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Symmetry Operation
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a turn rotation of a regular triangle about its center (geometry), center, a reflection (mathematics), reflection of a square across its diagonal, a translation (geometry), translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations. Each symmetry operation is performed with respect to some symmetry element (a point, line or plane). In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state. Two basic facts follow from this definition, which emphasizes its usefulness. # Physical properties must be Invariant (physics), invariant with respect to symmetry operations. # Symmetry operations can be collected together in groups which are isomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Improper Rotation
In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection (mathematics), reflection in a plane perpendicular to that axis. Reflection and Point reflection, inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.. It is used as a symmetry operation in the context of Symmetry (geometry), geometric symmetry, molecular symmetry and Crystallographic point group, crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have ''improper rotation symmetry''. Three dimensions In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the axis. For this reason ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiley-Interscience
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Water
Water is an inorganic compound with the chemical formula . It is a transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance. It is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a solvent). It is vital for all known forms of life, despite not providing food energy or organic micronutrients. Its chemical formula, , indicates that each of its molecules contains one oxygen and two hydrogen atoms, connected by covalent bonds. The hydrogen atoms are attached to the oxygen atom at an angle of 104.45°. In liquid form, is also called "water" at standard temperature and pressure. Because Earth's environment is relatively close to water's triple point, water exists on Earth as a solid, a liquid, and a gas. It forms precipitation in the form of rain and aerosols in the form of fog. Clouds consist of suspended droplets of water and ice, its solid state. When finely divided, crystalline ice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Around A Fixed Axis
Rotation around a fixed axis or axial rotation is a special case of rotational motion around an ''axis of rotation'' fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation (geometry), orientation and cannot describe such phenomena as nutation, wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result. This concept assumes that the rotation is also stable, such that no torque is required to keep it going. The kinematics and dynamics (mechanics), dynamics of rotation around a fixed axis of a rigid body are mathematically much simpler than those for Rigid body dynamics#Rigid-body angular momentum, free rotation of a rigid body; they are entirely analogous to those of linear motion along a single fixed di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Molecular Geometry
In a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are arccos(−) = 109.4712206...° ≈ 109.5° when all four substituents are the same, as in methane () as well as its heavier analogues. Methane and other perfectly symmetrical tetrahedral molecules belong to point group ''Td'', but most tetrahedral molecules have lower symmetry. Tetrahedral molecules can be chiral. Tetrahedral bond angle The bond angle for a symmetric tetrahedral molecule such as CH4 may be calculated using the dot product of two vectors. As shown in the diagram at left, the molecule can be inscribed in a cube with the tetravalent atom (e.g. carbon) at the cube centre which is the origin of coordinates, O. The four monovalent atoms (e.g. hydrogens) are at four corners of the cube (A, B, C, D) chosen so that no two atoms are at adjacent corners linked by only one cube edge. If the edge len ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclopentadienide
Sodium cyclopentadienide is an organosodium compound with the formula C5H5Na. The compound is often abbreviated as NaCp, where Cp− is the cyclopentadienide anion. Sodium cyclopentadienide is a colorless solid, although samples often are pink owing to traces of oxidized impurities. Preparation The first salt of cyclopentadienide to be reported was potassium cyclopentadienide, prepared by Johannes Thiele. In 1901 there was not much interest in the topic. Sodium cyclopentadienyl is prepared by treating cyclopentadiene with sodium: : The conversion can be conducted by heating a suspension of molten sodium in dicyclopentadiene.Tarun K. Panda, Michael T. Gamer, Peter W. Roesky "An Improved Synthesis of Sodium and Potassium Cyclopentadienide" Organometallics, 2003, 22, 877–878. In former times, the sodium was provided in the form of "sodium wire" or "sodium sand", a fine dispersion of sodium prepared by melting sodium in refluxing xylene and rapidly stirring. Sodium hydride ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ethylene
Ethylene (IUPAC name: ethene) is a hydrocarbon which has the formula or . It is a colourless, flammable gas with a faint "sweet and musky" odour when pure. It is the simplest alkene (a hydrocarbon with carbon–carbon bond, carbon–carbon double bonds). Ethylene is widely used in the chemical industry, and its worldwide production (over 150 million tonnes in 2016) exceeds that of any other organic compound. Much of this production goes toward creating polyethylene, which is a widely used plastic containing polymer chains of ethylene units in various chain lengths. Production greenhouse gas emissions, emits greenhouse gases, including methane from feedstock production and carbon dioxide from any non-sustainable energy used. Ethylene is also an important natural plant hormone and is used in agriculture to induce ripening of fruits. The hydrate of ethylene is ethanol. Structure and properties This hydrocarbon has four hydrogen atoms bound to a pair of carbon atoms that are con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Planar Molecular Geometry
In chemistry, the square planar molecular geometry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds. As the name suggests, molecules of this geometry have their atoms positioned at the corners. Examples Numerous compounds adopt this geometry, examples being especially numerous for transition metal complexes. The noble gas compound xenon tetrafluoride adopts this structure as predicted by VSEPR theory. The geometry is prevalent for transition metal complexes with d8 configuration, which includes Rh(I), Ir(I), Pd(II), Pt(II), and Au(III). Notable examples include the anticancer drugs cisplatin, tCl2(NH3)2 and carboplatin. Many homogeneous catalysts are square planar in their resting state, such as Wilkinson's catalyst and Crabtree's catalyst. Other examples include Vaska's complex and Zeise's salt. Certain ligands (such as porphyrins) stabilize this geometry. Splitting of d-orbitals A general d-orbital splitti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Molecular Geometry
In chemistry, octahedral molecular geometry, also called square bipyramidal, describes the shape of compounds with six atoms or groups of atoms or ligands symmetrically arranged around a central atom, defining the vertices of an octahedron. The octahedron has eight faces, hence the prefix '' octa''. The octahedron is one of the Platonic solids, although octahedral molecules typically have an atom in their centre and no bonds between the ligand atoms. A perfect octahedron belongs to the point group Oh. Examples of octahedral compounds are sulfur hexafluoride SF6 and molybdenum hexacarbonyl Mo(CO)6. The term "octahedral" is used somewhat loosely by chemists, focusing on the geometry of the bonds to the central atom and not considering differences among the ligands themselves. For example, , which is not octahedral in the mathematical sense due to the orientation of the bonds, is referred to as octahedral. The concept of octahedral coordination geometry was developed by Alfred ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion Operation
Inversion or inversions may refer to: Arts * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ''Inversions'' (novel) by Iain M. Banks * ''Inversion'' (video game), a 2012 third person shooter for Xbox 360, PlayStation 3, and PC * ''Inversions'' (EP), the 2014 extended play album by American rock music ensemble The Colourist * ''Inversions'' (album), a 2019 album by Belinda O'Hooley * ''Inversion'' (film), a 2016 Iranian film Linguistics and language * Inversion (linguistics), grammatical constructions where two expressions switch their order of appearance * Inversion (prosody), the reversal of the order of a foot's elements in poetry * Anastrophe, a figure of speech also known as an ''inversion'' Mathematics and logic * Additive inverse * Involution (mathematics), a function that is its own inverse (when applied twice, the starting value is obtained) * Inversion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
