NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 56

orbitals and density. This cycle of iterations to find the solution is common to
many methods including DFT and Hartree-Fock and therefore these methods are
referred to as Self-Consistent Field (SCF) methods.
It should be noted that no miracle is performed in DFT. In following the
Kohn-Sham method, a term called the exchange-correlation energy appears in
the equations, due to both the quantum mechanical nature of particles and the fact
that they are interacting. But this term is not known exactly. Therefore, the final
solution will be approximate. There is indeed ongoing work on finding improved
descriptions for the exchange-correlation energy in various systems [75]. One of
the most widely established methods is the so called local density approximation
(LDA). Here, the exchange-correlation energy for the system at each point in
space is taken to be equal to that of a system with a uniform density that is the
density in the actual system at that location.
Despite the fact that the lack of an exact exchange-correlation energy makes
the results obtained by DFT approximate (like other methods), it is important to
note that DFT is, in principle, a rigorous approach and is not based on a
simplified model for the system. As such, it provides a way not only to understand
what approximations one is making (when choosing the exchange-correlation
functional) but also to improve those approximations in a systematic manner. In
this regard, it also helps to put some of the other methods (that are based on
simplified models of the real problem) into perspective and to provide a means of
analyzing how accurate they are. For a very well written summary on DFT the
reader is encouraged to consult [75]. For more detail [80-82] can be used.
2.5.4.2. Semi-Empirical Methods. As pointed out before, starting from
first principles and solving the many-body quantum mechanical problem can
often be a formidable task even for the most advanced computers. Therefore,
methods have been developed that, in addition to theory, use empirical data to
provide simplified descriptions of many-particle systems. Molecular dynamics,
tight-binding, and Monte Carlo approaches are some of the popular methods of
this genre.
C LASSICAL M OLECULAR D YNAMICS . The basic idea in molecular dynamics is to
represent the interaction between each two atoms as a mean field arising from the
interaction of all individual electrons and nuclei. For example, in the case of two
carbon atoms instead of trying to calculate the force between them as the sum of
the attractive and repulsive forces between all the individual electrons and protons
(6 protons and 6 electrons in each atom, resulting in a total of 12 * 12=144
complex interactions), a semi-empirical model for the force between the two atoms
would be used. This force would be expressed as a function of the relative
positioning of the two atoms with a number of parameters that are obtained by
fitting the predictions of the model to available experimental data.
In very simple terms, the potential energy resulting from the interaction of two
atoms could be something like the curve in Figure 2.15. Since force is the negative
of the derivative of potential energy with respect to position, it can be seen that if