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Dihydrogen Cation
The hydrogen molecular ion, dihydrogen cation, or H+ 2, is the simplest molecular ion. It is composed of two positively charged protons and one negatively charged electron, and can be formed from ionization of a neutral hydrogen molecule. It is of great historical and theoretical interest because, having only one electron, the electronic Schrödinger equation
Schrödinger equation
for the system (in the clamped-nuclei approximation) can be solved in a relatively straightforward way due to the lack of electron–electron repulsion (electron correlation). The analytical solutions for the electronic energy eigenvalues are a generalization of the Lambert W function[1]which can be obtained using a computer algebra system within an experimental mathematics approach
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Gerhard Herzberg
Gerhard Heinrich Friedrich Otto Julius Herzberg, PC CC FRSC FRS[1] (December 25, 1904 – March 3, 1999) was a German-Canadian pioneering physicist and physical chemist, who won the Nobel Prize for Chemistry in 1971, "for his contributions to the knowledge of electronic structure and geometry of molecules, particularly free radicals".[2] Herzberg's main work concerned atomic and molecular spectroscopy. He is well known for using these techniques that determine the structures of diatomic and polyatomic molecules, including free radicals which are difficult to investigate in any other way, and for the chemical analysis of astronomical objects
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Helium Hydride Ion
The hydrohelium(1+) cation, HeH+, also known as the helium hydride ion or helium-hydride molecular ion, is a positively charged ion formed by the reaction of a proton with a helium atom in the gas phase, first produced in the laboratory in 1925. It is isoelectronic with molecular hydrogen.[2] It is the strongest known acid, with a proton affinity of 177.8 kJ/mol.[3] It has been suggested that HeH+ should occur naturally in the interstellar medium, but it has not yet been detected.[4] It is the simplest heteronuclear ion, and is comparable with the hydrogen molecular ion, H+ 2. Unlike H+ 2, however, it has a permanent dipole moment, which makes its spectroscopic characterization easier.[5] The calculated dipole moment of HeH+ is 2.26 or 2.84
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Proton
6973167262189800000♠1.672621898(21)×10−27 kg[1] 7002938272081300000♠938.2720813(58) MeV/c2[2] 7000100727646687900♠1.007276466879(91) u[2]Mean lifetime > 7036662709600000000♠2.1×1029 years (stable)Electric charge 6981160217648700000♠+1 e 6981160217662079999♠1.6021766208(98)×10−19 C[2]Charge radius 6999875100000000000♠0.8751(61) fm[2]Electric dipole moment < 6976540000000000000♠5.4×10−24 e⋅cmElectric polarizability 6997119999999999999♠1.20(6)×10−3 fm3Magnetic moment6974141060678730000♠1.4106067873(97)×10−26 J⋅T−1[2] 6997152103220530000♠1.5210322053(46)×10−3 μB[2] 7000279284735079999♠2.7928473508(85) μN[2]Magnetic polarizability 6996190000000000000♠1.9(5)×10−4 fm3Spin 1/2Isospin 1/2Parity +1Condensed I(JP) = 1/2(1/2+)A proton is a subatomic
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Computer Algebra
In computational mathematics, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although, properly speaking, computer algebra should be a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols. Software
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Bond Length
In molecular geometry, bond length or bond distance is the average distance between nuclei of two bonded atoms in a molecule. It is a transferable property of a bond between atoms of fixed types, relatively independent of the rest of the molecule.Contents1 Explanation 2 Bond lengths of carbon with other elements 3 Bond lengths in organic compounds 4 References 5 External linksExplanation[edit] Bond length
Bond length
is related to bond order: when more electrons participate in bond formation the bond is shorter. Bond length
Bond length
is also inversely related to bond strength and the bond dissociation energy: all other factors being equal, a stronger bond will be shorter. In a bond between two identical atoms, half the bond distance is equal to the covalent radius. Bond lengths are measured in the solid phase by means of X-ray diffraction, or approximated in the gas phase by microwave spectroscopy
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Partial Differential Equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives. PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems
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Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives
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Prolate Spheroidal Coordinates
Prolate
Prolate
spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate
Prolate
spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length. Prolate
Prolate
spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the z-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+
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Cosmic Ray
Cosmic rays are high-energy radiation, mainly originating outside the Solar System[1] and even from distant galaxies.[2] Upon impact with the Earth's atmosphere, cosmic rays can produce showers of secondary particles that sometimes reach the surface
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Helium Atom
A helium atom is an atom of the chemical element helium. Helium
Helium
is composed of two electrons bound by the electromagnetic force to a nucleus containing two protons along with either one or two neutrons, depending on the isotope, held together by the strong force. Unlike for hydrogen, a closed-form solution to the Schrödinger equation
Schrödinger equation
for the helium atom has not been found
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Triatomic Hydrogen
Triatomic hydrogen
Triatomic hydrogen
or H3 is an unstable triatomic molecule containing only hydrogen. Since this molecule contains only three atoms of hydrogen it is the simplest triatomic molecule[1] and it is relatively simple to numerically solve the quantum mechanics description of the particles. Being unstable the molecule breaks up in under a millionth of a second. Its fleeting lifetime makes it rare, but it is quite commonly formed and destroyed in the universe thanks to the commonness of the trihydrogen cation. The infrared spectrum of H3 due to vibration and rotation is very similar to that of the ion, H3+
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Molecular Term Symbol
In molecular physics, the molecular term symbol is a shorthand expression of the group representation and angular momenta that characterize the state of a molecule, i.e. its electronic quantum state which is an eigenstate of the electronic molecular Hamiltonian. It is the equivalent of the term symbol for the atomic case. However, the following presentation is restricted to the case of homonuclear diatomic molecules, or other symmetric molecules with an inversion centre. For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian
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Atomic And Molecular Astrophysics
Atomic astrophysics is concerned with performing atomic physics calculations that will be useful to astronomers and using atomic data to interpret astronomical observations. Atomic physics
Atomic physics
plays a key role in astrophysics as astronomers' only information about a particular object comes through the light that it emits, and this light arises through atomic transitions. Molecular astrophysics, developed into a rigorous field of investigation by theoretical astrochemist Alexander Dalgarno
Alexander Dalgarno
beginning in 1967, concerns the study of emission from molecules in space. There are 110 currently known interstellar molecules. These molecules have large numbers of observable transitions. Lines may also be observed in absorption—for example the highly redshifted lines seen against the gravitationally lensed quasar PKS1830-211
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List Of Quantum-mechanical Systems With Analytical Solutions
Much insight in quantum mechanics can be gained from understanding the closed-form solutions to the time-dependent non-relativistic Schrödinger equation
Schrödinger equation
in an appropriate configuration space
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ArXiv
arXiv (pronounced "archive")[2] is a repository of electronic preprints (known as e-prints) approved for publication after moderation, that consists of scientific papers in the fields of mathematics, physics, astronomy, computer science, quantitative biology, statistics, and quantitative finance, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million article milestone on October 3, 2008,[3][4] and hit a million by the end of 2014.[5][6] By October 2016 the submission rate had grown to more than 10,000 per month.[6][7]Contents1 History 2 Peer review 3 Submission formats 4 Access 5 Copyright status of files 6 Controversy 7 See also 8 Notes 9 References 10 External linksHistory[edit]A screenshot of the arXiv taken in 1994,[8] using the browser NCSA Mosaic
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