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
atomic positions:

(7.1) |

The specific form of the *V*_{2} term (in fact, only *V*_{2} and *V*_{3} 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/*r*^{n} interaction (a
``Lennard-Jones'' type), to the
interaction (a Morse
type) [92,93], or a combination of them
[18]. In these cases, a cutoff function is added to
limit the range of the potential and permit a reduction in
computational time.

J. Tersoff [93] 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 [93] and later for carbon
[47]. As for the Biswas and Hamann potential [92] ,
the Morse form is adopted, related to the exponential decay dependence
of the electronic density. It is written in the following form :

where the potential energy is decomposed into a site energy

(7.4) | |||

(7.5) |

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 *Si*_{2}, graphite, diamond, simple
cubic, and face-centered cubic structures. The *f*_{C} 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 *b*_{ij}term. 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 *b*_{ij}, which can accentuate or diminish the attractive force
relative to the repulsive force, according to the environment, such
that

(7.6) | |||

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
*r*_{ij} - *r*_{ik} 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 [47] 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*[55]. The following parameters (table 7.1) are reported
from paper [14] that shows a model of interatomic
potentials for multicomponent systems, taking
.

The potential for carbon atoms only [47] was tested by calculating the cohesive energy and the structure of diverse carbon geometries, the elastic constants and phonon frequencies, defect energies and migration barriers in diamond and graphite (the energy required for interstitial and vacancy migration). The results obtained with the parameters given in table 7.1 are in good agreement with experiment (for elastic constants and phonon dispersion) and with