Just as for the simple lattice gas that is equivalent to the Ising model, divide the system into cells such that, at most, only one molecule can occupy a given cell and a cell can either be occupied or empty. Let the total number of molecules be N and the total number of sites be M. Also, define a parameter that describes the occupancy of a cell as e _j = 0 if the jth cell is empty, and e _j = 1 if the jth cell contains a molecule. Note that

Instead of restricting ourselves to nearest neighbor interactions, let the potential energy of interaction of two atoms in two cells labeled i and j be ?v _ij. That is, it is still assumed that the system can be described by pairwise central interactions, but these include all pairs and not just nearest neighbors. Note that v _ij is a constant. We also assume that there is a binding energy of an atom to a cell given by ?v _j ^o and that this can be different for every cell. If either of the two cells i and j are empty, then the interaction energy and binding energy are both zero, so the Hamiltonian of the system is

The first term on the right is a double sum over all i and j that are not equal and is multiplied by 1/2...