Eqs. (1,3,5) describe the potential energy of the system, but we yet have to determine the electron density and have yet to consider the kinetic energy of the electrons. Density functional theory determines both of these quantities.
Within density functional theory a set of quantum mechanical Kohn-Sham orbitals describes the electrons. These are the electronic orbitals that you learn about in introductory chemistry class, each of which usually contains two electrons (one spin-up and one spin-down). In general, these orbitals may be complex, so that we must also consider the complex conjugates of the orbitals, . For the problems of interest in their course, the orbitals always turn out to be real so that . Thus, if you are unfamiliar or rusty with complex numbers you can simply ignore the *'s. We include them for those in the course who are familiar with quantum mechanics and who may have interest in problems where the orbitals can be complex.
Within quantum mechanics, the square magnitude of each orbital
gives the probability of finding an electron, when in that orbital, at
any point in space,
. The orbitals are not free to be any
functions whatsoever, but must
obey certain constraints. First, because the electron must be
somewhere in space, the probability must add up to unity,
The electron density and total kinetic energy come directly from the
orbitals. Because the square of each orbital gives the distribution
of the electrons in that orbital, the total electron density will be
the sum of squares of the orbitals time the number of electrons
in or ``filling'' each orbital,
There arises from advanced quantum mechanics one final subtle point.
The electron density defined in (9) is only an average. The
actual density fluctuates, resulting in relatively small but important
errors in Eqs. (5,10) due to correlations in
these fluctuations. In theory, we may correct for these errors
exactly, but this turns out to be quite difficult in practice. A very good
approximation to this exchange-correlation correction,
sufficient in practice to compute most properties to within a few
percent, is the local density approximation
That's it - this is all the quantum mechanics we need to predict accurately the behavior of matter!
Tomas Arias 2004-01-26