Resolvent of the Dirac Operator

One can formulate the resolvent of the Dirac operator in block form in terms of upper and lower components. We introduce the auxiliary operators... 

The same working equations are also obtained, in a somewhat more tedious way, starting from the resolvent of the Dirac operator, as suggested by Gesztesy et ah [6, 67]... 

Although the derivation of the MVD operator by means of a straightforward expansion is invalid, it can be saved by a more complicated argument considering the resolvent of the Dirac operator. It turns out that the MVD operator leads to correct first-order expectation values, which means that it can be favorably used in first-order perturbation theory, but not beyond. Practical applications show that the MVD operator indeed gives sufficiently accurate relativistic corrections in the first and second transition-metal row, but for the third row significant numerical deficiencies require a treatment of relativistic effects in higher than first order. 

The decoupling of the Dirac equation to the two-component form is a rather complicated process. It manipulates the resolvant operator which is transferred from the denominator to the numerator and then exposed to consecutive commutator relations in order to be shifted to the far right. Finally, a number of one-electron terms of the order 1/c2 is obtained some of them have no classical analogy and cannot be derived from non-relativistic theories. 

Figure 16.4 Total electronic densities of M(C2H2) with M=Ni,Pt from Hartree-Fock calculations with two-component ZORA, scalar-relativistic DKH10, and nonrelativistic Schrodinger one-electron operators subtracted from the four-component Dirac-Hartree-Fock reference densities (data taken from Ref. [880]). The molecular structure of the complexes is indicated by element symbols and lines positioned just below the atomic nuclei (top panel). Asymmetries in the plot are due to the discretization of the density on a cubic grid of points. The DKH densities have not been corrected for the picture-change effect and, hence, deviate from the four-component reference density in the closest proximity to the nuclei. But these effects can hardly be resolved on the numerical grid employed to represent the densities.

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