Three-dimensional Electronic Structures Research Articles

Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

We present an approach to solid-state electronic-structure calculations based on the finite-element method. In this method, the basis functions are strictly local, piecewise polynomials. Because the basis is composed of polynomials, the method is completely general and its convergence can be controlled systematically. Because the basis functions are strictly local in real space, the method allows for variable resolution in real space; produces sparse, structured matrices, enabling the effective use of iterative solution methods; and is well suited to parallel implementation. The method thus combines the significant advantages of both real-space-grid and basis-oriented approaches and so promises to be particularly well suited for large, accurate ab initio calculations. We develop the theory of our approach in detail, discuss advantages and disadvantages, and report initial results, including the first fully three-dimensional electronic band structures calculated by the method.

Electronic structure and energetics of pressure-induced two-dimensionalC60polymers

We report on the electronic structure and energetic stabilities of two-dimensional ${\mathrm{C}}_{60}$ polymers, in both tetragonal and rhombohedral phases, studied by using the local-density approximation in the framework of the density-functional theory. Owing to hybrid networks of ${\mathrm{sp}}^{2}$-like (threefold coordinated) and ${\mathrm{sp}}^{3}$-like (fourfold coordinated) carbon atoms, the electronic structure of these phases is considerably different from that of face-centered-cubic (fcc) ${\mathrm{C}}_{60}.$ Both systems are found to be elemental semiconductors having small indirect gaps. Furthermore, since the interlayer distance between adjacent polymerized planes for both phases is small, these systems are found to have three-dimensional electronic structures. From structural optimizations under the experimental lattice parameters, we reveal energetic high stabilities of these phases. In particular, the tetragonal phase is found to be considerably more stable in energy than the fcc phase. Its high stability is caused by the formation of intercluster bonds whose energy gain is larger than the energy loss due to the distortion of the carbon networks of ${\mathrm{C}}_{60}$ units upon polymerization.

Visualization of Three-Dimensional Delaunay Meshes

We describe an algorithm for the rapid display of three-dimensional Delaunay meshes (or selected portions thereof) on standard raster displays, without the use of special purpose graphics hardware. The algorithm allows the display of the interior structure as well as the surface of the mesh, and furthermore does not require that the meshed domain be convex, or even connected. The algorithm computes a depth ordering on the mesh elements. This ordering can be used to display subsets of the mesh, as well as isosurfaces induced by fields represented on the mesh. Furthermore, by utilizing mesh coherence, the depth ordering can be used to view the mesh from front to back as well as back to front. An implementation of the algorithm has been incorporated in a system for designing and analyzing the performance of three-dimensional semiconductor and electronic packaging structures. The system is in regular use and the mesh-display algorithm has been used to visualize both meshes and fields computed over the meshes.

ChemInform Abstract: THE BORANES AND THEIR RELATIVES

AbstractDer Artikel ist ein Abdruck der Rede, die der Nobelpreisträger für anorganische Chemie in Stockholm 1976 gehalten hat.

The Boranes and Their Relatives

This year, 1976, the Nobel Prize in Chemistry has been awarded for research in pure inorganic chemistry, in particular the boranes. May I say that I am most pleased and profoundly grateful. My own orientation to this field has been, as it has in all of my studies, the relationships of the chemical behavior of molecules to their three-dimensional geometrical and electronic structures. The early work on the molecular structures of boranes by X-ray diffraction led to a reasonable basis for a theory of chemical bonding different from that which is typical in carbon chemistry, and yielded an understanding of the pleasing polyhedral-like nature of these compounds. Assimilated by the preparative chemists, the principles helped to establish a large body of a hitherto unknown chemistry, which made a reality of the expectation that boron, next to carbon in the periodic table, should indeed have a complex chemistry. In these nearly thirty years both the theoretical and experimental methods have been applied by us and others to areas of inorganic, physical, organic and biochemistry. For examples, these areas include low temperature X-ray diffraction techniques, and the theoretical studies of multicentered chemical bonds including both delocalized and localized molecular orbitals. An early example is extended Huckel theory, originally developed for studies of the boranes, and even now one of the most widely applicable approximate methods for theoretical studies of bonding in complex molecules. More soundly based theories are presently in use by my research students for studying how enzymes catalyze reactions, details of which are based on the three-dimensional structures by X-ray diffraction methods. Besides illuminating particular problems, these developments may contribute toward the redefinition of areas of chemisttry, and thereby broaden the chemist’s view. Our research in the boranes and their related molecular species crosses areas of inorganic, experimental physical, theoretical and organic chemistry, and includes applications in biochemistry. More simply stated, the area is the study of the relationships of molecular structure to function. BORANES, AND EARLY STRUCTURE STUDIES By now, large numbers of chemical compounds related to polyborane chemistry exist: boron hydrides, carboranes, metalloboranes, metallocarboranes, mixed compounds with organic moieties, and others. These discoveries of preparative chemists are relatively recent. Long ago, Alfred Stock established borane chemistry. He developed the experimental techniques which were required