3. Quantum Mechanics of the Electron Gas

In chapter 2, we showed that the quantum mechanics of the electrons and nuclei which make up real systems can be simplified using the Born-Oppenheimer approximation to separate the motion of the nuclei and electrons. It is therefore possible to treat the nuclei as stationary and reduce the problem to that of a gas of interacting electrons moving in a static external potential due to the nuclei. We also showed that the many-electron wave-function must be antisymmetric under exchange of particles, and outlined the powerful variational method for finding the energy eigenvalues of the Hamiltonian.

In this chapter, we will first show how the problem of finding the ground-state energy can be simplified considerably by the use of density-functional theory, in which the electronic density, rather than the many-electron wave-function, plays the central rôle. Furthermore, it is possible to make a mapping from the system of interacting electrons to a fictitious system of non-interacting particles which has the same ground-state density. Thus the difficult interacting problem can be transformed into a simpler non-interacting problem. We will outline the local density approximation for the effects of exchange and correlation, which allows the theorems of density-functional theory to be applied, and gives surprisingly good results.

Exploiting these results, we will then describe the treatment of periodic
systems, and conclude the chapter with a discussion of the
pseudopotential approximation which eliminates the core electrons and strong
nuclear Coulomb potential from the problem.