|
Restricted Open-shell Hartree–Fock
Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. It uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons. This is the simple picture for open shell molecules but it is difficult to implement. The foundations of the ROHF method were first formulated by Clemens C. J. Roothaan in a celebrated paper and then extended by various authors, see e.g. for in-depth discussions. As with restricted Hartree–Fock theory for closed shell molecules, it leads to Roothaan equations written in the form of a generalized eigenvalue problem :\mathbf \mathbf = \mathbf \mathbf \mathbf Where F is the so-called Fock matrix (which is a function of C), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and \epsilon is the (diagonal, by convention) matrix of orbital energies. Unlike restricted Hartree–Fock theory for closed shell molecules, the form of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartree–Fock Method
In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method often assumes that the exact ''N''-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent (in the case of bosons) of ''N'' spin-orbitals. By invoking the variational method, one can derive a set of ''N''-coupled equations for the ''N'' spin orbitals. A solution of these equations yields the Hartree–Fock wave function and energy of the system. Especially in the older literature, the Hartree–Fock method is also called the self-consistent field method (SCF). In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Shell
In atomic physics and quantum chemistry, the electron configuration is the distribution of electrons of an atom or molecule (or other physical structure) in atomic or molecular orbitals. For example, the electron configuration of the neon atom is , meaning that the 1s, 2s and 2p subshells are occupied by 2, 2 and 6 electrons respectively. Electronic configurations describe each electron as moving independently in an orbital, in an average field created by all other orbitals. Mathematically, configurations are described by Slater determinants or configuration state functions. According to the laws of quantum mechanics, for systems with only one electron, a level of energy is associated with each electron configuration and in certain conditions, electrons are able to move from one configuration to another by the emission or absorption of a quantum of energy, in the form of a photon. Knowledge of the electron configuration of different atoms is useful in understanding the structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Molecular Orbitals
In chemistry, a molecular orbital is a mathematical function describing the location and wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The terms ''atomic orbital'' and ''molecular orbital'' were introduced by Robert S. Mulliken in 1932 to mean ''one-electron orbital wave functions''. At an elementary level, they are used to describe the ''region'' of space in which a function has a significant amplitude. In an isolated atom, the orbital electrons' location is determined by functions called atomic orbitals. When multiple atoms combine chemically into a molecule, the electrons' locations are determined by the molecule as a whole, so the atomic orbitals combine to form molecular orbitals. The electrons from the constituent atoms occupy the molecular orbitals. Mathematically, molecular orbitals are an approximate solution to the Schrödin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clemens C
Clemens is both a Late Latin masculine given name and a surname meaning "merciful". Notable people with the name include: Surname * Adelaide Clemens (born 1989), Australian actress. * Andrew Clemens (b. 1852 or 1857–1894), American folk artist * Aurelius Prudentius Clemens, 4th century Roman poet * Barry Clemens (born 1943), American basketball player * Bert A. Clemens (1874–1935), American politician * Brian Clemens (born 1931), British screenwriter and television producer * Clayton Clemens, American Professor of Government * Dan Clemens (1945–2019), American politician * Gabriel Clemens (born 1983), German darts player * George T. Clemens (1902–1992), American cinematographer * Harold W. Clemens (1918–1998), American politician * C. Herbert Clemens (born 1939), American mathematician * Isaac Clemens (1815–1880), Canadian farmer and politician * Jacob Clemens non Papa (c. 1510 to 1515–1555 or 1556), Franco-Flemish composer of the Renaissance * James Clemens (di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roothaan Equations
The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type. It applies to closed-shell molecules or atoms where all molecular orbitals or atomic orbitals, respectively, are doubly occupied. This is generally called Restricted Hartree–Fock theory. The method was developed independently by Clemens C. J. Roothaan and George G. Hall in 1951, and is thus sometimes called the ''Roothaan-Hall equations''. The Roothaan equations can be written in a form resembling generalized eigenvalue problem, although they are not a standard eigenvalue problem because they are nonlinear: :\mathbf \mathbf = \mathbf \mathbf \mathbf where F is the Fock matrix (which depends on the coefficients C due to electron-electron interactions), C is a matrix of coefficients, S is the overlap matrix of the basis functions, and \epsilon is the (diagonal, by convention) matrix of orbital energies. In the case of an ortho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Eigenvalue Problem
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form :\mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues : p\left(\lambda\right) = \det\le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fock Matrix
In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. It is most often formed in computational chemistry when attempting to solve the Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Because the Fock operator is a one-electron operator, it does not include the electron correlation energy. The Fock matrix is defined by the Fock operator. For the restricted case which assumes closed-shell orbitals and single- determinantal wavefunctions, the Fock operator for the ''i''-th electron is given by:Levine, I.N. (1991) ''Quantum Chemistry'' (4th ed., Prentice-Hall), p.403 :\hat F(i) = \hat h(i)+\sum_^ \hat J_j(i)-\hat K_j(i)/math> where: :\hat F(i) is the Fock operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlap Matrix
In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space. Orbital overlap can lead to bond formation. Linus Pauling explained the importance of orbital overlap in the molecular bond angles observed through experimentation; it is the basis for orbital hybridization. As ''s'' orbitals are spherical (and have no directionality) and ''p'' orbitals are oriented 90° to each other, a theory was needed to explain why molecules such as methane (CH4) had observed bond angles of 109.5°. Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp3 in the case of methane) which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogen orbitals, and can therefore form stronger C–H bonds.Pauling, Linus. (1960). ''The Nature Of The Chemical Bond''. Cornell University Press. A quantitative measure of the overlap of two atomic orbitals ΨA and ΨB o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrestricted Hartree–Fock
Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the Slater determinant, unrestricted Hartree–Fock theory uses different molecular orbitals for the α and β electrons. This has been called a ''different orbitals for different spins'' (DODS) method. The result is a pair of coupled Roothaan equations, known as the Pople–Nesbet–Berthier equations. :\mathbf^\alpha\ \mathbf^\alpha\ = \mathbf \mathbf^\alpha\ \mathbf^\alpha\ :\mathbf^\beta\ \mathbf^\beta\ = \mathbf \mathbf^\beta\ \mathbf^\beta\ Where \mathbf^\alpha\ and \mathbf^\beta\ are the Fock matrices for the \alpha\ and \beta\ orbitals, \mathbf^\alpha\ and \mathbf^\beta\ are the matrices of coefficients for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Contamination
In computational chemistry, spin contamination is the artificial mixing of different electronic spin-states. This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial parts of α and β spin-orbitals are permitted to differ. Approximate wave functions with a high degree of spin contamination are undesirable. In particular, they are not eigenfunctions of the total spin-squared operator, ''Ŝ''2, but can formally be expanded in terms of pure spin states of higher multiplicities (the contaminants). Open-shell wave functions Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals. For an open-shell system, the mean-field approach of Hartree–Fock theory gives rise to different equations for the α and β orbitals. Consequently, there are two approaches that can be taken – either to force double occupation of the lowest orbitals by constraining the α ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Post-Hartree–Fock
In computational chemistry, post-Hartree–Fock methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged. Details In general, the SCF procedure makes several assumptions about the nature of the multi-body Schrödinger equation and its set of solutions: * For molecules, the Born–Oppenheimer approximation is inherently assumed. The true wavefunction should also be a function of the coordinates of each of the nuclei. * Typically, relativistic effects are completely neglected. The momentum operator is assumed to be completely nonrelativistic. * The basis set is composed of a finite number of orthogonal functions. The true wavefunction is a linear combination of functions from a complete (infinite) basis set. * The energy eigenfunctions are assum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
