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Before we proceed developing program for the MD simulations of particles interacting via the Lennard-Jones (or any other) potential we have to derive the expression for the force, and for the characteristic time scale, to determine the integration time step .

(9) |

(10) |

The time scale can be determined by calculating the characteristic
oscillations frequency for two particles interacting
via the Lennard-Jones potential. Consider two particles
of the same mass that are bound together by the Lennard-Jones potential
with parameters and . Distance
between them is close to the equilibrium distance .As the force is central we can consider a one-dimensional problem

If we assume, that they oscillate around the *r*_{ij}^{0}
with a very small amplitude, we can expand the potential energy around *r*_{ij}^{0}
and find for the force:

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Equations of motion and characteristic frequency then are:

is the characteristic frequency of the oscillations, which sets a time scale to choose the time step. Usually for the Verlet algorithm .- energy has a discontinuity at the cut-off distance
*r*_{c}, and - force is then undefined at
*r*_{c}

This introduces the error in the numerical integration and energy
conservation. The cure is to use spline, that would make the
potential and force continuous. This, however precludes the cut-off
correction or makes it difficult. Simple way to cope with it is just
to ignore the error in the force if the *r*_{c} is large enough, and
to use the shifted potential energy *U* = *U* - *U*(*r*_{c}) to control
the energy conservation.

© 1997

Tue Nov 25 22:38:03 EST 1997