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In quantum mechanics, a Slater determinant (introduced by the American physicist John C. Slater[1]) is an expression describing the wavefunction of a many-fermion system which, by construction, satisfies the Pauli principle by being antisymmetric under an exchange of any pair of fermions. That is, if one were to exchange any fermion with a different one, the wavefunction would remain unchanged except for a reversal of its overall sign (i.e. it gets a minus sign). Fig. ...
John Clark Slater (1900-1976) was a major physicist and theoretical chemist. ...
This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...
In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ...
The Pauli exclusion principle, commonly referred to simply as the exclusion principle, is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. ...
In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ...
In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ...
In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ...
This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...
The plus and minus signs (+ and −) are used to represent the notions of positive and negative as well as the operations of addition and subtraction. ...
The Slater determinant arises from the consideration of a wavefunction for a collection of electrons. The wavefunction for each individual electron is known as a spin-orbital,  , where  indicates the position and spin of the electron. In quantum mechanics, a spin-orbital is a one-particle wavefunction taking both the position and spin angular momentum of a particle as its parameters. ...
Two-particle case
The simplest way to approximate the wavefunction of a many-particle system is to take the product of properly chosen one-electron wavefunctions of the individual particles. For the two-particle case, we have  This expression is used in the Hartree method as an ansatz for the molecular wavefunction and is known as a Hartree product. However, it is not satisfactory for fermions, such as electrons, because the wavefunction is not antisymmetric. An antisymmetric wavefunction can be mathematically described as follows: Ansatz (Ger. ...
Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ...
 Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products  where the coefficient is a normalization factor. This wavefunction is antisymmetric and no longer distinguishes between electrons. Moreover, it also goes to zero if any two wavefunctions or two electrons are the same. This is equivalent to satisfying the Pauli exclusion principle. The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ...
The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...
Generalization to the Slater determinant 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 In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
 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 assures an antisymmetrized function on the outset, symmetric functions are automatically rejected. In the same way, the use of Slater determinants assures the obeying of the Pauli principle. Indeed, the Slater determinant vanishes if the set {χi } is linearly dependent. In particular this is the case when two (or more) spinorbitals are the same. In chemistry one expresses this fact by stating that no two electrons can occupy the same spinorbital. 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. The Pauli exclusion principle, commonly referred to simply as the exclusion principle, is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. ...
In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...
In linear algebra, the Laplace expansion of the determinant of an n × n square matrix B expresses the determinant |B| as a sum of n determinants of (n-1) × (n-1) sub-matrices of B. There are 2n such expressions, one for each row and column of B. The Laplace...
In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...
A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree-Fock theory. In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed. 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. ...
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 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. ...
The word "detor" was proposed by S. F. Boys to describe the Slater determinant of the general type,[2] but this term is rarely used anymore.
References - ^ J.C. Slater,Theory of Complex Spectra, Phys. Rev. vol. 34, p. 1293 (1929)
- ^ Electronic wave functions I. A general method of calculation for the stationary states of any molecular system, S. F. Boys, p542, A200 (1950), Proc. Roy.Soc. (London).
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