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Encyclopedia > Ab initio quantum chemistry methods

Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. Computational chemistry is a branch of chemistry that uses the results of theoretical chemistry incorporated into efficient computer programs to calculate the structures and properties of molecules and solids, applying these programs to real chemical problems. ... Linus Pauling, as a pioneer of the valence bond theory, is one of the first quantum chemists. ...


The simplest type of ab initio electronic structure calculation is the Hartree-Fock (HF) scheme, in which the Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect is included in the calculation. This is a variational procedure, therefore the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree-Fock limit as the size of the basis is increased. Many types of calculations begin with a Hartree-Fock calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation. Møller-Plesset perturbation theory (MPn) and coupled cluster theory (CC) are examples of these post-Hartree-Fock methods. In some cases, particularly for bond breaking processes, the Hartree-Fock mehtod is inadequate and this single-determinant reference function is not a good basis for post-Hartree-Fock methods. It is then necessary to start with a wave function that includes more than one determinant such as Multi-configurational self-consistent field and methods have been developed that use these multi-determinant references for improvements. The current version of the article or section reads like an advertisement. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... The variational method is, in quantum mechanics, one way of finding approximations to the lowest energy eigenstate or ground state. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared... 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. ... Electronic correlation refers to the interaction between electrons in a quantum system whose electronic structure is being considered. ... Møller-Plesset perturbation theory is an implementation of perturbation theory in quantum chemistry, which provides a method for adding excitations to the Hartree-Fock wavefunction and therefore including the effect of electron correlation. ... This article needs to be cleaned up to conform to a higher standard of quality. ... 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 for diagonalizing the electronic Hamiltonian describing the electronic structure of molecules. ... In computational chemistry, the Multi-configurational self-consistent field or MCSCF method is a post-Hartree-Fock method which uses a linear combination of CSFs to approximate the true electronic wavefunction of an atom or molecule. ...


Almost always the basis set (which is usually built from the LCAO ansatz) used to solve the Schrödinger equation is not complete, and does not span the Hilbert space associated with ionization and scattering processes (see continuous spectrum for more details). In the Hartree-Fock method and the Configuration interaction method, this approximation allows one to treat the Schrödinger equation as a "simple" eigenvalue equation of the electronic molecular Hamiltonian, with a discrete set of solutions. In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. ... This article may be too technical for most readers to understand. ... Ansatz is a term (from German) often used by physicists. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... ... In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ... Mathematical formulation In mathematics and physics, the spectrum of an operator, in particular self adjoint or normal operators, can be classified via its spectral measures. ... 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. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, is the definition of energy of a quantum system. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... The electronic Hamiltonian for a multi-electron molecule in atomic units is: where is the vector position of electron with vector components in Bohr radii, is the charge of fixed nucleus a in units of the elementary charge, is the vector position of nucleus with vector components in Bohr radii. ... In mathematics and physics, discrete spectrum of an operator on Hilbert space is the part of the spectrum which corresponds to discrete spectral measures. ...

Contents

The most popular classes of ab initio electronic structure methods:


Hartree-Fock methods

  • Hartree-Fock (HF)
  • Restricted Open-shell Hartree-Fock (ROHF)
  • Unrestricted Hartree-Fock (UHF)

In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... Restricted Open-shell Hartree-Fock (ROHF) is a variant of Hartree-Fock theory for open shell molecules. ... 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. ...

Post-Hartree-Fock methods

Møller-Plesset perturbation theory is an implementation of perturbation theory in quantum chemistry, which provides a method for adding excitations to the Hartree-Fock wavefunction and therefore including the effect of electron correlation. ... 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. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Quadratic configuration interaction is an extension of Configuration interaction that corrects for size-consistency errors in the all singles and double excitation CI methods (CISD). ...

Multi-reference mehtods

Ab initio electronic structure methods have the advantage that they can be made to converge to the exact solution, when all approximations are sufficiently small in magnitude. In particular configuration interaction where all possible configurations are included (called "Full CI") tends to the exact non-relativistic solution of the Schrödinger equation. The convergence, however, is usually not monotonic, and sometimes the smallest calculation gives the best result for some properties. The downside of ab initio methods is their computational cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales nominally as N4 (N being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete and dash. However in practice it can scale closer to N3 as the program can indentify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably - MP2 as N5; MP4 as N6 and Couple cluster as N7. DFT methods scale similar to Hartree-Fock although they always take longer than an equivalent Hartree-Fock calculation. In computational chemistry, the Multi-configurational self-consistent field or MCSCF method is a post-Hartree-Fock method which uses a linear combination of CSFs to approximate the true electronic wavefunction of an atom or molecule. ... In computational chemistry, the multireference configuration interaction method consists in 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. ... In quantum chemistry, N-Electron Valence state Perturbation Theory (NEVPT) is a perturbative treatment applicable to multireference CASCI-type wavefunctions. ... The current version of the article or section reads like an advertisement. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, is the definition of energy of a quantum system. ... In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ... The current version of the article or section reads like an advertisement. ...


Linear scaling approaches

The problem of computational expense can be alleviated through simplification schemes. In the density fitting scheme, the four-index integrals used to describe the interaction between electron pairs are reduced to simpler two- or three-index integrals, by treating the charge densities they contain in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix "df", for example the density fitting MP2 is df-MP2 (lower-case is advisable to prevent confusion with DFT). In the local orbital approximation, the molecular orbitals, which are formally spread across the entire molecule, are restricted to localised domains. This eliminates the interactions between distant electron pairs and hence sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules. Methods employing this scheme are denoted by the prefix "L", e.g. LMP2. Both schemes can be employed together, as in the recently developed df-LMP2 method. In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... Møller-Plesset perturbation theory is an implementation of perturbation theory in quantum chemistry, which provides a method for adding excitations to the Hartree-Fock wavefunction and therefore including the effect of electron correlation. ... Density functional theory (DFT) is a quantum mechanical method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular molecules and the condensed phases. ... A representation of the 3D structure of myoglobin, showing coloured alpha helices. ...


Valence bond methods

Valence bond (VB) methods are generally ab initio although some semi-empirical versions have been proposed. Current VB approaches are:-

The Generalized Valence Bond (GVB) method is one of the simplest and oldest valence bond method that uses flexible orbitals in the general way used by Modern valence bond theory. ... Modern valence bond theory has been developed by several workers, including Gerratt, Cooper and Raimondi(1997); Li and McWeeny(2002); Song, Mo, Zhang and Wu (2005); and Shaik and Hiberty (2004). ...

Quantum Monte Carlo methods

A method that avoids making the variational overestimation of HF in the first place is Quantum Monte Carlo (QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. Such calculations can be very time-consuming, but they are probably the most accurate methods known today. This article or section is in need of attention from an expert on the subject. ... Monte Carlo methods are a widely used class of computational algorithms for simulating the behavior of various physical and mathematical systems. ...


 
 

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