Thomas–Fermi Approximation and Basics of the Density Functional Theory

As stated at the beginning of section 2.7 the total energy is a key function describing the basic physical and chemical properties of materials: the ground state. It consists of both kinetic (describing motion) and potential energy parts. To make a theoretical model realistic it is very important to incorporate all important contributions to both the parts of the total energy. In view of the big number of the particles involved into the model, this is very challenging for the first-principles theory. Different approximations are applied in order to achieve a trade-off between complexity and accuracy. Very successful in realistic modeling of the ground state is the density functional theory, DFT. In this chapter we present the basic ideas of the DFT and demonstrate both advantages and problems for optics within this method. Initially Thomas and Fermi (TF) in the 1920s suggested describing atoms as uniformly distributed electrons (negatively charged clouds) around nuclei in a sixdimensional phase space (momentum and coordinates). This is enormous simplification of the actual many-body problem. It is instructive to consider the basic ideas of the TF approximation before starting with a more accurate theory: the DFT. The basic ideas and results of the TF model in application for atoms are provided here. Following the TF approach the total energy of the system could be presented as a function (functional) of electron density [McQuarrie (1976); Parr and Yang (1989)]. Each h of the momentum space volume (h is the Planck constant) is occupied by two electrons and the electrons are moving in an effective potential field that is determined by nuclear charge and by assumed uniform distribution of electrons. The density of ΔN electrons in real space within a cube (nanoparticle) with a side l is given by