# 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
3*N* 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.

*© Peter H.
Nelson 1995-1997*. All rights reserved.

Last updated December 10, 1997