Relativistic four-component methodology

At a four-component level, several groups derived independently and almost simultaneously relativistic variants of Ramsey s equation within the fiamework of the one-electron Dirac equation [17,18,19]. The perturbing Hamiltonian, 

It is linear in the magnetic vector potential A, which incorporates the vector potential for both the external magnetic field and the nuclear magnetic dipole 6i is the vector containing the Dirac matrices. Inserting this into eq. 2, and using a 

The first numerical calculations within a four-component framework were done at the semi-empirical relativistic extended-Hiickel (REX) level [26]. This provided the first qualitative insights into SO contributions to the proton shieldings in the hydrogen halides and related cases (cf section 3). Attempts to obtain more quantitative results include the finite-perturbation Dirac-Fock implementation of Ishikawa, Nakatsuji and coworkers [21], and the more recent Dirac-Fock implementation of Visscher et al. [27]. The calculations of ref [21] used basis sets too small to provide useful numerical results (later calculations employed in roved basis sets [28]). Both methods were applied mainly to 

---

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

Four-component theories for the calculation of electronic structures as described in detail in this review have become mature particularly in the last decade as a highly accurate tool for any kind of system be it an atom, molecule or solid. Theoretical as well as methodological understanding gave detailed insight into the foundations of relativistic electronic structure theory. Some fundamental questions are still open as we have explicated, especially in the first two sections of this review, and they will certainly be tackled and answered in the years to come. Methodological advances will also continue to be made. 

The details of implementation of scalar relativity in GTOFF were presented in [41] and reviewed in [75], so we summarize the essential assumptions and methodological features here. First, all practical DFT implementations of relativistic corrections of which we are aware assume the validity (either explicitly or implicitly) of an underlying Dirac-Kohn-Sham four-component equation. We do also. The Hamiltonian is therefore a relativistic free particle Hamiltonian augmented by the usual non-relativistic potentials... 

Before the general methodology of contemporary relativistic EFG calculations is discussed some general aspects are outlined and early four-component hfs calculations are mentioned. 

---