Electrons single-particle equations

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

The basic idea in density functional theory is to replace the Schrodinger equation for the interacting electronic system with a set of single-particle equations whose density is the same as that of the original system. These equations are the Kohn-Sham equations[6], and may be written... 

DFT is then used to calculate the ground state electronic distribution for interacting electrons in this external potential. This is achieved by solving the Kohn-Sham single-particle equations,... 

There are only single-particle operators in (7.11). Therefore, the solution to the Schrodinger equation for a model system of noninteracting electrons can be written exactly as a single Slater determinant S = p, p2,.. , PnV where the single-particle orbitals pi are determined as solutions of the single-particle equation... 

In developing the one-electron picture of solids, we will not neglect the exchange and correlation effects between electrons, we will simply take them into account in an average way this is often referred to as a mean-field approximation for the electron-electron interactions. To do this, we have to pass from the many-body picture to an equivalent one-electron picture. We wiU first derive equations that look like single-particle equations, and then try to explore their meaning. 

The basic concept is that instead of dealing with the many-body Schrodinger equation, Fq. (2.1), which involves the many-body wavefunction P ( r ), one deals with a formulation of the problem that involves the total density of electrons (r). This is a huge simplification, since the many-body wavefunction need never be explicitly specified, as was done in the Hartree and Hartree-Fock approximations. Thus, instead of starting with a drastic approximation for the behavior of the system (which is what the Hartree and Hartree-Fock wavefunctions represent), one can develop the appropriate single-particle equations in an exact manner, and then introduce approximations as needed. 

In order to develop the pseudopotenfial for a specific atom we consider it as isolated, and denote by the single-particle states which are the solutions of the single-particle equations discussed earlier, as they apply to the case of an isolated atom. In principle, we need to calculate these states for all the electrons of the atom, using as an external potential that of its nucleus. Let us separate explicitly the single-particle states into valence and core sets, identified as and respectively. These satisfy the SchrMinger type equations... 

To put these arguments in quantitative form, consider the single-particle equations which involve a pseudopotential (we neglect for the moment all the electron interaction terms, which are anyway isotropic the full problem is considered in more detail in chapter 5) ... 

---