Interdisciplinary Applied Mathematics

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rn+1 = 2rnrn-1 + At2a(t) + O(At4).

Variants of the Verlet algorithm, such as the leap-frog scheme and the velocity Verlet algorithm, are popularly employed for time integration.

4. Data storage and analysis: After the equations of motion are integrated, the relevant properties of the system (e.g., temperature, pressure, volume) are calculated and stored.

In the rest of this section, we provide details on some key steps in MD. Since there are many excellent textbooks on MD (see, e.g., (Allen and Tildesley, 1994)), only the important details are highlighted.

16.1.1 Intermolecular Potentials

The definition of accurate intermolecular potentials is key to any atomistic simulation, and here we provide an overview of some of the intermolecular potentials developed in the past. In general, the potential energy (V) of a system consisting of N interacting particles can be expressed as:

V = £ n(‘.)+EE V2(ri, Vj ) + EE £ Vs(ri, Vj, Vk)+—-,

i    i j>i    i j>ik>j>i


where vi is the position of particle i. The first term on the right-hand side (Vi) is the potential energy due to the external fields, and the remaining terms, which are modeled by intermolecular potentials, represent the particle interactions (e.g., V2 is the potential between pairs of particles and V3 is the potential between particle triplets and so on (Sadus, 1999)). Typically, equation (16.2) is truncated after the second term; i.e., the three-body and higher-order interactions are neglected. The intermolecular interaction potentials have been discussed in detail in (Maitland et al., 1981), and (Stone, 1996). The many-body effects on the intermolecular interactions have been reviewed in (Elrod and Saykally, 1994).

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