Consider a pair of Lennard-Jones particles with
(typical for Argon).
Let the two molecules be situated at a distance of
from each other.
- Calculate the potential energy of this arrangement.
- Do the same calculation using and as units of
energy and length, respectively.
These parameters then vanish from the expression for the pair energy,
and the calculation is done with quantities of order .
- With the above units for energy and length, together with the
atomic mass unit, compute the metric value of the self-consistent unit
of time? Let one of the particles have a metric speed , typical
of the thermal velocities of atoms or small molecules. What is the value
of in self-consistent units?
As a first reusable module for a simulation program, write a code to
set up a cubic box of side length inhabited by up to particles
in a face-centered cubic arrangement. Use your favourite programming language
and make the code flexible enough to allow for easy change of volume
(i.e. density). Make sure that the lengths are measured in units
of . For later reference, let us call this subroutine
Advice: It is convenient to count the lower, left and front
face of the cube as belonging to the basic cell, while the three other
faces belong to the next periodic cells.
To test for correct arragement of the particles, compute the diagnostic
which is sometimes called ``melting factor''. For a fcc
configuration it should be equal to (why?).
By scaling all lengths, adjust the
volume such that the reduced number density becomes .
Augment the subroutine STARTCONF by a procedure that assigns random
velocities to the particles, making sure that the
sum total of each velocity component is zero.
The second subroutine will serve to compute the total potential
energy in the system, assuming a Lennard-Jones interaction and applying
the nearest image convention:
Write such a subroutine and call it ENERGY. Use it to compute the
energy in the system created by STARTCONF.
F. J. Vesely / University of Vienna