Scalar-Relativistic Approximations

For many applications an approximate consideration of relativistic effects is sufficient. In scalar-relativistic approaches, spin-orbit coupling is neglected, so that wave functions with non-relativistic symmetry are obtained. 

We define scalar-relativistic local basis states R8pL) 

Extended states kn) fulfill a scalar-relativistic Kohn-Sham-equation 

This equation is solved in complete analogy to the non-relativistic formalism. 

This section aims at demonstrating, for a number of typical examples, the importance of relativistic effects in solid state physics. The examples include results obtained with the described RFPLO method, comparison of this method with other relativistic codes, and further results from the literature. 

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Relativistic and electron correlation effects play an important role in the electronic structure of molecules containing heavy elements (main group elements, transition metals, lanthanide and actinide complexes). It is therefore mandatory to account for them in quantum mechanical methods used in theoretical chemistry, when investigating for instance the properties of heavy atoms and molecules in their excited electronic states. In this chapter we introduce the present state-of-the-art ab initio spin-orbit configuration interaction methods for relativistic electronic structure calculations. These include the various types of relativistic effective core potentials in the scalar relativistic approximation, and several methods to treat electron correlation effects and spin-orbit coupling. We discuss a selection of recent applications on the spectroscopy of gas-phase molecules and on embedded molecules in a crystal enviromnent to outline the degree of maturity of quantum chemistry methods. This also illustrates the necessity for a strong interplay between theory and experiment. 

The scalar-relativistic approximation is usually sufficient for a quantitative description of structural properties within a given variant of DFT. On the contrary, most magnetic effects can only be understood if s-o interaction is considered. This applies both to magnetic ground state properties discussed in this section and to excited state properties discussed in the following section. 

In systems with heavier elements, relativistic effects must be included. In the medium range of atomic numbers (up to about 54) the so called scalar relativistic scheme is often used [21], It describes the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term), but omits spin-orbit interaction. The latter becomes important for the heavy elements or when orbital magnetism plays a significant role. In the present version of WIEN2k the core states always are treated fully relativistically by numerically solving the radial Dirac equation. For all other states, the scalar relativistic approximation is used by default, but spin-orbit interaction (computed in a second-variational treatment [22]) can be included if needed [23]. 

Spin-orbit interaction is only included in the density functional framework if the proper fully relativistic formalism is invoked (Strange, 1998). Often an approximate treatment of relativistic effects is implemented, by solving for the kinetic energy in the scalar-relativistic approximation (Skriver, 1983a), and adding to the total energy functional the spin-orbit term as a perturbation of the form... 

W is a symmetric matrix and are anti-symmetric matrices. In the scalar-relativistic approximation, all terms that result from spin-orbit coupling (i.e., yyx,y,z) neglected. 

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