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About: Rhombus is a(n) research topic. Over the lifetime, 637 publication(s) have been published within this topic receiving 5900 citation(s). The topic is also known as: equilateral quadrilateral & diamond. more


Journal ArticleDOI: 10.1063/1.449927
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. more

Topics: Tetrahedral symmetry (54%), Ab initio quantum chemistry methods (53%), Basis set (52%) more

386 Citations

Journal ArticleDOI: 10.1017/S030500410002449X
William T. Tutte1Institutions (1)
01 Oct 1948-
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. more

240 Citations

Journal ArticleDOI: 10.1126/SCIENCE.1163338
14 Nov 2008-Science
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. more

Topics: Trihexagonal tiling (66%), Rhombille tiling (65%), Rhombus (51%)

189 Citations

Journal ArticleDOI: 10.1063/1.450259
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. more

Topics: Rhombus (55%), Ground state (52%), Electronic structure (51%)

113 Citations

25 May 1999-
Abstract: GaN-type LED or LD made on a (0001)GaN single crystal substrate having natural cleavage planes on sides. A GaN/GaN LED has a shape of a equilateral triangle, parallelogram, trapezoid, equilateral hexagon or rhombus. A GaN/GaN LD has a shape of a parallelogram with cleavage planes on two ends and two sides. Another GaN/GaN LD has a shape of a square with cleavage planes on two ends. more

Topics: Gallium nitride (52%), Rhombus (52%)

109 Citations

No. of papers in the topic in previous years

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Topic's top 5 most impactful authors

Christian Krattenthaler

11 papers, 211 citations

Ernesto Pérez-Chavela

6 papers, 132 citations

Mihai Ciucu

6 papers, 118 citations

Gleb A. Koshevoy

4 papers, 6 citations

Peter H. Beton

4 papers, 248 citations

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