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Hückel Method
The Hückel method or Hückel molecular orbital theory, proposed by Erich Hückel in 1930, is a simple method for calculating molecular orbitals as linear combinations of atomic orbitals. The theory predicts the molecular orbitals for π-electrons in π-delocalized molecules, such as ethylene, benzene, butadiene, and pyridine. It provides the theoretical basis for Hückel's rule that cyclic, planar molecules or ions with 4n+2 π-electrons are aromatic. It was later extended to conjugated molecules such as pyridine, pyrrole and furan that contain atoms other than carbon and hydrogen (heteroatoms). A more dramatic extension of the method to include σ-electrons, known as the extended Hückel method (EHM), was developed by Roald Hoffmann. The extended Hückel method gives some degree of quantitative accuracy for organic molecules in general (not just planar systems) and was used to provide computational justification for the Woodward–Hoffmann rules. To distinguish the origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erich Hückel
Erich Armand Arthur Joseph Hückel (August 9, 1896, Berlin – February 16, 1980, Marburg) was a German physicist and physical chemist. He is known for two major contributions: *The Debye–Hückel theory of electrolytic solutions *The Hückel method of approximate molecular orbital (MO) calculations on π electron systems. Hückel was born in the Charlottenburg suburb of Berlin. He studied physics and mathematics from 1914 to 1921 at the University of Göttingen. On receiving his doctorate, he became an assistant at Göttingen, but soon became an assistant to Peter Debye at Zürich. It was there that he and Debye developed their theory (the Debye–Hückel theory, in 1923) of electrolytic solutions, elucidating the behavior of strong electrolytes by considering interionic forces, in order to account for their electrical conductivity and their thermodynamic activity coefficients. After spending 1928 and 1929 in England and Denmark, working briefly with Niels Bohr, Hückel joi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Hückel Method
The extended Hückel method is a semiempirical quantum chemistry method, developed by Roald Hoffmann since 1963. It is based on the Hückel method but, while the original Hückel method only considers pi orbitals, the extended method also includes the sigma orbitals. The extended Hückel method can be used for determining the molecular orbitals, but it is not very successful in determining the structural geometry of an organic molecule. It can however determine the relative energy of different geometrical configurations. It involves calculations of the electronic interactions in a rather simple way for which the electron-electron repulsions are not explicitly included and the total energy is just a sum of terms for each electron in the molecule. The off-diagonal Hamiltonian matrix elements are given by an approximation due to Wolfsberg and Helmholz that relates them to the diagonal elements and the overlap matrix element. H_ = KS_ \dfrac ''K'' is the Wolfsberg–Helmholz c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degenerate Energy Level
Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to describe modern art ** Decadent movement, often associated with degeneracy * Dégénération, a single by Mylène Farmer * ''Degeneration'' (Nordau), an 1892 book by Max Nordau * '' Resident Evil: Degeneration'', a 2008 film * "Degenerate", a song by Blink-182 from the album '' Dude Ranch'' * "Degenerates", a song by A Day to Remember from the album ''You're Welcome'' Science, mathematics, and medicine Mathematics * Degeneracy (mathematics), a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class * Degeneracy (graph theory), a measure of the sparseness of a graph * Degeneration (algebraic geometry), the act of taking a limit of a family of varieties * Degenerate form, bilinear form ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Method (quantum Mechanics)
In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle. The method consists of choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method, Density matrix renormalization group, and Ritz method apply the variational method. Description Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H . Ignoring complications about continuous spectra, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of simu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Electron
In chemistry, sigma bonds (σ bonds) are the strongest type of covalent chemical bond. They are formed by head-on overlapping between atomic orbitals. Sigma bonding is most simply defined for diatomic molecules using the language and tools of symmetry groups. In this formal approach, a σ-bond is symmetrical with respect to rotation about the bond axis. By this definition, common forms of sigma bonds are s+s, pz+pz, s+pz and dz2+dz2 (where z is defined as the axis of the bond or the internuclear axis). Quantum theory also indicates that molecular orbitals (MO) of identical symmetry actually mix or ''hybridize''. As a practical consequence of this mixing of diatomic molecules, the wavefunctions s+s and pz+pz molecular orbitals become blended. The extent of this mixing (or hybridization or blending) depends on the relative energies of the MOs of like symmetry. For homodiatomics (homonuclear diatomic molecules), bonding σ orbitals have no nodal planes at which the wavefunction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pi Electron
In chemistry, pi bonds (π bonds) are covalent chemical bonds, in each of which two lobes of an orbital on one atom overlap with two lobes of an orbital on another atom, and in which this overlap occurs laterally. Each of these atomic orbitals has an electron density of zero at a shared nodal plane that passes through the two bonded nuclei. This plane also is a nodal plane for the molecular orbital of the pi bond. Pi bonds can form in double and triple bonds but do not form in single bonds in most cases. The Greek letter π in their name refers to p orbitals, since the orbital symmetry of the pi bond is the same as that of the p orbital when seen down the bond axis. One common form of this sort of bonding involves p orbitals themselves, though d orbitals also engage in pi bonding. This latter mode forms part of the basis for metal-metal multiple bonding. Pi bonds are usually weaker than sigma bonds. The C-C double bond, composed of one sigma and one pi bond, has a bond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugated System
In theoretical chemistry, a conjugated system is a system of connected p-orbitals with delocalized electrons in a molecule, which in general lowers the overall energy of the molecule and increases stability. It is conventionally represented as having alternating single and multiple bonds. Lone pairs, radicals or carbenium ions may be part of the system, which may be cyclic, acyclic, linear or mixed. The term "conjugated" was coined in 1899 by the German chemist Johannes Thiele. Conjugation is the overlap of one p-orbital with another across an adjacent σ bond (in transition metals, d-orbitals can be involved). A conjugated system has a region of overlapping p-orbitals, bridging the interjacent locations that simple diagrams illustrate as not having a π bond. They allow a delocalization of π electrons across all the adjacent aligned p-orbitals. The π electrons do not belong to a single bond or atom, but rather to a group of atoms. Molecules containing conjugated syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Back-of-the-envelope Calculation
A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions. A similar phrase in the U.S. is "back of a napkin", also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet". History In the natural sciences, ''back-of-the-envelope calculation'' is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations, which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems. Fermi was known f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Organic Chemistry
Physical organic chemistry, a term coined by Louis Hammett in 1940, refers to a discipline of organic chemistry that focuses on the relationship between chemical structures and reactivity, in particular, applying experimental tools of physical chemistry to the study of organic molecules. Specific focal points of study include the rates of organic reactions, the relative chemical stabilities of the starting materials, reactive intermediates, transition states, and products of chemical reactions, and non-covalent aspects of solvation and molecular interactions that influence chemical reactivity. Such studies provide theoretical and practical frameworks to understand how changes in structure in solution or solid-state contexts impact reaction mechanism and rate for each organic reaction of interest. Application Physical organic chemists use theoretical and experimental approaches work to understand these foundational problems in organic chemistry, including classical and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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