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Coupled Cluster
Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry, but it is also used in nuclear physics. Coupled cluster essentially takes the basic Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation. Some of the most accurate calculations for small to medium-sized molecules use this method. The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear-physics phenomena, but became more frequently used when in 1966 Jiří Čížek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation. CC theory is simply the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-body System
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provide an accurate description of the system. ''Many'' can be anywhere from three to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as few-body systems. In general terms, while the underlying physical laws that govern the motion of each individual particle may (or may not) be simple, the study of the collection of particles can be extremely complex. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excited State
In quantum mechanics, an excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). Excitation refers to an increase in energy level above a chosen starting point, usually the ground state, but sometimes an already excited state. The temperature of a group of particles is indicative of the level of excitation (with the notable exception of systems that exhibit negative temperature). The lifetime of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited state, returning the system to a state with lower energy (a less excited state or the ground state). This return to a lower energy level is often loosely described as decay and is the inverse of excitation. Long-lived excited states are often calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir Fock. In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles. The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory. Quantum many-bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain. Overview Any physical law which can be expressed as a variational principle describes a self-adjoint operator. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. History Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Consistency
In quantum chemistry, size consistency and size extensivity are concepts relating to how the behaviour of quantum chemistry calculations changes with size. Size consistency (or strict separability) is a property that guarantees the consistency of the energy behaviour when interaction between the involved molecular system is nullified (for example, by distance). Size-extensivity, introduced by Bartlett, is a more mathematically formal characteristic which refers to the correct (linear) scaling of a method with the number of electrons. Let A and B be two non-interacting systems. If a given theory for the evaluation of the energy is size consistent, then the energy of the supersystem A+B, separated by a sufficiently large distance so there is essentially no shared electron density, is equal to the sum of the energy of A plus the energy of B taken by themselves (E(A+B) = E(A) + E(B)). This property of size consistency is of particular importance to obtain correctly behaving dissociation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extensive Quantity
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one whose magnitude is independent of the size of the system, whereas an extensive quantity is one whose magnitude is additive for subsystems. The terms ''intensive and extensive quantities'' were introduced into physics by German writer Georg Helm in 1898, and by American physicist and chemist Richard C. Tolman in 1917. An intensive property does not depend on the system size or the amount of material in the system. It is not necessarily homogeneously distributed in space; it can vary from place to place in a body of matter and radiation. Examples of intensive properties include temperature, ''T''; refractive index, ''n''; density, ''ρ''; and hardness, ''η''. By contrast, extensive properties such as the mass, volume and entropy of sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brueckner Orbitals
Brueckner is a surname of German origin. Notable people with the surname include: *Benedikt Brueckner (born 1990), German professional ice hockey player * Georg F Brueckner (1930–1992), German martial arts pioneer and inventor * Guenter Brueckner (1934–1998), American solar physicist *Keith Brueckner Keith Allen Brueckner (March 19, 1924 – September 19, 2014) was an American theoretical physicist who made important contributions in several areas of physics, including many-body theory in condensed matter physics, and laser fusion. Biography ... (1924–2014), American theoretical physicist German toponymic surnames {{surname, Brueckner German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-configurational Self-consistent Field
Multi-configurational self-consistent field (MCSCF) is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate (e.g., for molecular ground states which are quasi-degenerate with low-lying excited states or in bond-breaking situations). It uses a linear combination of configuration state functions (CSF), or configuration determinants, to approximate the exact electronic wavefunction of an atom or molecule. In an MCSCF calculation, the set of coefficients of both the CSFs or determinants and the basis functions in the molecular orbitals are varied to obtain the total electronic wavefunction with the lowest possible energy. This method can be considered a combination between configuration interaction (where the molecular orbitals are not varied but the expansion of the wave function) and Hartree–Fock (where there is only one determinant, but the molecular orbitals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Configuration Interaction
Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematically, ''configuration'' simply describes the linear combination of Slater determinants used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)2(2s)2(2p)1...), ''interaction'' means the mixing (interaction) of different electronic configurations (states). Due to the long CPU time and large memory required for CI calculations, the method is limited to relatively small systems. In contrast to the Hartree–Fock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of configuration state functions (CSFs) built from spin orbitals (denoted by the superscript ''SO''), : \Psi = \sum_ c_ \Phi_^ = c_0\Phi_0^ + c_1\Phi_1^ + where Ψ i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slater Determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electrons (or other fermions).Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P. W. Atkins, Oxford University Press, 1977, . Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity. The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital \chi(\mathbf), where \mathbf denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere. The Slater determinant is nam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. Use An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations). After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
