Dirac equation magnetic potentials

In summary, the OP-term introduced by Brooks and coworkers has been transferred to a corresponding potential term in the Dirac equation. As it is demonstrated this approach allows to account for the enhancement of the spin-orbit induced orbital magnetic moments and related phenomena for ordered alloys as well as disordered. systems by a corresponding extension of the SPR-KKR-CPA method. 

Symmetrically opposite recipes are valid for a Hamiltonian operator in momentum space.) When magnetic fields are present, then the momentum vector receives an additional term, the vector potential A. At relativistic speeds the Dirac equation shall be used. 

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. 

For heavier atoms the pure fine-structure effect is expected to break down due to relativistic effects. In the very heavy open-shell target atom thallium (Z=81) the ground-state atoms populate only one of the fine-structure levels, and the effect may be important at low energies. In an R-matrix calculation using magnetic potentials derived from the Dirac equation, Goerss, Nordbeck and Bartschat (1991) showed that... 

In the absence of interactions, electrons are described by the Dirac equation (1928), which rules out the quantum relativistic motion of an electron in static electric and magnetic fields E= yU and B = curl A (where U and A are the scalar and vectorial potentials, respectively) [43-45]. As the electrons involved in a solid structure are characterized by a small velocity with respect to the light celerity c (v/c 10 ) a 1/c-expansion of the Dirac equation may be achieved. More details are given in a paper published by one of us [46]. At the zeroth order, the Pauli equation (1927), in which the electronic spin contribution appears, is retrieved then conferring to this last one a relativistic origin. At first order the spin-orbit interaction arises and is described by the following Hamiltonian... 

Here q is the electric charge of the particle described by the Dirac equation. Particles in a magnetic field are described by the potential matrix... 

This corresponds to the principle of minimal coupling, according to which the interaction with a magnetic field is described by replacing in the Hamiltonian operator the canonical momentum p by the kinetic momentum 11 = p — f A(x). Other types of external-field interactions include scalar or pseudoscalar fields and anomalous magnetic moment interactions. The classification of external fields rests on the behavior of the Dirac equation rmder Lorentz transformations. A brief description of these potential matrices will be given below. 

In the presence of the electromagnetic potential a new term appears in the quadratic form of the Dirac equation. This allows an introduction of the intrinsic magnetic moment of an electron, generated by its spin, which interacts with the external magnetic field. 

The Kohn-Sham-Dirac equation (28) has to be solved self consistently, since the crystal potential and the XC-field depend via the (magnetization) density on its solutions. For a local orbital method it is advantageous to use a strictly local language for all relevant quantities, so that computationally expensive transformations between different numerical representations are avoided during the self consistency cycle. In the (R)FPLO method, the density n(r) and the magnetization density m(r) = m r)z are represented as lattice sums... 

To have the electron magnetic moment show up, it is necessary to make it interact with an external magnetic field and to have its spin momentum appear, it has to be combined with an orbital momentum. Equation (2.11) was thus extended to include interactions with an electromagnetic field. Let us call A4 and A the scalar and vector potentials in MKSA units (in earlier formulations of the Dirac equation [5, 6], A was divided by c due to the use of cgs units). We can write... 

The introduction of the generalized momentum operator in the one-electron kinetic energy part of the Dirac equation leads to three new interaction terms, as shown in eq. (8.29) and (8.30). It should be noted that the last two terms will also show up in a non-relativistic treatment when the magnetic vector potential is included, and only the s B term should be considered a relativistic effect. 

Vignale and Rasolt derived the Dirac-Kohn-Sham equation incorporating the vector potential from the nonrelativistic Dirac equation neglecting the magnetic effect, i.e., the Zeeman interaction term, in Eq. (6.101) as (Vignale and Rasolt 1987, 1988)... 

We do not receive a full description of excited states and potential energy curves without the spin-orbit terms. Spin-orbit effect arises due to the interaction of the magnetic dipole of the electronic spin and the movement of electrons in its orbit. For the nonrelativistic case, angular momentum I and spin s are normal constants of motion and they both commute with the nonrelativistic Hamiltonian. For the relativistic case and the Dirac equation neither s nor 1 are normal constants of motion for this case, but the total angular momentum operator j = 1 + sis. 

The second, third and foiuth terms inside the first summation in equation (3) are the perturbations introduced into the hamiltonian by the effects of the external fields. The fourth term, describing the electric field perturbation, is linear in the external potential or electric field. The second and third terms give rise to linear and quadratic responses ro a constant, uniform magnetic field. Smaller terms, arising from the Dirac equation, which represent spin-orbit coupling etc. have been omitted. 

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