# Quelle:Ry/Biosym MSI - Integration Algo 1995

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 Titel Molecular Dynamics - Integration Algorithms Jahr 1995 Anmerkung Chapter from "The Discover Program - Documentation" Release 95.0/3.00, October 1995 URL http://www.uoxray.uoregon.edu/local/manuals/biosym/discovery/General/Dynamics/Integration_Algo.html Literaturverz. nein Fußnoten nein Fragmente 2

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At its simplest, molecular dynamics solves Newton’s equation of motion:

$\mathrm{F}_i(t)=m_i \, \mathrm{a}_i (t)=m_i \, \frac{\partial ^{2}\mathrm{r}_i}{\partial t_i^{2}}$ (2.1)

where Fi is the force, mi is the mass and ai is the acceleration of atom i.

At its simplest, molecular dynamics solves Newton's familiar equation of motion:

Eq. 5-1: $\mathbf{F}_i(t)=m_i \mathbf{a}_i (t)$

where Fi is the force, mi is the mass, and ai is the acceleration of atom i.

 Anmerkungen Also refer to the next page Ry/Fragment 035 01. Sichter

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The force on atom i can be computed directly from the derivative of the potential energy V with respect to the coordinates ri:

$\mathrm{F}_i=-\frac{\partial V}{\partial r_i}$ (2.2)

The classical equations of motion are deterministic, which means that once the initial coordinates and velocities and other dynamic information at time t are known, the positions and velocities at time t + Δt are determined (calculated). The coordinates and velocities for a complete dynamics run are called the trajectory. The time step Δt depends on the integration method as well as the system itself. Time step should be small enough in order to avoid integration errors. Because of fast vibrational motions of the atoms, a time step of 1 fs is usually used. Although the initial coordinates are known, the initial velocities are randomly generated at the beginning of a dynamics run, according to the desired temperature.

Analytical integration of equations such as (2.1) is not possible for large systems, so numerical integration should be performed. Molecular dynamics is usually applied to large systems. Energy evaluation is time consuming and the memory requirements are large. To generate the correct statistical ensembles, energy conservation is also important. Thus, the basic criteria for a good integrator for molecular simulations are as follows:

• It should be fast, ideally requiring only one energy evaluation per time step.
• It should require little computer memory.
• It should permit the use of a relatively long time step.
• It must show good conservation of energy.

Variants of the Verlet [24] algorithm of integrating the equations of motion (Equation (2.1)) are perhaps the most widely used method in molecular dynamics. The advantages of Verlet integrators is that these methods require only one energy evaluation per step, require only modest memory, and also allow a relatively large time step to be used.

24. Verlet, L. Phys. Rev. 1967 , 159.

The force on atom i can be computed directly from the derivative of the potential energy V with respect to the coordinates ri:

Eq. 5-2:

Notice that classical equations of motion are deterministic. That is, once the initial coordinates and velocities are known, the coordinates and velocities at a later time can be determined. The coordinates and velocities for a complete dynamics run are called the trajectory.

A standard method of solving ordinary differential equation such as Eq. 5-2 numerically is the finite difference method. The general idea is as follows. Given the initial coordinates and velocities and other dynamic information at time t, the positions and velocities at time t + Δt are calculated. The timestep Δt depends on the integration method as well as the system itself. All the solution methods offered by the Discover program belong to this category.

Although the initial coordinates are determined in the input file or from a previous operation such as minimization, the initial velocities are randomly generated at the beginning of a dynamics run, according to the temperature and a random number seed. [...]

Criteria of Good Integrators in Molecular Dynamics

Molecular dynamics is usually applied to a large system. Energy evaluation is time consuming and the memory requirement is large. To generate the correct statistical ensembles, energy conservation is also important. Thus, the basic criteria for a good integrator for molecular simulations are as follows:

• It should be fast, ideally requiring only one energy evaluation per timestep.
• It should require little computer memory.
• It should permit the use of a relatively long timestep.
• It must show good conservation of energy.

[...]

Verlet Leapfrog Integrator

Variants of the Verlet (1967) algorithm of integrating the equations of motion (Eq. 5-2) are perhaps the most widely used method in molecular dynamics. The Discover program uses the leapfrog version in release 2.9.5 and the velocity version for release 95.0. The advantages of Verlet algorithms is that it requires only one energy evaluation per step, requires only modest memory, and also allows a relatively large timestep to be used.

Verlet, L. Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules", Phys. Rev., 159, 98-103 (1967).

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