The basic assumption made in the present calculation is that the
dynamics can be treated classically and that the atoms are spherical
and chemically inert. Many mathematical models were proposed to
simulate the interatomic potential energy, and from it, the
interaction forces. A common feature of these models is the
resemblance to the Taylor expansion of the energy as a function of the
The specific form of the V2 term (in fact, only V2 and V3 are significant because any external interactions are not usually taken into account and adding more terms in the expansion will make the computation impracticable) varies from the 1/rn interaction (a ``Lennard-Jones'' type), to the interaction (a Morse type) [92,93], or a combination of them . In these cases, a cutoff function is added to limit the range of the potential and permit a reduction in computational time.
J. Tersoff  abandoned the use of N-body potential
form and proposed a new approach by effectively coupling two body and
higher multi atom correlations into the model. The central idea is
that in real systems, the strength of each bond depends on the local
environment, i.e. an atom with many neighbors forms weaker bonds than
an atom with few neighbors. Then, J. Tersoff developed a pair
potential the strength of which depends on the environment. It was
calibrated firstly for silicon  and later for carbon
. As for the Biswas and Hamann potential  ,
the Morse form is adopted, related to the exponential decay dependence
of the electronic density. It is written in the following form :
It has to be noticed that the parameters R and D are not
systematically optimized but are chosen so as to include the
first-neighbor shell only for several selected high-symmetry bulk
structure of silicon, namely for Si2, graphite, diamond, simple
cubic, and face-centered cubic structures. The fC function, thus,
decreases from 1 to 0 in the range R-D<r<R+D.
The main feature of this potential is the presence of the bijterm. As explained before, the basic idea is that the strength of
each bond depends upon the local environment and is lowered when the
number of neighbors is relatively high. This dependence is expressed
by bij, which can accentuate or diminish the attractive force
relative to the repulsive force, according to the environment, such
The term defines the effective coordination number of atom i, i.e. the number of nearest neighbors, taken into account the relative distance of two neighbors rij - rik and the bond-angle . The function has a minimum for , the parameter d determines how sharp the dependence on angle is, and c expresses the strength of the angular effect.
This potential and the parameters were chosen to fit theoretical and experimental data obtained for realistic and hypothetical silicon configurations, namely the cohesive energy of several high-symmetry bulk structures mentioned above, the lattice constant and bulk modulus of the silicon lattice in the diamond configuration. It has later been calibrated for carbon atoms by J. Tersoff  in the same manner, constraining the vacancy formation energy in diamond to have at least 4 eV, close to the value found by Bernholc et al. The following parameters (table 7.1) are reported from paper  that shows a model of interatomic potentials for multicomponent systems, taking .