Dirac equation modified fields

In the presence of electric and magnetic fields the Dirac equation is modified to... 

The latest major achievement in the field of elinnination techniques for the small component is due to Dyall and has been worked out to an efficient computational tool for quantum chemistry within the last few years [36-39]. This method is commonly dubbed normalised elimination of the small component (NESC) and is based on the modified Dirac equation [40,41], where the small component (f> of the 4-spinor 4> is replaced by a pseudolarge component defined by the relation... 

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... 

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. 

Even with the high laser intensities employed for the study of multiphoton processes, the electric fields they produce are seldom comparable to intramolecular coulombic fields. As such, the interaction operator [9] can be treated as a perturbation on the molecular states. The result of these perturbations is to modify the form of the Schrodinger equation so that its usual eigenfunctions no longer represent stationary states, and hence transitions occur. The quantum amplitude for a transition between a given initial state i and a final state f is given by the following result from time-dependent perturbation theory, expressed in Dirac notation ... 

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