Symmetry Considerations

Abstract

Many problems in solid state theory consist in solving the time-independent Schrödinger equation

$$H{\psi _n} = {E_n}{\psi _n}$$

(A1)

to obtain the energy eigenvalues E _n together with the corresponding eigenfunctions ψ _n . Since one is dealing with a large number of particles (~10²³ electrons + nuclei) a solution of this many-particle problem can only be obtained after various approximations and simplifications. Without going into detail we shall simply mention the usual approximations made in solving Schrödinger’s equation:

• (i) Since our main interest is in the electronic system, we separate the motion of electrons and nuclei and consider the latter as being fixed (Born-Oppenheimer approximation). This approximation of course is invalid if we are interested in finite temperature effects. It is, however, in most cases sufficient to treat finite temperature effects, like electron-phonon coupling by perturbation.

• (ii) The remaining many-electron problem can be reduced to a one-electron problem by defining an averaged potential, like the Hartree or Hartree-Fock potential. Exchange and correlation can be treated in several local or non-local approximations according to their importance in the particular problem. The resulting one-electron potential can be iterated in a self-consistent way until convergence is reached.

• (iii) Spin and other relativistic effects are usually introduced through the standard two-component Pauli matrix formalism. This approximation of Dirac’s treatment is valid in the low energy region we are concerned with.