eq. (6.29) with the theory of energy representation, 200-ps QM/MM simulation was carried out to constract the energy distribution functions p(e) in the solution system, while 400-ps simulation was devoted for the distributions po(e) and Xoi O in the pure solvent system. In the constmction of these distribution functions the solvent molecules of which mass centers were within a sphere Q. of radius 11 A were considered, where the center of the sphere was placed at the mass center of the solute. In the computation of S/j. in Eq. (6.29), our simulations were conducted over 300 ps to yield the energy distribution functions defined in Eqs. (6.31) and (6.32), where all the solvent molecules are involved with the minimum image convention. 

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