Transformations modified Dirac

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

The transformed two-electron operator bears a striking resemblance to the operator from the modified Dirac equation given in (15.43). We need only define [f = and = [c /( p- -mc )],4 [f andtheidentityiscomplete. Theanalysisoftheterms of the modified Dirac equation into scalar and spin-orbit terms in section 15.4 can then be transferred directly to the above equation. The kinematic factors are reintroduced at the end to obtain the final expressions. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

Finally, we mention the regular modified Dirac equation of Sadlej and Snijders (1994), which is exact and follows the lines of chapter 15. The regular transformation of the small component. 

This is essentially the nonrelativistic operator multiplied by fi. We must transform this operator to the modified Dirac representation, in which we have... 

Practical approaches to construct the decoupling transformation start from the modified Dirac equation [81-84]... 

This transformation has to be applied to the left and the right of the Dirac Hamiltonian to obtain the modified Hamiltonian. The same applies to operators for various molecular properties, which must also be modified the unmodified form is simply multiplied on the left and the right by the transformation operator T to obtain the modified form. We consider here both an operator defined by a scalar potential and an operator defined by a vector potential. 

It is clear from this section that in eliminating the small component by a Foldy-Wouthuysen transformation we have arrived at operators that are much more difficult to handle than those of either the original Dirac equation or the modified... 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

---