2.4 Reduced units

Consider a pair of Lennard-Jones particles with $\epsilon=1.6537\cdot 10^{-21} J$ and $\sigma=3.405 \cdot 10^{-10} m$ (typical for Argon). Let the two molecules be situated at a distance of $3.2 \cdot10^{-10} m$ from each other.

- Calculate the potential energy of this arrangement.

- Do the same calculation using $\epsilon$ and $\sigma$ 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 $1$.

- 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 $v=500 m/s$, typical of the thermal velocities of atoms or small molecules. What is the value of $v$ in self-consistent units?


PROJECT MC/MD: As a first reusable module for a simulation program, write a code to set up a cubic box of side length $L$ inhabited by up to $N=4 m^{3}$ 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 $\sigma_{LJ}$. For later reference, let us call this subroutine STARTCONF.

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

$$S_{0} \equiv \frac{1}{3}\sum_{i=1}^{N} \left[ \cos (4 m \pi x_{i}/L)+\cos (4 m \pi y_{i}/L)+ \cos (4 m \pi z_{i}/L) \right]$$

which is sometimes called ``melting factor''. For a fcc configuration it should be equal to $N$ (why?).

By scaling all lengths, adjust the volume such that the reduced number density becomes $\rho^{*}=0.6$.


PROJECT MD: 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.


PROJECT MC/MD: 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:

$$E_{pot}=\frac{1}{2}\sum_{i}\sum_{j \neq i} u_{LJ}(r_{ij}) = \sum_{i=1}^{N-1} \sum_{j=i+1}^{N} u_{LJ}(r_{ij})$$

Write such a subroutine and call it ENERGY. Use it to compute the energy in the system created by STARTCONF.