NANOSCALE DEVICES: APPLICATIONS AND MODELING - Bio-Inspired and Nanoscale Integrated Computing - page 55

interactions as they move in their complex orbitals), it can often provide a good
insight into the physics of a problem and even make relatively good quantitative
predictions as well. For instance, this method has shown to be quite effective in
finding the occupied orbitals of carbon nanotubes and ionization energies of
several molecular systems [76, 77].
T HE D ENSITY F UNCTIONAL T HEORY . The density functional theory (DFT) is
perhaps the most celebrated of approaches to solving the quantum many-body
problem and has become increasingly popular in modeling and simulating solid
materials including nanostructures. DFT provides a framework to accurately treat
the quantum many-body problem. So, in principle it could be classified as a first-
principles approach. However, later we will see that in practical implementations
of DFT one needs to use approximate descriptions for the so called exchange-
correlation energy. Therefore, DFT is not always necessarily considered an ab
initio technique. Nonetheless, due to the fact that it does not essentially rely on
empirical data, it is included in this section.
A paper published by Hohenberg and Kohn in 1964 [78] demonstrates that in
treating an N-body problem, instead of working with the coordinates of all
individual particles (electrons and nuclei), one can use a particle density distribu-
tion in space. This remarkably simplifies the problem and brings it to a much more
manageable level. This is accomplished on the basis of the two following theorems,
proven in that seminal paper [75, 78]:
1. The external potential acting on a fully interacting many-particle system in
the ground state is determined uniquely by its electron density distribution
in space. (Obviously we can add a constant to the potential everywhere
without any real effect).
2. The ground state energy of the system (with the correct density) is smaller
than the energy in any other configuration (with any other trial density).
The first theorem enables one to transform the many-body Schro¨ dinger
equation to an equation for the particle density functional. The second theorem
provides a variational principle to solve the equation by finding the density
distribution that minimizes the resulting energy for the system, which will be its
ground state. A formal approach to finding the solution was proposed by Kohn
and Sham in 1965 [79]. The key to this approach is that, using the introduction of
an effective potential, the equation for the interacting many-body system is
rewritten in such a way that it looks like the equation for a system of
noninteracting particles. Then this equation is solved in a self-consistent manner
as follows: one makes a guess for the density distribution, uses it to find the
effective potential, uses the effective potential to find the orbitals (which is feasible
since the equation looks like the one for noninteracting particles), and then
calculates the electron density again from these orbitals. This process is continued
until the density converges, and then one will have the ground state energy,