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| '''Electronic correlation''' is the interaction between [[electron]]s in the [[electronic structure]] of a [[quantum mechanics|quantum]] system.
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| == Atomic and molecular systems ==
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| [[Image:Electron correlation.png|thumb|right|380px|Electron correlation energy in terms of various levels of theory of solutions for the Schrödinger equation.]]
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| Within the [[Hartree–Fock method]] of [[quantum chemistry]], the antisymmetric [[wave function]] is approximated by a single [[Slater determinant]]. Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account Coulomb correlation, leading to a total electronic energy different from the exact solution of the non-relativistic [[Schrödinger equation]] within the [[Born–Oppenheimer approximation]]. Therefore the [[Hartree–Fock method#Variational optimization of orbitals|Hartree–Fock limit]] is always above this exact energy. The difference is called the ''correlation energy'', a term coined by [[Per-Olov Lowdin|Löwdin]].<ref>{{cite journal | first = Per-Olov | last = Löwdin | title = Quantum Theory of Many-Particle Systems. III. Extension of the Hartree–Fock Scheme to Include Degenerate Systems and Correlation Effects | journal = [[Physical Review]] | volume = 97 | issue = 6 | pages = 1509–1520 | publisher = [[American Physical Society]] | date = March 1955 | doi = 10.1103/PhysRev.97.1509 |bibcode = 1955PhRv...97.1509L }}</ref>
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| A certain amount of electron correlation is already considered within the HF approximation, found in the [[exchange interaction|electron exchange]] term describing the correlation between electrons with parallel spin. This basic correlation prevents two parallel-spin electrons from being found at the same point in space and is often called [[Fermi correlation]]. Coulomb correlation, on the other hand, describes the correlation between the spatial position of electrons due to their Coulomb repulsion. There is also a correlation related to the overall symmetry or total spin of the considered system.
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| The word correlation energy has to be used with caution. First it is usually defined as the energy difference of a correlated method relative to the Hartree–Fock energy. But this is not the full correlation energy because some correlation is already included in HF. Secondly the correlation energy is highly dependent on the [[basis set (chemistry)|basis set]] used. The "exact" energy is the energy with full correlation and full basis set.
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| Electron correlation is sometimes divided into dynamical and non-dynamical (static) correlation. Dynamical correlation is the correlation of the movement of electrons and is described under [[electron correlation dynamics]]<ref> J.H. McGuire, "Electron Correlation Dynamics in Atomic Collisions", Cambridge University Press, 1997</ref> and also with the [[configuration interaction]] (CI) method. Static correlation is important for molecules where the ground state is well described only with more than one (nearly-)degenerate determinant. In this case the Hartree–Fock wavefunction (only one determinant) is qualitatively wrong. The [[multi-configurational self-consistent field]] (MCSCF) method takes account of this static correlation but not on the dynamical correlation.
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| If one wants to calculate excitation energies (energy differences between the ground and [[excited state]]s) one has to be careful that both states are equally balanced (e.g., [[Multireference configuration interaction]]).
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| === Methods ===
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| In simple terms the molecular orbitals of the Hartree–Fock method are optimized by evaluating the energy of an electron in each molecular orbital moving in the mean field of all other electrons, rather than including the instantaneous repulsion between electrons.
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| To account for electron correlation there are many [[post-Hartree–Fock]] methods, including:
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| * [[configuration interaction]] (CI)
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| One of the most important methods for correcting for the missing correlation is the [[configuration interaction]] (CI) method. Starting with the Hartree–Fock wavefunction as the ground determinant one takes a linear combination of the ground and excited determinants <math> \Phi_I</math> as the correlated wavefunction and optimizes the weighting factors <math>c_I</math> according to the [[Variational method (quantum mechanics)|Variational]] Principle. When taking all possible excited determinants one speaks of Full-CI. In a Full-CI wavefunction all electrons are fully correlated. For non-small molecules Full-CI is much too computationally expensive. One truncates the CI expansion and gets well-correlated wavefunctions and well-correlated energies according to the level of truncation.
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| * [[Møller–Plesset perturbation theory]] (MP2, MP3, MP4, etc.)
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| Perturbation theory gives correlated energies, but no new wavefunctions. PT is not variational. This means the calculated energy is not an upper bound for the exact energy.
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| * [[multi-configurational self-consistent field]] (MCSCF)
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| There are also combinations possible. E.g. one can have some nearly degenerate determinants for the [[multi-configurational self-consistent field]] method to account for static correlation and/or some truncated CI method for the biggest part of dynamical correlation and/or on top some perturbational ansatz for small perturbing (unimportant) determinants. Examples for those combinations are [[CASPT2]] and SORCI.
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| == Crystalline systems ==
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| In [[condensed matter physics]], electrons are typically described with reference to a periodic lattice of atomic nuclei. Non-interacting electrons are therefore typically described by [[Bloch waves]], which correspond to the delocalized, symmetry adapted molecular orbitals used in molecules (while [[Wannier function]]s correspond to localized MOs). A number of important theoretical approximations have been proposed to explain electron correlations in these crystalline systems.
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| The [[Fermi liquid]] model of correlated electrons in metals is able to explain the temperature dependence of resistivity by electron-electron interactions. It also forms the basis for the [[BCS theory]] of [[superconductivity]], which is the result of phonon-mediated electron-electron interactions.
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| Systems that escape a Fermi liquid description are said to be '''strongly-correlated'''. In them, interactions plays such an important role that qualitatively new phenomena emerge.<ref name="2009-Quintanilla-Hooley">{{Cite journal
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| | last1 = Quintanilla|first1= Jorge |last2= Hooley|first2= Chris
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| | title = The strong-correlations puzzle
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| | journal = [[Physics World]]
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| | year = 2009
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| | volume = 22
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| | pages = 32–37
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| |issn=0953-8585
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| | url = http://physicsworldarchive.iop.org/index.cfm?action=summary&doc=22%2F06%2Fphwv22i06a38%40pwa-xml&qt=
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| }}</ref> This is the case, for example, when the electrons are close to a metal-insulator transition. The [[Hubbard model]] is based on the [[Tight binding (physics)|tight-binding approximation]], and can explain conductor-insulator transitions in [[Mott insulators]] such as [[transition metal oxides]] by the presence of repulsive Coulombic interactions between electrons. Its one-dimensional version is considered an archetype of the strong-correlations problem and displays many dramatic manifestations such as quasi-particle [[fractionalization]]. However there is no exact solution of the Hubbard model in more than one dimension.
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| The [[RKKY|RKKY Interaction]] can explain electron spin correlations between unpaired inner shell electrons in different atoms in a conducting crystal by a second-order interaction that is mediated by conduction electrons.
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| The [[Luttinger liquid|Tomonaga Luttinger liquid]] model approximates second order electron-electron interactions as bosonic interactions.
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| == Mathematical viewpoint ==
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| For two independent electrons ''a'' and ''b'',
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| :<math>\rho(\mathbf{r}_a,\mathbf{r}_b) \sim \rho(\mathbf{r}_a)\rho(\mathbf{r}_b), \, </math>
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| where ''ρ('''r'''<sub>a</sub>,'''r'''<sub>b</sub>)'' represents the joint electronic density, or the probability density of finding electron ''a'' at '''''r'''<sub>a</sub>'' and electron ''b'' at '''''r'''<sub>b</sub>''. Within this notation, ''ρ('''r'''<sub>a</sub>,'''r'''<sub>b</sub>) d'''r'''<sub>a</sub> d'''r'''<sub>b</sub>'' represents the probability of finding the two electrons in their respective volume elements ''d'''r'''<sub>a</sub>'' and ''d'''r'''<sub>b</sub>''.
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| If these two electrons are correlated, then the probability of finding electron ''a'' at a certain position in space depends on the position of electron ''b'', and vice versa. In other words, the product of their independent density functions does not adequately describe the real situation. At small distances, the uncorrelated pair density is too large; at large distances, the uncorrelated pair density is too small (i.e. the electrons tend to "avoid each other").
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| == References ==
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| {{reflist}}
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| == See also ==
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| * [[Configuration interaction]]
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| * [[Coupled cluster]]
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| * [[Hartree–Fock]]
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| * [[Møller–Plesset perturbation theory]]
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| * [[Post-Hartree–Fock]]
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| * [[Quantum Monte Carlo]]
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| * [[Strongly correlated material]]
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| [[Category:Atomic physics]]
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| [[Category:Quantum chemistry]]
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| [[Category:Electron]]
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