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Full Configuration Interaction
Full configuration interaction (or full CI) is a linear variational approach which provides numerically exact solutions (within the infinitely flexible complete basis set) to the electronic time-independent, non-relativistic Schrödinger equation. Explanation It is a special case of the configuration interaction method in which ''all'' Slater determinants (or configuration state functions, CSFs) of the proper symmetry are included in the variational procedure (i.e., all Slater determinants obtained by exciting all possible electrons to all possible virtual orbitals, orbitals which are unoccupied in the electronic ground state configuration). This method is equivalent to computing the eigenvalues of the electronic molecular Hamiltonian within the basis set of the above-mentioned configuration state functions. In a minimal basis set a full CI computation is very easy. But in larger basis sets this is usually just a limiting case which is not often attained. This is because ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variational Method (quantum Mechanics)
In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle. The method consists of choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method, density matrix renormalization group, and Ritz method apply the variational method. Description Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H . Ignoring complications about continuous spectr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Davidson Correction
The Davidson correction is an energy correction often applied in calculations using the method of truncated configuration interaction, which is one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry. It was introduced by Ernest R. Davidson. It allows one to estimate the value of the full configuration interaction energy from a limited configuration interaction expansion result, although more precisely it estimates the energy of configuration interaction up to quadruple excitations (CISDTQ) from the energy of configuration interaction up to double excitations (CISD). It uses the formula :\Delta E_Q = (1 - a_0^2)(E_ - E_), \ :E_ \approx E_ + \Delta E_Q, \ where ''a''0 is the coefficient of the Hartree–Fock wavefunction in the CISD expansion, ''E''CISD and ''E''HF are the energies of the CISD and Hartree–Fock wavefunctions respectively, and Δ''EQ'' is the correction to estimate ''E''CISDTQ, the energy of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Nicholas C
Nicholas is a male name, the Anglophone version of an ancient Greek name in use since antiquity, and cognate with the modern Greek , . It originally derived from a combination of two Greek words meaning 'victory' and 'people'. In turn, the name means "victory of the people." The name has been widely used in countries with significant Christian populations, owing in part to the veneration of Saint Nicholas, which became increasingly prominent in Western Europe from the 11th century. Revered as a saint in many Christian denominations, the Eastern Orthodox, Catholic, and Anglican Churches all celebrate Saint Nicholas Day on December 6. In maritime regions throughout Europe, the name and its derivatives have been especially popular, as St Nicholas is considered the protector saint of seafarers. This remains particularly so in Greece, where St Nicholas is the patron saint of the Hellenic Navy. Origins The name derives from the . It is understood to mean 'victory of the people', bei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Møller–Plesset Perturbation Theory
Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post-Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset. Rayleigh–Schrödinger perturbation theory The MP perturbation theory is a special case of RS perturbation theory. In RS theory one considers an unperturbed Hamiltonian operator \hat_, to which a small (often external) perturbation \hat is added: :\hat = \hat_ + \lambda \hat. Here, ''λ'' is an arbitrary real parameter that controls the size of the perturbation. In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator, which thus serves as the unperturbed operator. The perturbation is the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multireference Configuration Interaction
In quantum chemistry, the multireference configuration interaction (MRCI) method consists of a configuration interaction expansion of the eigenstates of the electronic molecular Hamiltonian in a set of Slater determinants which correspond to excitations of the ground state electronic configuration but also of some excited states. The Slater determinants from which the excitations are performed are called reference determinants. The higher excited determinants (also called configuration state functions (CSFs) or shortly configurations) are then chosen either by the program according to some perturbation theoretical ansatz according to a threshold provided by the user or simply by truncating excitations from these references to singly, doubly, ... excitations resulting in MRCIS, MRCISD, etc. For the ground state using more than one reference configuration means a better electronic correlation, correlation and so a lower energy. The problem of size consistency, size inconsistency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Basis Set (chemistry)
In theoretical chemistry, theoretical and computational chemistry, a basis set is a set of Function (mathematics), functions (called basis functions) that is used to represent the Wave function, electronic wave function in the Hartree–Fock method or Density functional theory, density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer. The use of basis sets is equivalent to the use of an approximate resolution of the identity: the Atomic orbital, orbitals , \psi_i\rangle are expanded within the basis set as a linear combination of the basis functions , \psi_i\rangle \approx \sum_\mu c_ , \mu\rangle, where the expansion coefficients c_ are given by c_ = \sum_\nu \langle \mu, \nu \rangle^ \langle \nu , \psi_i \rangle. The basis set can either be composed of atomic orbitals (yielding the linear combination of atomic orbitals approach), which is the usual choice within the qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Electronic 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] |
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Eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |