Molecular dynamics finite difference methods

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

In molecular dynamics (MD) simulation atoms are moved in space along their lines of force (which are determined from the first derivative of the potential energy function) using finite difference methods [27, 28]. At each time step the evolution of the energy and forces allow the accelerations on each atom to be determined, in turn allowing the atom changes in velocities and positions to be evaluated and hence allows the system clock to move forward, typically in time steps of the order of a few fs. Bulk system properties such as temperature and pressure are easily determined from the atom positions and velocities. As a result simulations can be readily performed at constant temperature and volume (NVT ensemble) or constant temperature and pressure (NpT ensemble). The constant temperature and pressure constraints can be imposed using thermostats and barostat [29-31] in which additional variables are coupled to the system which act to modify the equations of motion. 

Molecular dynamics [MD] is a well-established technique for simulating the structure and properties of materials in the solid, liquid, and gas phases and will only be briefly overviewed here [1]. In these simulations, N atoms are placed in a simulation cell with an initial set of positions and interact via an interatomic potential 7[r]. The force on each particle is determined by its interaction with all other atoms to within an interaction cutoff specified by 7[r]. For a given set of initial particle positions, velocities, and a specification of the position- or time-dependent forces acting on the particles, MD simulations solve the classical Newton s equations of motions numerically via finite-difference methods to calculate the time evolution of the particle trajectories. 

A common approach in the various finite difference methods used to integrate the equations of motions for classical molecular dynamics simulations is that it is assumed that the positions, velocities, and accelerations (as well as all other dynamic properties) can be approximated using Taylor series expansions ... 

One of the most widely used finite difference methods in classical molecular dynamics simulations is the Verlet algorithm (Verlet 1967). In the Verlet algorithm, the positions and accelerations at time t and the positions from the previous time step R(f - dt) are used to calculate the updated positions R(f + df) using the equation ... 

Unfortunately, molecular dynamics are computationally very expensive which makes simulating radiation kinetics very difficult. This problem is further compounded by the necessity to perform many realisations to obtain statistically significant results something which is not practical at present. In order to solve the ordinary differential equations of motion to generate a trajectory, a range of finite different methods are available (for example the velocity Verlet algorithm [26]). 

K. Sharp,/. Comput. Chem., 12, 454 (1991). Incorporating Solvent and Ion Screening into Molecular Dynamics Using the Finite-Difference Poisson-Boltzmann Method. 

Numerous approaches to handling molecular solute-continuum solvent electrostatic interactions, are described in detail in several recent reviews. - The methods most widely used and most often applied to Brownian dynamics simulations, however, fall in the category of finite difference solutions to the Poisson-Boltzmann equation. So, here we concentrate on that approach, providing a review of the basic theory along with the state-of-the-art methods in calculating potentials, energies, and forces. 

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