Relativistic charge densities

Comparison with (110) allows us to identify the relativistic charge density p and current density j as... 

We may now try to obtain non-relativistic expressions for charge and current densities based on the electric and magnetic interaction energies as we did for the relativistic case. The non-relativistic charge density is straightforwardly given by... 

Unfortunately, it is not possible to calculate a (i.e., ACommon practice is a semi-empirical calibration The shift S is measured (with the same source) for two compounds to be considerd rather ionic with the resonant atom in two different charge states. Pertinent examples would be NpFj and NpF. Contact densities calculated with a self-consistent procedure for the free ions are used to obtain Ap(0). The constant a is then determined according to eq. (22). One certainly can raise a number of objections towards this method but it is still the most universal one. It is important, especially for the lanthanides and actinides, that relativistic charge densities p(0) are used. Details concerning isomer... 

Relativistic quantum mechanics yields the same type of expressions for the isomer shift as the classical approach described earlier. Relativistic effects have to be considered for the calculation of the electron density. The corresponding contributions to i/ (0)p may amount to about 30% for iron, but much more for heavier atoms. In Appendix D, a few examples of correction factors for nonrelativistically calculated charge densities are collected. Even the nonrelativistically calculated p(0) values accurately follow the chemical variations and provide a reliable tool for the prediction of Mossbauer properties [16]. 

Core electrons are highly relativistic and DFT methods may show systematic errors in calculating the charge density at the nucleus because of the inherent approximations. Fortunately, this does not hamper practical calculations of isomer shifts of unknown compounds, because only differences of li//(o)P are involved. In practice, the reliability of the results depends more on the number of compounds used for calibration and how wide the spread of their isomer shift values was. The isomer shift scale for several Mossbauer isotopes has been calibrated by this approach, among which are Au [1], Sn [4], and Fe [5-9]. For details on practical calculation of Mossbauer isomer shifts, see Chap. 5. 

An expression for 8 in terms of the source and absorber nonrelativistic s electron densities at the origin, s(0) and a(0), respectively, can be obtained by considering the electrostatic interaction between the s electrons and a nucleus with a uniform charge density. A relativistic calculation yields (7) ... 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

Relativistic charge-current densities expressed in terms of G-spinor basis sets for stable and economical numerical calculations [2]. 

It should be mentioned that in the approach with nonzero electric divergence, the photon mass is also related to the space charges in vacuo. Now, in the approach with a / 0, we have j = ctE but jeff = 0. Let us now assume j = aE and j 7 0, which means fs 0. In such a case, jo is assumed to be associated with p, where p is the charge density in vacuo. So, in such an approach one can think of the existence of a kind of space charge in vacuo that is to be considered to be associated to nonzero electric field divergence. This will result in a displacement current in vacuum similar to that measured by Bartlett and Corle [43]. The assumption of the existence of space charge in vacuo makes our theory not only fully relativistic but also helps us to understand gauge condition. In the conventional framework of Maxwell s equations... 

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

As we are not considering relativistic rates of transport, the potential can adjust Itself Instantaneously to the space charge density. Therefore, although the Poisson equation is essentially of a static nature, it is also valid under the dynamic conditions considered here. 

Since the results obtained using MuUiken population analysis depend on the choice of basis-set size, we used the same size in both nonrelativistic and relativistic calculations for each molecule. The atomic potentials for generating the basis functions were derived from the spherical average of the molecular charge density around the nuclei. The computational details of the non-relativistic... 

In this work, we first calculate the geometric structure of the Au-dimer with an ab-initio all-electron fully relativistic density functional method [17]. The total energy is expressed as a functional of the charge density... 

Here J°/c is identical to the ordinary electronic charge density p, while the other three components represent the electronic current density j. is the central quantity of relativistic density functional theory. All properties of the system are determined by J. ... 

Considering the outermost atomic orbitals, the effects of relativistic corrections on one-electron binding energies and the spatial distribution of the radial charge densities are illustrated by the results displayed in Fig. 4. From the strong... 

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