Dirac equation without spin-orbit

The Dirac Equation Without Spin-Orbit Coupling... 

The radial Dirac equations without spin-orbit interaction (9.35) are in a form which, except for the term -(k + l)v /(rv )P, is identical to the equations (4-90,4-91) solved by Louaks [9.10]. In the subroutine WAVEFC a technique similar to that used by Louoks is therefore applied to solve the case without spin-orbit interaction. 

C SOLVES THE DIRAC EQUATIONS WITHOUT SPIN-ORBIT COUPLING AND ... 

The third step is to generate radial wave functions and the corresponding potential parameters. To this end, the programme solves the Dirac equation without the spin-orbit interaction (Sect.9.6.1) using the trial potential. Hence, the programme includes the important relativistic mass-velocity and Darwin shifts. The potential parameters are calculated from (3.33-35) and then converted to standard parameters by the formulae in Sect. 4.6. The energy derivatives are calculated from the solutions of the Dirac equation at two energies, E + e and E - e, where e is some small fraction of the relevant bandwidth. 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

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. 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

---