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| In [[quantum mechanics]], a '''Slater determinant''' is an expression that describes the [[wavefunction]] of a multi-[[fermionic]] system that satisfies [[Skew-symmetric matrix|anti-symmetry]] requirements and consequently the [[Pauli exclusion principle]] by changing [[Plus and minus signs|sign]] upon exchange of fermions.<ref name="Atkins">Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> It is named for its discoverer, [[John C. Slater]], who published Slater determinants as a means of ensuring the antisymmetry of a wave function through the use of [[Matrix (mathematics)|matrices]].<ref>{{cite journal| last1=Slater| first1=J.| last2=Verma| first2=HC| title=The Theory of Complex Spectra| journal=Physical Review| volume=34| issue=2| pages=1293–1295| year=1929| pmid=9939750| doi=10.1103/PhysRev.34.1293|bibcode = 1929PhRv...34.1293S }}</ref> The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the [[spin-orbital]], <math>\chi(\mathbf{x})</math>, where <math>\mathbf{x}</math> denotes the position and spin of the singular electron. Two electrons within the same spin orbital result in no wave function.
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| == Resolution ==
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| === Two-particle case ===
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| The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen [[Orthogonality (quantum mechanics)|orthogonal]] wave functions of the individual particles. For the two-particle case with spatial coordinates <math>\mathbf{x}_1</math> and <math>\mathbf{x}_2</math>, we have
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| :<math>
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| \Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2).
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| </math>
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| This expression is used in the [[Hartree–Fock method]] as an [[ansatz]] for the many-particle wave function and is known as a [[Hartree product]]. However, it is not satisfactory for [[fermions]] because the wave function above is not antisymmetric, as it must be for [[fermion]]s from the [[Pauli exclusion principle]]. An antisymmetric wave function can be mathematically described as follows:
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| :<math>
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| \Psi(\mathbf{x}_1,\mathbf{x}_2) = -\Psi(\mathbf{x}_2,\mathbf{x}_1)
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| </math>
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| which does not hold for the Hartree product. Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a [[linear combination]] of both Hartree products
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| :<math>
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| \Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\}
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| </math>
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| :<math>
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| = \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix}
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| </math>
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| where the coefficient is the [[normalization factor]]. This wave function is now antisymmetric and no longer distinguishes between fermions, that is: one cannot indicate an ordinal number to a specific particle and the indices given are interchangeable. Moreover, it also goes to zero if any two wave functions of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.
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| === Generalizations ===
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| The expression can be generalised to any number of fermions by writing it as a [[determinant]]. For an ''N''-electron system, the Slater determinant is defined as <ref name="Atkins" />
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| :<math>
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| \Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) =
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| \frac{1}{\sqrt{N!}}
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| \left|
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| \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\
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| \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\
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| \vdots & \vdots & \ddots & \vdots \\
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| \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N)
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| \end{matrix} \right|\equiv \left| \begin{matrix}
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| \chi _1 & \chi _2 & \cdots & \chi _N \\
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| \end{matrix}
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| \right|,
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| </math>
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| where in the final expression, a compact notation is introduced: the normalization constant and labels for the fermion coordinates are understood – only the wavefunctions are exhibited. The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for ''N'' = 2. It can be seen that the use of Slater determinants ensures an antisymmetrized function at the outset; symmetric functions are automatically rejected. In the same way, the use of Slater determinants ensures conformity to the [[Pauli principle]]. Indeed, the Slater determinant vanishes if the set {χ<sub>i</sub> } is [[linearly dependent]]. In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons can occupy the same spin orbital. In general the Slater determinant is evaluated by the [[Laplace expansion]]. Mathematically, a Slater determinant is an antisymmetric tensor, also known as a [[wedge product]].
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| A single Slater determinant is used as an approximation to the electronic wavefunction in [[Hartree–Fock method|Hartree–Fock theory]]. In more accurate theories (such as [[configuration interaction]] and [[MCSCF]]), a linear combination of Slater determinants is needed.
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| The word "'''detor'''" was proposed by S. F. Boys to describe the Slater determinant of the general type,<ref>{{ cite journal| title=Electronic wave functions I. A general method of calculation for the stationary states of any molecular system|author-link=S. Francis Boys|first=S. F. |last=Boys|page=542| volume=A200 |year=1950|journal= Proceedings of the Royal Society}}</ref> but this term is rarely used.
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| Unlike [[fermions]] that are subject to the Pauli exclusion principle, two or more [[bosons]] can occupy the same quantum state of a system.
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| Wavefunctions describing systems of identical [[bosons]] are symmetric under the exchange of particles
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| and can be expanded in terms of [[permanent]]s.
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| ==See also ==
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| * [[Antisymmetrizer]]
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| * [[Atomic orbital|Electron orbital]]
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| * [[Fock space]]
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| * [[Quantum electrodynamics]]
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| * [[Quantum mechanics]]
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| * [[Physical chemistry]]
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| * [[Hund's rule]]
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| * [[Hartree–Fock method]]
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| * [http://www.cond-mat.de/events/correl13/manuscripts/koch.pdf Many-Electron States] in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6
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| [[Category:Quantum mechanics]]
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| [[Category:Quantum chemistry]]
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| [[Category:Theoretical chemistry]]
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| [[Category:Computational chemistry]]
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| [[Category:Determinants]]
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| [[Category:Pauli exclusion principle]]
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