Modeling Materials

The aim here is to calculate equilibrium properties of a material, given only the intermolecular potential. (e.g. for an atomic system the Lennard-Jones 6-12 potential could be used).

From statistical mechanics it can be shown that many properties of a material can be calculated from:

which is the ensemble average value of the property X. -this is a weighted average (using exp(-U/kT)) over all possible configurations. U is the configurational energy of the system. kT is the Boltzmann constant times the temperature and the integral is over all possible coordinates of the Lennard-Jones particles (there are 3N coordinates for N particles in three dimensions).

We could calculate <X> by choosing random configurations and then weight weigth them with exp(-U/kT) but this is extremely inefficient. An alternate method is to choose the configurations with weight exp(-U/kT) and then weight them evenly, i.e.:

where configuration j is chosen with weight exp(-U(j)/kT), where U(j) is the configurational energy of configuration j.

On the next page we will discuss how this can be done efficiently.