6.1.2 Technical Matters

- What units?
- What about boundaries?
- What initial confiuration?
- How to adjust density and temperature

To avoid the handling of small numbers, choose units appropriate to the model.

where and .

As the third mechanical unit, choose the atomic mass .

The time unit is now the combination .

For hard spheres of diameter the customary standard density is ; thus: .

To avoid ``wall effects''we surround the basic cell containing the particles by periodic images of itself.

For each particle store, instead of any coordinate , the quantity

(with the side length of the cell). When a particle leaves the cell to the right, it is automatically replaced by a particle entering from the left, etc.

Nearest Image Convention: In computing

The NIC rule may be implemented by a sequence of IF commands or by the more compact code line

where nint() denotes the rounded value of , i.e. the integer nearest to . This formulation of NIC is also better suited for vectorising compilers.

On an

Population number in a cubic cell with face-centered cubic arrangement: , with . Therefore typical particle numbers in simulations are etc.

Given , the

The

(with ) or, in reduced units,

we first take the average of over a number of MD steps to determine the actual temperature of the simulated system. Then we scale each velocity component according to

Since is a fluctuating quantity it can be adjusted only approximately.

- 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?

*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

By scaling all lengths, adjust the
volume such that the reduced number density becomes .

__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:

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

See also: