Particle in a ringIn quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle ) is Using polar coordinates on the 1-dimensional ring of radius R, the wave function depends only on the angular coordinate, and so Requiring that the wave function be periodic in with a period (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions and Under these conditions, the solution to the Schrödinger equation is given by The energy eigenvalues are quantized because of the periodic boundary conditions, and they are required to satisfy or The eigenfunction and eigenenergies are where Therefore, there are two degenerate quantum states for every value of (corresponding to ).
Resonance (chemistry)In chemistry, resonance, also called mesomerism, is a way of describing bonding in certain molecules or polyatomic ions by the combination of several contributing structures (or forms, also variously known as resonance structures or canonical structures) into a resonance hybrid (or hybrid structure) in valence bond theory. It has particular value for analyzing delocalized electrons where the bonding cannot be expressed by one single Lewis structure.
Diatomic moleculeDiatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide () or nitric oxide (), the molecule is said to be heteronuclear. The bond in a homonuclear diatomic molecule is non-polar.
Atomic orbitalIn atomic theory and quantum mechanics, an atomic orbital (ˈɔːrbɪtəl) is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital.
Hydrogen bondIn chemistry, a hydrogen bond (or H-bond) is a primarily electrostatic force of attraction between a hydrogen (H) atom which is covalently bound to a more electronegative "donor" atom or group (Dn), and another electronegative atom bearing a lone pair of electrons—the hydrogen bond acceptor (Ac). Such an interacting system is generally denoted , where the solid line denotes a polar covalent bond, and the dotted or dashed line indicates the hydrogen bond.
Single bondIn chemistry, a single bond is a chemical bond between two atoms involving two valence electrons. That is, the atoms share one pair of electrons where the bond forms. Therefore, a single bond is a type of covalent bond. When shared, each of the two electrons involved is no longer in the sole possession of the orbital in which it originated. Rather, both of the two electrons spend time in either of the orbitals which overlap in the bonding process. As a Lewis structure, a single bond is denoted as AːA or A-A, for which A represents an element.
Linear combination of atomic orbitalsA linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In a mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e.
Frontier molecular orbital theoryIn chemistry, frontier molecular orbital theory is an application of MO theory describing HOMO/LUMO interactions. In 1952, Kenichi Fukui published a paper in the Journal of Chemical Physics titled "A molecular theory of reactivity in aromatic hydrocarbons." Though widely criticized at the time, he later shared the Nobel Prize in Chemistry with Roald Hoffmann for his work on reaction mechanisms. Hoffman's work focused on creating a set of four pericyclic reactions in organic chemistry, based on orbital symmetry, which he coauthored with Robert Burns Woodward, entitled "The Conservation of Orbital Symmetry.
Valence bond theoryIn chemistry, valence bond (VB) theory is one of the two basic theories, along with molecular orbital (MO) theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of the dissociated atoms combine to give individual chemical bonds when a molecule is formed. In contrast, molecular orbital theory has orbitals that cover the whole molecule. In 1916, G. N. Lewis proposed that a chemical bond forms by the interaction of two shared bonding electrons, with the representation of molecules as Lewis structures.
Bent bondIn organic chemistry, a bent bond, also known as a banana bond, is a type of covalent chemical bond with a geometry somewhat reminiscent of a banana. The term itself is a general representation of electron density or configuration resembling a similar "bent" structure within small ring molecules, such as cyclopropane (C3H6) or as a representation of double or triple bonds within a compound that is an alternative to the sigma and pi bond model.
Homo ergasterHomo ergaster is an extinct species or subspecies of archaic humans who lived in Africa in the Early Pleistocene. Whether H. ergaster constitutes a species of its own or should be subsumed into H. erectus is an ongoing and unresolved dispute within palaeoanthropology. Proponents of synonymisation typically designate H. ergaster as "African Homo erectus" or "Homo erectus ergaster". The name Homo ergaster roughly translates to "working man", a reference to the more advanced tools used by the species in comparison to those of their ancestors.
Homo erectusHomo erectus (ˌhoʊmoʊ_ə'rɛktəs; meaning "upright man") is an extinct species of archaic human from the Pleistocene, with its earliest occurrence about 2 million years ago. Its specimens are among the first recognizable members of the genus Homo. Several human species, such as H. heidelbergensis and H. antecessor, appear to have evolved from H. erectus, and Neanderthals, Denisovans, and modern humans are in turn generally considered to have evolved from H. heidelbergensis. H.
