Elimination regular approximation

In the mid-eighties another method to eliminate the small component has been developed in order to arrive at regular expansions for the Hamiltonian [13,26]. These regular approximations are based on the general theory of effective Hamiltonians [27,28], where the full problem under consideration is projected onto a smaller, suitably chosen model space with an effective Hamiltonian, which comprises all desired properties of the problem sufficiently well. In the case of the Dirac Hamiltonian the basic idea, being as simple as ingenious, is to rewrite the expression for w in the form... 

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

These problems are avoided if one uses regular Hamiltonians which are bounded from below. Many applications are based on the so-called zero order regular approximation (ZORA), which has been extensively investigated by the Amsterdam group [46-50]. It can be viewed as the first term in a clever expansion of the elimination of the small component, an expansion which already covers, at zeroth order, a substantial part of the relativistic effects. In fact ZORA is a rediscovery of the so-called CPD Hamiltonian (named after the authors. Ref. [60]). 

The relativistic elimination of small components (RESC) is another two-component scheme that was applied to the DKS problem [69,102]. After transforming the DKS Hamiltonian in the same fashion as in the regular approximations [based on the generator X(e), Eq. (44)], the potential dependence of the relativistic terms is decomposed using the identity... 

M. Filatov, D. Cremer. Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum chemistry. 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

Tetrahydrocannabinol is metabolized in the liver to form active metabolites which are further metabolized to inactive polar compounds these are excreted in the urine. Some metabolites are excreted into the bile and then recycled via the enterohepatic circulation. Because of their high lipophilicity, most active metabolites are widely distributed in fat deposits and the brain, from which sources they are only slowly eliminated. The half-life of elimination for many of the active metabolites has been calculated to be approximately 30 hours. Accordingly, accumulation occurs with regular, chronic dosing. Traces of the cannabinoids can be detected in the blood and urine of users for many days after the last administration. There is some evidence of metabolic tolerance occurring after chronic use of the drug. THC and related cannabinoids readily penetrate the placental barrier and may possibly detrimentally affect foetal development. 

The existence of an approximate relation of this kind is indeed to be expected for a series of regular solutions. For if we eliminate X2 between the two equations (28.25) we find... 

There are two major approaches to reduce the Dirac Hamiltonian to two-component form. The various regular elimination techniques have been developed to highly sophisticated and very successful methods, which are widely used by the community. The other approach comprises the various unitary transformation methods, which amount either to expansions in 1 /c as for the FW transformation or to expansions in powers of the external potential V as for the DK approximations. In addition, we have presented a very general extension of the traditional DK approximation to arbitrary unitary transformations. 

Salt deposition occurred at the constricted area of the torch injector when the SDS solution was introduced in the ICP-MS, with the regular torch positioned horizontally. This produced a significant diminution of the analytical signal with time. After approximately one hour, the injector was completely blocked. Several torch modifications were attempted, including a 2-mm-i.d. injector with a reduced taper and an injector with the taper region moved close to the injector inlet. A straight-tube injector with a 1.5-mm i.d. was also tried. Salt deposition still occurred. Finally, the deposition was eliminated by using a Leeman torch with a removable injector. 

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