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| {{Electronic structure methods}}
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| In [[computational physics]] and [[Computational chemistry|chemistry]], the '''Hartree–Fock''' ('''HF''') method is a method of approximation for the determination of the [[wave function]] and the energy of a [[Many-body problem|quantum many-body system]] in a [[stationary state]].
| | There's no question that laptop animation will carry on to prosper, but an exploration of remaining squeezed into extinction in the video video game industry might be warranted. I found it three weeks later on, then once again virtually 4 yrs right after that. With any luck , the remaining retail launch of Titanfall will seem [http://www.mightylittlestartups.com/groups-2/a-track-record-in-no-fuss-clash-of-clans-secrets/ clash of clans hack] device internet site greater, but even the beta test seemed respectable sufficient to be immersive.<br><br>'Brave' is the tale of a younger 11th Century Scottish princess, Merida (Kelly Macdonald) whose spirit is as wild and fiery as her head of unruly pink hair. I introduced ahead a previous distinguished businessman and a trainer. If they are productive, the create new types of social group. Even if you are a hardened vet to the genre, it is really a first rate way to waste a pair hours.<br><br>But with the war and enjoy tales in the movie, do you believe that modern-day audiences can nevertheless relate to it. I then established groups to type a improved perimeter. Our deduced genome and conduct indicate or else. Standing in the sweltering, wet-wool-blanket air of DFW on a Friday morning in July, I commenced to stress at the largely empty condition of my pockets.<br><br>FP: I have a person film that's probably heading to appear out at the end of this 12 months, termed 'Desert Dancer. I was to begin with skeptical about Titanfall, notably considering the fact that quite a few games have experimented with and unsuccessful to do well in the crowded shooter genre, but just after getting my hands on with Titanfall for the duration of the beta examination, I am confident that Titanfall will be a program seller for Xbox A person.<br><br>'Brave' examines the connection among a mom and a daughter and the bond that is at times stretched to the limit involving them. Nanea Reeves, COO if Machinima, says there exists a dismissal of players of pill and cell game titles as "casual. Which is why she stated she wishes she could ride a horse freely.<br><br>This presents every single and each and every specific a very good option to group up with their counterparts from other areas of the world which is also great way to establish even more backlinks. FP: It was exciting, because he's these kinds of an animated actor. If you like trading card games and are receiving exhausted of a lot more competitive and expensive games like Yu-Gi-Oh, but nonetheless want to enjoy some thing as fun, and uncomplicated as Pokemon, try the Naruto TCG.<br><br>Basically you can kind an alliance identified as a Clan with your good friends or just strangers. In the course of the beta take a look at, 3 activity modes were accessible to enjoy: Attrition, Really hard Level, and Past Titan Standing. six million customers, with City - Ville 2, Chef - Ville and Bubble Safari rounding out the leading 5 games. |
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| 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 [[fermion]]s) or by a single [[permanent]] (in the case of [[boson]]s) of ''N'' [[spin-orbital]]s. 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.
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| 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 "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution. The solutions to the non-linear Hartree–Fock equations also behave as if each particle is subjected to the mean field created by all other particles (see the [[Hartree–Fock#The Fock operator|Fock operator]] below) and hence the terminology continued. The equations are almost universally solved by means of an [[Iterative method|iterative]] method, although the [[fixed-point iteration]] algorithm does not always converge.<ref>{{citation
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| |journal = Computer Physics Communication
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| |title = General HF program
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| |last = Froese Fischer | first = Charlotte
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| |volume = 43 | pages =355–365 | year = 1987 }}</ref>
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| This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method.
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| The Hartree–Fock method finds its typical application in the solution of the [[Molecular Hamiltonian|Schrödinger equation]] for atoms, molecules, nanostructures<ref>{{cite journal |first=Mudar A. |last=Abdulsattar |title=SiGe superlattice nanocrystal infrared and Raman spectra: A density functional theory study |journal=[[Journal of Applied Physics|J. Appl. Phys.]] |volume=111 |issue=4 |pages=044306 |year=2012 |doi=10.1063/1.3686610 |bibcode = 2012JAP...111d4306A }}</ref> and solids but it has also found widespread use in [[nuclear physics]]. (See [[Bogoliubov transformation|Hartree–Fock–Bogoliubov method]] for a discussion of its application in [[nuclear structure#Nuclear pairing phenomenon|nuclear structure]] theory). In [[atomic structure]] theory, calculations may be for a spectrum with many excited energy levels and consequently the [[Hartree–Fock method for atoms]] assumes the wave function is a single [[atomic configuration state function|configuration state]] function with well-defined [[quantum numbers]] and that the energy level is not necessarily the [[ground state]].
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| For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately.
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| The rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case.
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| The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals (atomic or molecular) doubly occupied. [[Open shell|Open-shell]] systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods:
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| * [[Restricted open-shell Hartree–Fock]] (ROHF)
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| * [[Unrestricted Hartree–Fock]] (UHF)
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| == Brief history ==
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| The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the [[Schrödinger equation]] in 1926. In 1927 [[Douglas Hartree|D. R. Hartree]] introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s (by E. Fues, [[Robert Bruce Lindsay|R. B. Lindsay]], and himself) set in the [[old quantum theory]] of Bohr.
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| In the [[Bohr model]] of the atom, the energy of a state with [[principal quantum number]] n is given in atomic units as <math> E = -1 / n^2</math>. It was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula. By introducing the [[quantum defect]] ''d'' as an empirical parameter, the energy levels of a generic atom were well approximated by the formula <math> E = -1/(n+d)^2</math>, in the sense that one could reproduce fairly well the observed transitions levels observed in the [[X-ray]] region (for example, see the empirical discussion and derivation in [[Moseley's law]]). The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which clearly does not exist in the isolated hydrogen atom. This repulsion resulted in partial [[screening effect|screening]] of the bare nuclear charge. These early researchers later introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data.
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| Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e., [[ab initio quantum chemistry methods|ab initio]]. His first proposed method of solution became known as the '''Hartree method'''. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, and its connection to the solution of the many-body Schrödinger equation was unclear. However, in 1928 [[John C. Slater|J. C. Slater]] and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the [[variational principle]] to an [[ansatz]] (trial wave function) as a product of single-particle functions.
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| In 1930 Slater and [[Vladimir Fock|V. A. Fock]] independently pointed out that the Hartree method did not respect the principle of [[exchange symmetry|antisymmetry]] of the wave function. The Hartree method used the [[Pauli exclusion principle]] in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of [[quantum statistics]].
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| It was then shown that a [[Slater determinant]], a [[determinant]] of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the [[exchange symmetry|antisymmetric]] property of the exact solution and hence is a suitable [[ansatz]] for applying the [[variational principle]]. The original Hartree method can then be viewed as an approximation to the Hartree–Fock method by neglecting [[exchange symmetry|exchange]]. Fock's original method relied heavily on [[group theory]] and was too abstract for contemporary physicists to understand and implement. In 1935 Hartree reformulated the method more suitably for the purposes of calculation.
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| The Hartree–Fock method, despite its physically more accurate picture, was little used until the advent of electronic computers in the 1950s due to the much greater computational demands over the early Hartree method and empirical models. Initially, both the Hartree method and the Hartree–Fock method were applied exclusively to atoms, where the spherical symmetry of the system allowed one to greatly simplify the problem. These approximate methods were (and are) often used together with the [[central field approximation]], to impose that electrons in the same shell have the same radial part, and to restrict the variational solution to be a spin eigenfunction. Even so, solution by hand of the Hartree–Fock equations for a medium sized atom were laborious; small molecules required computational resources far beyond what was available before 1950.
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| == Hartree–Fock algorithm ==
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| The Hartree–Fock method is typically used to solve the time-independent Schrödinger equation for a multi-electron atom or molecule as described in the [[Born–Oppenheimer approximation]]. Since there are no known solutions for many-electron systems ([[Hydrogen atom|hydrogenic atoms]] and the diatomic hydrogen cation being notable one-electron exceptions), the problem is solved numerically. Due to the nonlinearities introduced by the Hartree–Fock approximation, the equations are solved using a nonlinear method such as [[iteration]], which gives rise to the name "self-consistent field method."
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| === Approximations ===
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| The Hartree–Fock method makes five major simplifications in order to deal with this task:
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| * The [[Born–Oppenheimer approximation]] is inherently assumed. The full molecular wave function is actually a function of the coordinates of each of the nuclei, in addition to those of the electrons.
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| * Typically, [[Special Relativity|relativistic]] effects are completely neglected. The [[momentum|momentum operator]] is assumed to be completely non-relativistic.
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| * The variational solution is assumed to be a [[linear combination]] of a finite number of [[basis set (chemistry)|basis functions]], which are usually (but not always) chosen to be [[orthogonal]]. The finite basis set is assumed to be approximately [[completeness|complete]].
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| * Each [[energy eigenfunction]] is assumed to be describable by a single [[Slater determinant]], an antisymmetrized product of one-electron wave functions (i.e., orbitals).
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| * The [[mean field theory|mean field approximation]] is implied. Effects arising from deviations from this assumption, known as [[electron correlation]], are completely neglected for the electrons of opposite spin, but are taken into account for electrons of parallel spin.<ref>{{cite book |title=Modelling Molecular Structures|last=Hinchliffe|first=Alan|authorlink= |edition=2nd |year=2000|publisher=John Wiley & Sons Ltd|location=Baffins Lane, Chichester, West Sussex PO19 1UD, England|isbn=0-471-48993-X |page=186|url= |accessdate=}}</ref><ref name="Szabo">{{cite book
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| | last1 = Szabo
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| | first1 = A.
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| | last2= Ostlund| first2= N. S.
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| | title = Modern Quantum Chemistry
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| | publisher = Dover Publishing | year = 1996 | location = Mineola, New York
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| | isbn = 0-486-69186-1}}</ref> (Electron correlation should not be confused with electron exchange, which is fully accounted for in the Hartree–Fock method.)<ref name="Szabo"/>
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| Relaxation of the last two approximations give rise to many so-called [[post-Hartree–Fock]] methods.
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| [[Image:Hartree-Fock.png|thumb|right|325px|Greatly simplified algorithmic flowchart illustrating the Hartree–Fock method]]
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| === Variational optimization of orbitals ===
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| The [[Variational method (quantum mechanics)|variational theorem]] states that for a time-independent Hamiltonian operator, any trial wave function will have an energy [[expectation value]] that is greater than or equal to the true [[ground state]] wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground state energy of a given molecule. In the context of the Hartree–Fock method, the best possible solution is at the ''Hartree–Fock limit''; i.e., the limit of the Hartree–Fock energy as the basis set approaches [[completeness]]. (The other is the ''full-CI limit'', where the last two approximations of the Hartree–Fock theory as described above are completely undone. It is only when both limits are attained that the exact solution, up to the Born–Oppenheimer approximation, is obtained.) The Hartree–Fock energy is the minimal energy for a single Slater determinant.
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| The starting point for the Hartree–Fock method is a set of approximate one-electron wave functions known as ''[[spin-orbital]]s''. For an [[atomic orbital]] calculation, these are typically the orbitals for a hydrogenic atom (an atom with only one electron, but the appropriate nuclear charge). For a [[molecular orbital]] or crystalline calculation, the initial approximate one-electron wave functions are typically a [[linear combination of atomic orbitals]] (LCAO).
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| The orbitals above only account for the presence of other electrons in an average manner. In the Hartree–Fock method, the effect of other electrons are accounted for in a [[mean-field theory]] context. The orbitals are optimized by requiring them to minimize the energy of the respective Slater determinant. The resultant variational conditions on the orbitals lead to a new one-electron operator, the [[Fock operator]]. At the minimum, the occupied orbitals are eigensolutions to the Fock operator via a [[unitary transformation]] between themselves. The Fock operator is an effective one-electron Hamiltonian operator being the sum of two terms. The first is a sum of kinetic energy operators for each electron, the internuclear repulsion energy, and a sum of nuclear-electronic [[Coulomb's law|Coulombic]] attraction terms. The second are Coulombic repulsion terms between electrons in a mean-field theory description; a net repulsion energy for each electron in the system, which is calculated by treating all of the other electrons within the molecule as a smooth distribution of negative charge. This is the major simplification inherent in the Hartree–Fock method, and is equivalent to the fifth simplification in the above list.
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| Since the Fock operator depends on the orbitals used to construct the corresponding [[Fock matrix]], the eigenfunctions of the Fock operator are in turn new orbitals which can be used to construct a new Fock operator. In this way, the Hartree–Fock orbitals are optimized iteratively until the change in total electronic energy falls below a predefined threshold. In this way, a set of self-consistent one-electron orbitals are calculated. The Hartree–Fock electronic wave function is then the Slater determinant constructed out of these orbitals. Following the basic postulates of quantum mechanics, the Hartree–Fock wave function can then be used to compute any desired chemical or physical property within the framework of the Hartree–Fock method and the approximations employed.
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| == Mathematical formulation ==
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| === The Fock operator ===
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| {{Main|Fock matrix}}
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| Because the electron-electron repulsion term of the [[electronic molecular Hamiltonian]] involves the coordinates of two different electrons, it is necessary to reformulate it in an approximate way. Under this approximation, (outlined under [[Hartree–Fock#Hartree–Fock algorithm|Hartree–Fock algorithm]]), all of the terms of the exact Hamiltonian except the nuclear-nuclear repulsion term are re-expressed as the sum of one-electron operators outlined below, for closed-shell atoms or molecules (with two electrons in each spatial orbital).<ref name=Levine403>Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Englewood Cliffs, New Jersey: Prentice Hall. p. 403. ISBN 0-205-12770-3.</ref> The "(1)" following each operator symbol simply indicates that the operator is 1-electron in nature.
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| ::<math>\hat F[\{\phi_j\}](1) = \hat H^{\text{core}}(1)+\sum_{j=1}^{N/2}[2\hat J_j(1)-\hat K_j(1)]</math>
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| where
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| ::<math>\hat F[\{\phi_j\}](1)</math>
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| is the one-electron Fock operator generated by the orbitals <math>\phi_j</math>, and
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| ::<math>\hat H^{\text{core}}(1)=-\frac{1}{2}\nabla^2_1 - \sum_{\alpha} \frac{Z_\alpha}{r_{1\alpha}}</math>
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| is the one-electron core [[Hamiltonian (quantum mechanics)|Hamiltonian]]. Also
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| ::<math>\hat J_j(1)</math>
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| is the [[Coulomb operator]], defining the electron-electron repulsion energy due to each of the two electrons in the ''j''th orbital.<ref name=Levine403/> Finally
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| ::<math>\hat K_j(1)</math>
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| is the [[exchange operator]], defining the electron exchange energy due to the antisymmetry of the total n-electron wave function.<ref name=Levine403/> Finding the Hartree–Fock one-electron wave functions is now equivalent to solving the eigenfunction equation:
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| ::<math>\hat F(1)\phi_i(1)=\epsilon_i \phi_i(1)</math>
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| where <math>\phi_i\;(1)</math> are a set of one-electron wave functions, called the Hartree–Fock molecular orbitals.
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| === Linear combination of atomic orbitals ===
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| {{main|basis set (chemistry)|basis set}}
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| Typically, in modern Hartree–Fock calculations, the one-electron wave functions are approximated by a [[linear combination of atomic orbitals]]. These atomic orbitals are called [[Slater-type orbital]]s. Furthermore, it is very common for the "atomic orbitals" in use to actually be composed of a linear combination of one or more [[Gaussian orbital|Gaussian-type orbitals]], rather than Slater-type orbitals, in the interests of saving large amounts of computation time.
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| Various [[basis set (chemistry)|basis sets]] are used in practice, most of which are composed of Gaussian functions. In some applications, an orthogonalization method such as the [[Gram–Schmidt process]] is performed in order to produce a set of orthogonal basis functions. This can in principle save computational time when the computer is solving the [[Roothaan equations|Roothaan–Hall equations]] by converting the [[overlap matrix]] effectively to an [[identity matrix]]. However, in most modern computer programs for molecular Hartree–Fock calculations this procedure is not followed due to the high numerical cost of orthogonalization and the advent of more efficient, often sparse, algorithms for solving the [[generalized eigenvalue problem]], of which the [[Roothaan equations|Roothaan–Hall equations]] are an example.
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| == Numerical stability ==
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| [[Numerical stability]] can be a problem with this procedure and there are various ways of combating this instability. One of the most basic and generally applicable is called ''F-mixing'' or damping. With F-mixing, once a single electron wave function is calculated it is not used directly. Instead, some combination of that calculated wave function and the previous wave functions for that electron is used—the most common being a simple linear combination of the calculated and immediately preceding wave function. A clever dodge, employed by Hartree, for atomic calculations was to increase the nuclear charge, thus pulling all the electrons closer together. As the system stabilised, this was gradually reduced to the correct charge. In molecular calculations a similar approach is sometimes used by first calculating the wave function for a positive ion and then to use these orbitals as the starting point for the neutral molecule. Modern molecular Hartree–Fock computer programs use a variety of methods to ensure convergence of the Roothaan–Hall equations.
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| == Weaknesses, extensions, and alternatives ==
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| Of the five simplifications outlined in the section "Hartree–Fock algorithm", the fifth is typically the most important. Neglecting electron correlation can lead to large deviations from experimental results. A number of approaches to this weakness, collectively called [[post-Hartree–Fock]] methods, have been devised to include electron correlation to the multi-electron wave function. One of these approaches, [[Møller–Plesset perturbation theory]], treats correlation as a [[perturbation theory|perturbation]] of the Fock operator. Others expand the true multi-electron wave function in terms of a linear combination of Slater determinants—such as [[multi-configurational self-consistent field]], [[configuration interaction]], [[quadratic configuration interaction]], and [[multi-configurational self-consistent field#Complete active space SCF|complete active space SCF (CASSCF)]]. Still others (such as [[Variational Monte Carlo|variational quantum Monte Carlo]]) modify the Hartree–Fock wave function by multiplying it by a correlation function ("Jastrow" factor), a term which is explicitly a function of multiple electrons that cannot be decomposed into independent single-particle functions.
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| An alternative to Hartree–Fock calculations used in some cases is [[density functional theory]], which treats both exchange and correlation energies, albeit approximately. Indeed, it is common to use calculations that are a hybrid of the two methods—the popular B3LYP scheme is one such [[hybrid functional]] method.
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| Another option is to use [[modern valence bond]] methods.
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| == Software packages ==
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| For a list of software packages known to handle Hartree–Fock calculations, particularly for molecules and solids, see the [[list of quantum chemistry and solid state physics software]].
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| == See also ==
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| {{multicol}}
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| '''Related fields'''
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| * [[Quantum chemistry]]
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| * [[Computational chemistry]]
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| * [[Computational physics]]
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| * [[Quantum chemistry computer programs]]
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| {{multicol-break}}
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| '''Concepts'''
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| * [[Born–Oppenheimer approximation]]
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| * [[Roothaan equations]]
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| * [[Koopmans' theorem]]
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| * [[Post-Hartree–Fock]]
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| * [[DIIS|Direct Inversion of Iterative Subspace]]
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| {{multicol-break}}
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| '''People'''
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| * [[Douglas Hartree]]
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| * [[Vladimir Aleksandrovich Fock]]
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| * [[Clemens C. J. Roothaan|Clemens Roothaan]]
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| * [[George G. Hall]]
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| * [[John Pople]]
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| * [[Reinhart Ahlrichs]]
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| {{multicol-end}}
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| == References ==
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| {{Reflist}}
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| == Sources ==
| |
| *{{cite book
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| | last = Levine | first = Ira N.
| |
| | title = Quantum Chemistry |edition = 4th
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| | publisher = Prentice Hall | year = 1991 | location = Englewood Cliffs, New Jersey
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| | pages = 455–544 | isbn = 0-205-12770-3}}
| |
| *{{cite book
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| | last = Cramer | first = Christopher J.
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| | title = Essentials of Computational Chemistry
| |
| | publisher = John Wiley & Sons, Ltd. | year = 2002 | location = Chichester
| |
| | pages = 153–189 | isbn = 0-471-48552-7}}
| |
| *{{cite book
| |
| | last1 = Szabo
| |
| | first1 = A.
| |
| | last2= Ostlund|first2= N. S.
| |
| | title = Modern Quantum Chemistry
| |
| | publisher = Dover Publishing | year = 1996 | location = Mineola, New York
| |
| | isbn = 0-486-69186-1}}
| |
| | |
| == External links ==
| |
| * [http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html Introduction by C. David Sherrill]
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| <!-- Comp Chem not redundant - that and Quant Chem are sister cats undet Theor Chem -->
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| {{DEFAULTSORT:Hartree-Fock method}}
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| <!-- [[Category:Computational physics]] redundant -->
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| <!-- [[Category:Atomic physics]] redundant -->
| |
| [[Category:Electronic structure methods]]
| |
| [[Category:Quantum chemistry]]
| |
| [[Category:Theoretical chemistry]]
| |
| [[Category:Computational chemistry]]
| |
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