Abstract
In atomic and many-particle physics, Green functions often occur as propagators to formally represent the (integration over the) complete spectrum of the underlying Hamiltonian. However, while these functions are very crucial to describing many second- and higher-order perturbation processes, they have hardly been considered and classified for complex atoms. Here, we show how relativistic (many-electron) Green functions can be approximated and systematically improved for few- and many-electron atoms and ions. The representation of these functions is based on classes of virtual excitations, or so-called excitation schemes, with regard to given bound-state reference configurations, and by applying a multi-configuration Dirac-Hartree-Fock expansion of all atomic states involved. A first implementation of these approximate Green functions has been realized in the framework of Jac, the Jena Atomic Calculator, and will facilitate the study of various multi-photon and/or multiple electron (emission) processes.
Highlights
Various non-linear processes have been observed during the past years but could often not be calculated in good detail for many ions, atoms or molecules of interest
Having an approximate Green function characterized by a number of symmetry channels, that is, properly constructed sets of many-electron atomic state functions (ASF) with well-defined symmetry J and energetic order, we just need to deal with those subspaces that have to be taken into account for a particular application
We have shown how relativistic Green functions can be approximated and systematically enlarged to few- and many-electron atoms and ions
Summary
Various non-linear (second- as well as higher-order perturbation) processes have been observed during the past years but could often not be calculated in good detail for many ions, atoms or molecules of interest. After a short theoretical account of a few selected properties of Green functions, and especially on useful excitation schemes for atoms and ions with complex shell structures, emphasis is placed in Section 3 upon the representation and implementation of these functions in terms of proper data structures These data structures are designed in order to support the application of the approximate (Green) functions for different atomic processes and for most, if not all, atoms or ions across the periodic table. Α and β are the well-known Dirac matrices, I 4 the 4 × 4 unit-matrix, and where—as usual in atomic structure theory—the rest energy m c 2 is not incorporated into the (total) electronic energy ε of the hydrogenic atom or ion Solutions to this equation are often discussed in the literature in terms of the Whittaker or the Kummer functions of the first and second kind, but they can be formally expressed by means of their spectral decomposition. For N-electron atoms and ions, in contrast, the 3 N infinities of the associated Green functions are a (very) serious challenge, and any truncation of these infinities must be based on good physical insight into the particular application as well as into other approximations that need to be done in order to keep computations feasible
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