Rhombus

About: Rhombus is a research topic. Over the lifetime, 637 publications have been published within this topic receiving 5900 citations. The topic is also known as: equilateral quadrilateral & diamond.

Theoretical study of small silicon clusters: Equilibrium geometries and electronic structures of Sin (n=2–7,10)

Abstract: The geometries and energies of small silicon clusters have been investigated in a systematic manner by means of accurate ab initio calculations. The effects of polarization functions and electron correlation have been included in these calculations. Several geometrical arrangements and electronic states have been considered for each cluster. All the geometries considered have been completely optimized within the given symmetry constraints with several basis sets at the Hartree–Fock level of theory. Single point calculations have been performed at these geometries using complete fourth‐order perturbation theory with the polarized 6‐31G* basis set. The effects of larger basis sets including multiple sets of polarization functions have been considered for Si2 and Si3. Singlet ground states are found for Si3–Si7 with the associated geometries corresponding to a triangle, a planar rhombus, a trigonal bipyramid, an edge‐capped trigonal bipyramid, and a tricapped tetrahedron, respectively. The best calculated structure for Si10 corresponds to a tetracapped octahedral arrangement where alternate faces of the octahedron have been capped to yield a structure with overall tetrahedral symmetry. All the geometries are considerably different from those derived from microcrystal fragments. Binding energies have been computed for all clusters and used to interpret the distribution and fragmentation patterns of small silicon cluster ions observed recently.

The dissection of equilateral triangles into equilateral triangles

Abstract: In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math . J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N 1 , N 2 , N 3 . We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

Random Tiling and Topological Defects in a Two-Dimensional Molecular Network

Abstract: A molecular network that exhibits critical correlations in the spatial order that is characteristic of a random, entropically stabilized, rhombus tiling is described. Specifically, we report a random tiling formed in a two-dimensional molecular network of p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite. The network is stabilized by hexagonal junctions of three, four, five, or six molecules and may be mapped onto a rhombus tiling in which an ordered array of vertices is embedded within a nonperiodic framework with spatial fluctuations in a local order characteristic of an entropically stabilized phase. We identified a topological defect that can propagate through the network, giving rise to a local reordering of molecular tiles and thus to transitions between quasi-degenerate local minima of a complex energy landscape. We draw parallels between the molecular tiling and dynamically arrested systems, such as glasses.

Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization

Abstract: Nonreciprocal non-Hermitian higher-order topological skin states are realized in $L\phantom{\rule{0}{0ex}}C$ electric circuits with diodes. Here, the authors give examples with rhombus geometry of the nonreciprocal honeycomb system and rhombohedron geometry of the diamond lattice system. In the accompanying image, red balls represent zero-energy topological corner states.

Isomers and excitation energies of C4

Abstract: Coupled‐cluster (CC) and many‐body perturbation theory (MBPT) studies of the rhombic and linear structures of C4 are reported. For each isomer, the electronic spectra is obtained, and comparisons are made with experimental matrix‐isolated ESR and electronic spectra. The closed‐shell 1Ag rhombic ground state is found to be more stable than the 3∑−g state of the linear isomer by 5 kcal/mol at the highest level of calculation performed (CCSDT‐1). However, the predicted spectrum for linear C4 is in reasonable agreement with the observed results. An allowed electronic transition for the rhombus is predicted to lie in the same region, suggesting the possibility that both isomers could coexist in the experiment. Finally, vibrational frequencies for the rhombic isomer are calculated using analytical second‐order MBPT second derivatives to aid in the experimental identification of this transient species.