|
Kato Theorem
The Kato theorem, or Kato's cusp condition (after Japanese mathematician Tosio Kato), is used in computational quantum physics. It states that for generalized Coulomb potentials, the electron density has a cusp at the position of the nuclei, where it satisfies : Z_k = - \frac \frac , _ Here \mathbf denotes the positions of the nuclei, Z_k their atomic number and a_o is the Bohr radius. For a Coulombic system one can thus, in principle, read off all information necessary for completely specifying the Hamiltonian directly from examining the density distribution. This is also known as E. Bright Wilson's argument within the framework of density functional theory (DFT). The electron density of the ground state of a molecular system contains cusps at the location of the nuclei, and by identifying these from the total electron density of the system, the positions are thus established. From Kato's theorem, one also obtains the nuclear charge of the nuclei, and thus the externa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tosio Kato
was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis. Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption of the Second World War, he received his doctorate in 1951 from the University of Tokyo, where he became a professor in 1958. From 1962, he worked as a professor at the University of California at Berkeley in the United States. Many works of Kato are related to mathematical physics. In 1951, he showed the self-adjointness of Hamiltonians for realistic (singular) potentials. He dealt with nonlinear evolution equations, the Korteweg–de Vries equation (Kato smoothing effect in 1983) and with solutions of the Navier-Stokes equation. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electron Density
In quantum chemistry, electron density or electronic density is the measure of the probability of an electron being present at an infinitesimal element of space surrounding any given point. It is a scalar quantity depending upon three spatial variables and is typically denoted as either \rho(\textbf r) or n(\textbf r). The density is determined, through definition, by the normalised N-electron wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Conversely, the density determines the wave function modulo up to a phase factor, providing the formal foundation of density functional theory. According to quantum mechanics, due to the uncertainty principle on an atomic scale the exact location of an electron cannot be predicted, only the probability of its being at a given position; therefore electrons in atoms and molecules act as if they are "smeared out" in space. For one-electron systems, the electron density at any point is proportional to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp (singularity)
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation :\begin x &= f(t)\\ y &= g(t), \end a cusp is a point where both derivatives of and are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope \lim (g'(t)/f'(t))). Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , in contrast to self-intersection points that involve more than one value. In some contexts, the condition on the directional derivative may be omitted, although, in this case, the singularity may look like a regular point. For a curve defined by an implicit equation :F(x,y) = 0, which is smooth, cusps are points where the terms of lowest degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every atom of that element. The atomic number can be used to uniquely identify ordinary chemical elements. In an ordinary uncharged atom, the atomic number is also equal to the number of electrons. For an ordinary atom, the sum of the atomic number ''Z'' and the neutron number ''N'' gives the atom's atomic mass number ''A''. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of the nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in unified atomic mass units (making a quantity called the "relative isotopic mass"), is within 1% of the whole number ''A''. Atoms with the same atomic number but dif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bohr Radius
The Bohr radius (''a''0) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. Its value is The number in parenthesis denotes the uncertainty of the last digits. Definition and value The Bohr radius is defined as a_0 = \frac = \frac = \frac , where * \varepsilon_0 is the permittivity of free space, * \hbar is the reduced Planck constant, * m_ is the mass of an electron, * e is the elementary charge, * c is the speed of light in vacuum, and * \alpha is the fine-structure constant. The CODATA value of the Bohr radius (in SI units) is History In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction. The original derivation posited that electrons have orbital angular momentum in integer multiples of the reduced Planck co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgar Bright Wilson
Edgar Bright Wilson Jr. (December 18, 1908 – July 12, 1992) was an American chemist. Wilson was a prominent and accomplished chemist and teacher, recipient of the National Medal of Science in 1975, Guggenheim Fellowships in 1949 and 1970, the Elliott Cresson Medal in 1982, and a number of honorary doctorates. He was also the Theodore William Richards Professor of Chemistry, Emeritus at Harvard University. One of his sons, Kenneth G. Wilson, was awarded the Nobel Prize in physics in 1982. E. B. Wilson was a student and protégé of Nobel laureate Linus Pauling and was a coauthor with Pauling of ''Introduction to Quantum Mechanics'', a graduate level textbook in Quantum Mechanics. Wilson was also the thesis advisor of Nobel laureate Dudley Herschbach. Wilson was elected to the first class of the Harvard Society of Fellows. Wilson made major contributions to the field of molecular spectroscopy. He developed the first rigorous quantum mechanical Hamiltonian in internal coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Functional Theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. DFT has been very popular for calculations in solid-state physics since the 1970s. However, DFT was not considered accurate enough for calculations in quantum chemistry until the 1990s, when the approximations used in the theory were greatly refined to better model the exchange and correlation interactions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Hamiltonian
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity. The elementary parts of a molecule are the nuclei, characterized by their atomic numbers, ''Z'', and the electrons, which have negative elementary charge, −''e''. Their interaction gives a nuclear charge of ''Z'' + ''q'', where , with ''N'' equal to the number of electrons. Electrons and nuclei are, to a very good approximation, point charges and point masses. The molecular Hamiltonian is a sum of several terms: its major terms are the kinetic energies of the electrons and the Coulomb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Born–Oppenheimer Approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics. The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but even then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
