Slater determinant - Citizendia

In quantum mechanics, a Slater determinant is an expression which describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry ( $Φ = 0$ ) requirements and subsequently the Pauli exclusion principle by changing sign upon exchange of fermions. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 The plus and minus signs ( + and &minus) are Mathematical symbols used to represent the notions of positive and negative as well as the operations 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 matrices. John Clarke Slater (1900-1976 was a noted American physicist and theoretical chemist. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally ^[1]^[2] 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, $\chi(\mathbf{x})$ , where $\mathbf{x}$ denotes the position and spin of the singular electron; two electrons within the same spin orbital resulting in no wave function (Pauli's Exclusion Principle). 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

1 Resolution
- 1.1 Two-particle case
- 1.2 Generalizations
2 See also
3 References

Resolution

Two-particle case

The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen wave functions of the individual particles. For the two-particle case, we have

$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2).$

This expression is used in the Hartree method as an ansatz for the many-particle wave function and is known as a Hartree product. In physics and mathematics an ansatz ( Ger, "anset onset" today "setup" plural Ansätze) is an educated guess that is However, it is not satisfactory for fermions, such as electrons, because the wave function is not antisymmetric. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. An antisymmetric wave function can be mathematically described as follows:

$\Psi(\mathbf{x}_1,\mathbf{x}_2) = -\Psi(\mathbf{x}_2,\mathbf{x}_1).$

Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products

$\Psi(\mathbf{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\}$

$= \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix}$

where the coefficient is the normalization factor. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics The concept of a normalizing constant arises in Probability theory and a variety of other areas of Mathematics. This wave function is antisymmetric and no longer distinguishes between fermions. Moreover, it also goes to zero if any two wave functions or two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925

Generalizations

The expression can be generalised to any number of fermions by writing it as a determinant. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n For an N-electron system, the Slater determinant is defined as

$\Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \left| \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots && \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{matrix} \right|.$

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. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Indeed, the Slater determinant vanishes if the set {χ_i } is linearly dependent. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors 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. In Linear algebra, the Laplace expansion of the Determinant ofan n × n square matrix B expresses the determinant Mathematically, a Slater determinant is an antisymmetric tensor, also known as a wedge product.

A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree-Fock theory. In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed. Configuration interaction ( CI) is a Post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Multi-configurational self-consistent field (MCSCF is a method in Quantum chemistry used to generate qualitatively correct reference states of molecules in cases where

The word "detor" was proposed by S. F. Boys to describe the Slater determinant of the general type,^[3] but this term is rarely used.

References

^ Slater, John. In Quantum mechanics, an antisymmetrizer \mathcal{A} (also known as antisymmetrizing operator) is a linear operator that makes a wave function of N identical Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Physical chemistry, is the application of Physics to macroscopic microscopic atomic subatomic and particulate phenomena in chemical systems It is mostly defined as a large In Atomic physics, Hund's rules refer to a simple set of rules used to determine which is the Term symbol that corresponds to the ground state of a multi- Electron In Computational physics and Computational chemistry, the Hartree-Fock ( HF) method is an approximate method for the determination of the ground-state C. (1929). Theory of Complex Spectra Physics. Review. vol. 34 Retrieved on 13 August 2007 from "APS Physics: Physics Review Online Archive" on http://prola.aps.org/abstract/PR/v34/i10/p1293_1
^ Slater, John. Events 3114 BC - According to the Lounsbury correlation the start of the Maya calendar. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. C. (1929). , p 1293
^ 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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Contents

Resolution

Two-particle case

Generalizations

See also

References