Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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synthetic polymers, biopolymers), in addition to non-bonding forces (van der Waals and electrostatic), it is necessary to consider the intramolecular bonding interactions illustrated in Figure 6.4b. The elementary models of this kind include contributions due to: (i) bond stretching: 2-atoms vibrations about the equilibrium bond length (6.8) 1 (ii) angle bending: 3-atoms vibrations about the equilibrium angle they define (6.8) 2 1 2 ∑ bonds k α i j r i j − r 0 i j 2 , 1 2 ∑ bendangles k β i jk θ i jk − θ 0 i jk 2 , (6.8) (iii) torsion (dihedrals, proper and improper, 4-atoms torsional vibrations), and (iv) various cross-terms [29]. These so-called bonded interactions are not further discussed in this introduction. The details are available in [29], [30]. Further considerations of empirical potentials go beyond the objectives of this intro duction and can be found in literature (e.g., [24], [30]; and many others). In the last thirty years, empirical potentials have been developed in a targeted way - for specific material systems with a range of applicability in mind. The ultimate test of any empirical potential is its success in simulating properties of interest. However, it seems appropriate to conclude this brief review by adding that in the constant competition between the more sophisticated and the spatially larger MD models, under the constraints imposed by computational capabilities, the latter are still considered more advisable in terms of meaningful results. In other words, it is generally accepted that it is better to increase the size of the MD system and simplify the potential, than to do the opposite. This trend has resulted in the development of parallel processing [11], [31], [32] without which MD simulations in the The empirical potentials presented in Chapter 6.2.2 have an unlimited range. In MD simulations, it is a common custom to establish the cut-off distance ( r cut ) and to neglect interatomic interactions for distances that exceed it because the corresponding forces are insignificant (Figure 6.5). This neglect of interatomic action in the range of potential asymptotic approach to zero leads to program simplification and huge computational savings due to a drastic reduction in the number of interacting atomic pairs. However, a simple shortening of the potential would lead to a new problem: whenever the mutual distance between pairs of atoms "crossed" over the cutting distance, there would be a small, abrupt change in the energy of the system. A large number of such events could, on the one hand, have an impact on the law of conservation of energy, and on the other, affect the physics of subtle micro-processes that depend on the details of the local energy state. Therefore, limiting the range of potentials is most often done by a smooth transition in the attractive range, for example, by using a cubic spline. As an example, the approach of Holian et al. [27] is based on the Lennard-Jones potential (6.5) interrupted at r spl ≈ 1 . 109 r 0 and replaced by the cubic spline in r 2 φ spl ( r ) = − A ( r 2 cut − r 2 ) 2 + B ( r 2 cut − r 2 ) 3 which reaches zero at r cut . The spline parameters are chosen to ensure continuity of coordinates, inclination and curvature ( C 0, C 1, and C 2) at the point of intersection ( r = r spl ): contemporary research cannot be imagined. Shortening the range of potential