Dirac one-electron

Matrix theory for Dirac one-electron problems was set up in the last section, and we shall now generalize this, first for closed-shell atoms and then for the general open-shell case. We use the effective Hamiltonian of (95) as the starting... 

Things become very complex when we try to switch on the electronic interaction. First, what should we use as the interaction eneigy operator Nobody knows, but one idea might be to add to the sum of the Dirac one-electron operators the Coulombic interaction operators of all the particles. This is what is known as the Dirac-Coulomb (DC) model... 

One of the major fundamental difference between nonrelativistic and relativistic many-electron problems is that while in the former case the Hamiltonian is explicitly known from the very beginning, the many-electron relativistic Hamiltonian has only an implicit form given by electrodynamics [13,37]. The simplest relativistic model Hamiltonian is considered to be given by a sum of relativistic (Dirac) one-electron Hamiltonians ho and the usual Coulomb interaction term ... 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

Many-electron wave functions correct to oi may be expanded in a set of CSFs that spans the entire N-electron positive-energy space j (7/J 7r), constructed in terms of Dirac one-electron spinors. Individual CSFs are eigenfimctions of the total angular momentum and parity operators and are linear combinations of antisymmetrized products of positive-energy spinors (g D(+ ). The one-electron spinors are mutually orthogonal so the CSFs / (7/J 7r) are mutually orthogonal. The un-... 

For studies of the Fermi surface in the lanthanide compounds, it is necessary to develop a reliable theoretical method in which hybridization of the 4f electrons with other electrons as well as the relativistic effect can be taken into account quantitatively. For that purpose, the relativistic APW method proposed by Loucks (1967) provides a good starting basis. Loucks derived his original method from the Dirac one-electron equation, which is a natural extension of Slater s non-relativistic APW method (Slater 1937). It proved to be a powerful method comparable to a relativistic KKR method (Onodera and Okazaki 1966, Takada 1966). Loucks method does not accocunt for the symmetrization of the wave functions by group theory, nor it is a self-consistent method. These shortcomings are serious limitations for calculations of the energy band structure in the lanthanide compounds. 

Yamagami and Hasegawa carried out a self-consistent calculation of the energy band structure by solving the Kohn-Sham-Dirac one-electron equation by the density-functional theory in a local-density approximation (LDA). This self-consistent, symmetrized relativistic APW approach was applied to many lanthanide compounds and proved to give quite accurate results for the Fermi surface. 

Relativity affects the kinetic term and the exchange-correlation potential in the Kohn-Sham equation. As investigated in detail for the uranium atom and the cerium atom, the relativistic effect on the exchange correlation potential is rather small and therefore we use /Ac[ ( )] in n relativistic band structure calculation. The relativistic effect on the kinetic term is appreciably large and can be taken into account by adopting the Kohn-Sham-Dirac one-electron equation instead of eq. (3) as follows ... 

Overall, there are approximately 12n unique quantities in the unmodified Dirac one-electron matrices. There are irP unique quantities in the modified spin-fi ee Dirac one-electron matrices if the basis is contracted, and 2n quantities if the basis is uncontracted. In a nonrelativistic calculation, where there is no small component, the number of unique quantities is approximately il2rp. Thus, the spin-free modified Dirac method has twice as many one-electron integrals in a contracted calculation and only 33% more integrals in an uncontracted calculation than a nonrelativistic calculation. 

---