Non-relativistic limit

In the non-relativistic limit the small component of the wave function (eq. (8.14)) is... 

Kutzelnigg, W. (1989) Perturbation theory of relativistic corrections 1. The non-relativistic limit of the Dirac equation and a direct perturbation expansion. Zeitschrifi fur Physik D, 11, 15-28. 

To understand the physical implications of these equations it is customary to consider the non-relativistic limit of their stationary solutions, assuming... 

In the non-relativistic limit both E and e(j> are small compared to me2 and hence... 

In order to develop a fully consistent non-relativistic theory for particles and fields we must first determine the non-relativistic limit of Maxwell s equations. This is to some extent hampered by the fact that the equations, and in particular the occurrence of the speed of light c, change according to the unit system chosen, a situation that has been analyzed by Jackson [25] and Kutzelnigg [15,35]. In STbased atomic units, the non-relativistic limit is obtained directly as the limit c 00 and leads to the equations... 

In the non-relativistic domain the Coulomb term is chosen as the two-electron operator which is fully consistent with the non-relativistic limit of... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

The second term on the right-hand side of the equation gives for point nuclei directly the one-electron spin-orhit operator (2) of the Breit-Pauli Hamiltonian and can he eliminated to give a spin-free equation that becomes equivalent to the Schrddinger equation in the non-relativistic limit. In a quaternion formulation of the Dirac equation the elimination becomes particularly simple. The algebra of the quaternion units is that of the Pauli spin matrices... 

These operators may be contrasted with the property operators obtained at the non-relativistic level. The Levy-Leblond equation given in (29) is consistent with the rigorous non-relativistic limit in that only scalar potentials are included. We now follow the standard, pragmatic approach and consider the operator form... 

Table II gives calculated results for copper porphine and experimental values for copper tetraphenylporphine. Results at the non-relativistic limit (c = >) were obtained from the same program by setting the speed of light to a very large number (10 a.u.) ...

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