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Fock Operator
The Fock matrix is defined by the Fock operator. In its general form the Fock operator writes: :\hat F(i) = \hat h(i)+\sum_^ hat J_j(i)-\hat K_j(i)/math> Where ''i'' runs over the total ''N'' spin orbitals. In the closed-shell case, it can be simplified by considering only the spatial orbitals. Noting that the \hat J terms are duplicated and the exchange terms are null between different spins. 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 for the ''i''-th electron in the system, :(i) is the one-electron Hamiltonian 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-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 two, two, and six electrons, respectively. Electronic configurations describe each electron as moving independently in an orbital, in an average field created by the nuclei and all the other electrons. Mathematically, configurations are described by Slater determinants or configuration state functions. According to the laws of quantum mechanics, a level of energy is associated with each electron configuration. 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 structure o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic Orbital
In quantum mechanics, an atomic orbital () is a Function (mathematics), function describing the location and Matter wave, wave-like behavior of an electron in an atom. This function describes an electron's Charge density, charge distribution around the Atomic nucleus, atom's nucleus, and can be used to calculate the probability of finding an electron in a specific region around the nucleus. Each orbital in an atom is characterized by a set of values of three quantum numbers , , and , which respectively correspond to electron's energy, its angular momentum, orbital angular momentum, and its orbital angular momentum projected along a chosen axis (magnetic quantum number). The orbitals with a well-defined magnetic quantum number are generally complex-valued. Real-valued orbitals can be formed as linear combinations of and orbitals, and are often labeled using associated Spherical harmonics#Harmonic polynomial representation, harmonic polynomials (e.g., ''xy'', ) which describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb Operator
The Coulomb operator, named after Charles-Augustin de Coulomb, is a quantum mechanical operator used in the field of quantum chemistry. Specifically, it is a term found in the Fock operator. It is defined as: : \widehat J_j (1) f_i(1)= f_i(1) \int ^2 \frac\,dr_2 where : \widehat J_j (1) is the one-electron Coulomb operator defining the repulsion resulting from electron ''j'', : f_i(1) is the one-electron wavefunction of the i^ electron being acted upon by the Coulomb operator, : \varphi_j(2) is the one-electron wavefunction of the j^ electron, : r_ is the distance between electrons (i) and (j) . See also * Core Hamiltonian * Exchange operator In quantum mechanics, the exchange operator \hat, also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles descri ... References {{DEFAULTSORT:Coulomb Operator Quantum chemis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exchange Operator
In quantum mechanics, the exchange operator \hat, also known as permutation operator, is a quantum mechanical operator that acts on states in Fock space. The exchange operator acts by switching the labels on any two identical particles described by the joint position quantum state \left, x_1, x_2\right\rangle. Since the particles are identical, the notion of exchange symmetry requires that the exchange operator be unitary. Construction In three or higher dimensions, the exchange operator can represent a literal exchange of the positions of the pair of particles by motion of the particles in an adiabatic process, with all other particles held fixed. Such motion is often not carried out in practice. Rather, the operation is treated as a "what if" similar to a parity inversion or time reversal operation. Consider two repeated operations of such a particle exchange: :\hat\left, x_1, x_2\right\rangle = \left, x_2, x_1\right\rangle :\hat^2\left, x_1, x_2\right\rangle = \ ... [...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 method is named after Douglas Hartree and Vladimir Fock. 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. Hartree–Fock approximation is an instance of mean-field theory, where neglecting higher-order fluctuations in order parameter allows interaction terms to be replaced with quadratic terms, obtaining exactly solvable Hamiltonians. Especially ... [...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] |
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 \mathbf is the so-called Fock matrix (which is a function of \mathbf), \mathbf is a matrix of coefficients, \mathbf 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atomic, Molecular, And Optical Physics
Atomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions, at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO theory includes classical, semi-classical and quantum treatments. Typically, the theory and applications of emission, absorption, scattering of electromagnetic radiation (light) from Excited state, excited atoms and molecules, analysis of spectroscopy, generation of lasers and masers, and the optical properties of matter in general, fall into these categories. Atomic and molecular physics Atomic physics is the subfield of AMO that studies atoms as an isolated system of electrons and an atomic nuclei, atomic nucleus, while molecular physics is the study of the physical properties of molecules. The term ''atomic physics'' is often associated with nuclear power and nuclear bombs, due to the synonymous use of ''atomic'' and ''nuclear'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of Molecule, molecules, Material, materials, and solutions at the atomic level. These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed Wave function, wave functions as well as to observable properties such as structures, spectra, and Thermodynamics, thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics. Chemists rely heavily on spectroscopy through which information regarding the Quantization (physics), quantization of energy on a molecular scale can be obtained ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
