Direct Relativistic Effects

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

Polarization. The central cone of the synchrotron beam from a bending magnet and, in general, the beam from insertion devices is polarized in the plane of the orbit (i.e., horizontally). Due to relativistic effects the cone of the radiation characteristics is narrow even if the beam is emitted from a bending magnet (cf. [10], p. 9-13 and Sect. 2.2.2). If necessary, polarization correction should be carried out directly at the synchrotron radiation facility by means of the locally available computer programs. 

Four-component relativistic molecular calculations are based directly on the Dirac equation. They include both scalar relativistic effects and spin-orbit... 

The basic mechanism of spin-orbit coupling is magnetic induction. It is therefore a truly relativistic effect, as will be discussed shortly. The potentials of a moving charge can be found from Maxwell s equations, as direct solutions or from Lorentz transformations of potentials of a static charge to a moving frame. Maxwell s equations can be divided into the homogeneous parr... 

We should note that the Schrodinger equation is non-relativistic since we derived it from the non-relativistic expression for the energy eqn (2.26). The Dirac equation is the relativistic analogue that is based on the relativistic expression for the energy, namely eqn (26). It led directly to the novel concept of electron spin. Since the valence electrons, which control the cohesive and structural properties of materials, usually travel with velocity v c, they are adequately described by the Schrodinger equation. For the heavier elements, such as the lanthanides and actinides, relativistic effects can be included perturbatively when necessary. Photons, the quanta of the... 

Although the increased electron affinity associated with the heavier elements usually manifests itself only indirectly (via electronegativity, etc.), it is directly responsible for the fact that cesium auride, Cs Au, is an ionic salt rather than an alloy. Both the increased ionization energy and increased electron affinity in these elements result from relativistic effects. 

If elements heavier than the first row transition elements are considered, relativistic effects must be accounted for. This can be done quite easily using the ECP formalism, since the relativistic effects (excluding the spin-orbit coupling) can be accommodated directly into the ECP operators (see also the next section). 

Relativistic effects on calculated NMR shieldings and chemical shifts have sometimes been divided into "direct" and "indirect" effects. According to this point of view, indirect effects are those that result from relativistic changes of the molecular geometry (the well-known relativistic bond contraction (55) in particular) whereas direct effects refer to a fixed geometry. 

In this chapter, we use exclusively relativistically optimized or experimental geometries. Hence, we concentrate on direct relativistic effects only. They can be separated into scalar and spin-orbit/Fermi contact effects. In addition, there are, in both cases, core and valence contributions. 

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