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ATOMIC AND MOLECULAR SYSTEMS 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 Limit is always above this exact energy. The difference is called the ''correlation energy'', a term coined by Löwdin . A certain amount of electron correlation is already considered within the HF approximation, found in the 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 with opposite spin due to their Coulomb repulsion. There is also a correlation related to the overall symmetry or total spin of the considered system. CRYSTALLINE SYSTEMS In Condensed Matter Physics , electrons are typically described with reference to a periodic lattice of atomic nucleii. Non-interacting electrons are therefore typically described by Bloch Waves rather than the orbitals used in atomic and molecular systems. A number of important theoretical approximations have been proposed to explain electron correlations in these crystalline systems. The Fermi Liquid model of correlated electrons in metals is able to explain the temperature dependence of resistivity by electron-electron interactions. Superconductivity is the result of electron correlations. The Hubbard Model model is based on the 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. The 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. The Tomonaga Luttinger Liquid model approximates second order electron-electron interactions as bosonic interactions. MATHEMATICAL VIEWPOINT For two independent electrons ''a'' and ''b'', :, where ''ρ(ra,rb)'' represents the joint electronic density, or the probability of finding electron ''a'' at ''ra'' and electron ''b'' at ''rb''. Within this notation, ''ρ(ra,rb)dradrb'' represents the probability of finding the two electron in the respective volume elements ''dra'' and ''drb''. 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, and too small at large distances - the electrons tend to "avoid each other". SEE ALSO
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