Chemistry Reference
In-Depth Information
t + Δ. From Taylor series expansion we have r(t +Δt) = 2r(t) - r(t- Δt) + aΔt
2,
i.e.
the new position at t+Δ.
The Verlet algorithm, even with long time steps, yields good energy-conserving
properties, are very compact and simple to program. Some disadvantages includes
the fact that numerical errors are introduced and the awkward handling of
velocities. Improvements of Verlet algorithms include the Beeman's algorithm
[566] (better energy conservation and more accurate expression for velocities) as
well as the 'Leap-Frog' algorithm (velocities are explicitly calculated) [567].
Typically, the directions of simulation in time of the methods are arbitrary,
i.e.
time reversible.
In order to increase the efficiency and stability of MD simulations various models
are used. SHAKE [568], for example, can be used to reduce computer time as a
constraint algorithm. Additional forces are assigned to the atoms maintaining
bond lengths at fixed equilibrium values. It is thus not necessary to calculate
stretching energies for frozen bonds. Simulations can be done for longer periods.
With improvement of algorithms, the number of degrees of freedom decreased
and the number of time steps increased. Other improvement in algorithms include
RATTLE, SETTLE, LINCS, WIGGLE [569-572] and others.
Real size of most systems contain on the order of Avogadro's number
(6.023x10
23
) of particles. Despite great advances computers do not model more
than millions of atoms at a time. Consequently, the size of the system is limited
when MD is performed and only a limited number of particles are tracked. This
can yield effects due to interactions of the container wall with the atoms.
Periodic boundary conditions can be used to eliminate surface effects by placing
the system into periodic space filling shaped (cubic, rhombic, dodecahedrom and
truncated octahedron) boxes whereas the simulation box is replicated infinitely
through space (lattice). The periodic image in every box moves with the same
orientation, as in the central box. Upon leaving this box, one of its images will
enter through the opposite faces with the same velocity. The central box is a
convenient coordinate system as there are no walls or surfaces. The number of
particles (central box) in the entire system is conserved.
