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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, a Slater determinant is an expression which describes the wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 of a multi-fermionic system that satisfies anti-symmetry
Skew-symmetric matrix

In linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:...
  requirements and subsequently the Pauli exclusion principle
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
 by changing sign
Plus and minus signs

The plus and minus signs are mathematical symbols used to represent the notions of Negative and non-negative numbers as well as the operations of addition and subtraction....
 upon exchange of fermions. It is named for its discoverer, John C. Slater
John C. Slater

John Clarke Slater was a noted American physics and theoretical chemistry.Slater studied at the University of Rochester, earning his B.S. in 1920....
 who published Slater determinants as a means of ensuring the antisymmetry of a wave function through the use of matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
. 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
Spin-orbital

In quantum mechanics, a spin-orbital is a one-particle wavefunction taking both the position and spin of a particle as its parameters.The spinorbital of a single electron, for example, is a complex-valued function of four real variables: the three scalars used to define its position, and a fourth scalar, ms, which can be eit...
, , where denotes the position and spin of the singular electron; two electrons within the same spin orbital resulting in no wave function.

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.






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In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, a Slater determinant is an expression which describes the wavefunction
Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a mathematical space that maps the possible states of the system into the complex numbers....
 of a multi-fermionic system that satisfies anti-symmetry
Skew-symmetric matrix

In linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation:...
  requirements and subsequently the Pauli exclusion principle
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
 by changing sign
Plus and minus signs

The plus and minus signs are mathematical symbols used to represent the notions of Negative and non-negative numbers as well as the operations of addition and subtraction....
 upon exchange of fermions. It is named for its discoverer, John C. Slater
John C. Slater

John Clarke Slater was a noted American physics and theoretical chemistry.Slater studied at the University of Rochester, earning his B.S. in 1920....
 who published Slater determinants as a means of ensuring the antisymmetry of a wave function through the use of matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
. 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
Spin-orbital

In quantum mechanics, a spin-orbital is a one-particle wavefunction taking both the position and spin of a particle as its parameters.The spinorbital of a single electron, for example, is a complex-valued function of four real variables: the three scalars used to define its position, and a fourth scalar, ms, which can be eit...
, , where denotes the position and spin of the singular electron; two electrons within the same spin orbital resulting in no wave function.

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

This expression is used in the Hartree method as an ansatz
Ansatz

Ansatz is a German noun with several meanings in the English language. The fact that the word Ansatz is found in the English language today suggests that it has been carried by those who have used it frequently,, such as mathematicians and physicists....
 for the many-particle wave function and is known as a Hartree product. However, it is not satisfactory for fermions, such as electrons, because the wave function is not antisymmetric. An antisymmetric wave function can be mathematically described as follows:

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

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field , with some generalisations given at the end of the article....
 of both Hartree products

where the coefficient is the normalization factor. 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
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
.

Generalizations


The expression can be generalised to any number of fermions by writing it as a determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
. For an N-electron system, the Slater determinant is defined as

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 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
Laplace expansion

In linear algebra, the Laplace expansion of the determinant ofan n × n square Matrix B expresses the determinant |B| as a sum of n determinants of  ×  sub-matrices of B....
. 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
Hartree-Fock

In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....
. In more accurate theories (such as configuration interaction
Configuration interaction

Configuration interaction is a post Hartree-Fock linear variational method for solving the nonrelativistic Schr?dinger equation within the Born-Oppenheimer approximation for a Quantum chemistry multi-electron system....
 and MCSCF), a linear combination of Slater determinants is needed.

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

See also

  • Antisymmetrizer
    Antisymmetrizer

    In quantum mechanics, an antisymmetrizer is a linear operator that makes a wave function of N identical particles fermions antisymmetric under the exchange of the coordinates of any pair of fermions....
  • Electron orbital
    Electron orbital

    An electron orbital may refer to:* An atomic orbital, describing the behaviour of an electron in an atom* A molecular orbital, describing the behaviour of an electron in a molecule...
  • Quantum Electrodynamics
    Quantum electrodynamics

    Quantum electrodynamics is a relativity theory quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s....
  • Quantum Mechanics
    Quantum mechanics

    Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
     and Physical Chemistry
    Physical chemistry

    Physical chemistry is the application of physics to macroscopic, microscopic, atomic, subatomic, and particulate phenomena in chemical systems within the field of chemistry traditionally using the principles, practices and concepts of thermodynamics, quantum chemistry, statistical mechanics and kinetics....
  • Hund's Rule
  • Hartree-Fock
    Hartree-Fock

    In computational physics and computational chemistry, the Hartree-Fock method is an approximate method for the determination of the Stationary state wavefunction and Stationary state energy of a Many-body problem....