Fast Solution of the Kohn-Sham Equation in 3 D

We propose to generalize a rapidly converging algorithm for solving the Kohn-Sham equation to geometries with two- and three-dimensional symmetry breaking. The work is to be carried out in a collaboration between the many- body theory group at the Institute for Theoretical physics, and the Institute for Analysis and Computational Mathematics, at the Johannes Kepler University Linz. Our algorithm has its roots the Hohenberg-Kohn theorem and solves directly for the electron density; single- particle wave functions are used only as auxiliary quantities. The method has been successfully implemented for simple, one-dimensional geometries; preliminary results in axially symmetric geometries indicate that good convergence can also be achieved with a new "state space" version of the algorithm. This "state space" version is also suitable for implementation in situations with three dimensional symmetry breaking and promises to be a so- called O(N) log (N) algorithm, in other words the computational effort scales proportional to N log (N), where N is the number of meshpoints (not particles !) The major new effort will be to implement and to test algorithms for the generation of the single-particle basis. A number of options shall be tried. We intend to apply our tools to a selected number of experimentally relevant problems such as the study of photoemission spectra from clusters, and the energetics and spectroscopy of ionic implants in fullerenes.

The study of small electronic systems is at the heart of modern nano-science and -technology. Such small systems are, for example, encountered in modern semiconductor devices. Both theo-retical and experimental methods are nowadays advanced to a point where the size of experimen-tally accessible systems has been reduced to a level where they are accessible by modern theoreti-cal methods. Here, the task of theoretical physics is to explain rather than describe the behaviour of physical systems from the most fundamental laws of nature. A significant faction of modern theoretical physics is based on the availability of large-scale com-putational resources. In interplay between technical and algorithmic development, much progress has been made in the computational simulation of the above-mentioned nano-systems. Our project is a paradigm of such advances, where physical understanding has led to algorithmic innovations which would, of course, not been possible without suitable computational resources. We have developed new program packages for calculating the electronic structure of small elec-tronic systems ranging from a few to a few hundred electrons. These packages implement a num-ber of innovative algorithmic ideas. One part of our method has its roots in the Hohenberg-Kohn theorem and solves directly for the density of electrons in such devices, others provide new meth-ods for calculating the properties of single electrons subject to arbitrary external forces (Schrödinger equation solver). One of the most interesting problems in the physics of small metal particles is the interplay be-tween the quantum mechanical motion of the electrons and the and classical degrees of freedom of the ion cores. We have studied this effect very carefully and found the degree by which elec-tronic properties deform an underlying ion lattice. Quasi-Two-Dimensional electronic systems are found in many semiconductor devices. These sys-tems have often irregular shapes and are, moreover, potentially contaminated by random impuri-ties. In a series of papers, we have carried out extensive studies on how shape, impurity distribu-tion, and external magnetic fields influence the regularity of electronic features in such systems.