Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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of the periodic cell. Therefore, several alternative MD methods have been proposed as potential solutions for above mentioned problems in a series of papers [48]-[51]. In these studies, the shape of a periodic cell is treated as a phenomenological variable subject to change over time.

Figure 6.9: Periodic boundary conditions and the simulation cell.

The time required to calculate the interactions between the N atoms of a 2D system using algorithm (6.9) is proportional to N 2 [5]. In order to improve the speed of program execution, it is very useful to maintain a list of neighbors of each atom that Verlet originated in his classic work [52]. For example, the list of neighbors of the i -th atom is formed by including all atoms at a distance less than the prescribed cut-off distance. Between periodic updates of the neighbor list (typically, several calculation cycles), the program does not check all other atoms in the system but only those that appear in the list for a given atom. 6.2.6 Temperature and Pressure Control In the absence of dissipative forces, the equation of motion of classical mechanics results in the conservation of the total energy of the system which corresponds to a microcanonical ( N , V , E ) ensemble. The calculated temperature (6.16) and pressure in the standard MD formulation are not constrained and can vary significantly during the simulation (Figure 6.7). Most often, it is of utmost importance to evaluate these temperature and pressure changes. On the other hand, sometimes it is of interest in MD simulations to constrain temperature or pressure or both. The importance of temperature and pressure control has made the topics of virtual thermostats and barostats very popular among researchers, resulting in an abundance of different methods (e.g., [53]-[55]).