Slater determinant
Encyclopedia
In quantum mechanics
, a Slater determinant is an expression that describes the wavefunction
of a multi-fermionic system that satisfies anti-symmetry
requirements and consequently the Pauli exclusion principle
by changing sign
upon exchange of fermions . 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
. 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,
, where 
denotes the position and spin of the singular electron. Two electrons within the same spin orbital result in no wave function.
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, 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; that is to say: on the one hand, the interchange of fermions must give rise to negation of the wave function because the fermions are different, yet on the other hand, they should still be indistinguishable. This problem can be overcome by taking a linear combination
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 of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle
.
. 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 ensures an antisymmetrized function on 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 {χi } 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.
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.
The word "detor" was proposed by S. F. Boys to describe the Slater determinant of the general type, but this term is rarely used.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, a Slater determinant is an expression that describes the wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...
of a multi-fermionic system that satisfies anti-symmetry
Skew-symmetric matrix
In mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...
requirements and consequently the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
by changing sign
Plus and minus signs
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous...
upon exchange of fermions . It is named for its discoverer, John C. Slater
John C. Slater
John Clarke Slater was a noted American physicist who made major contributions to the theory of the electronic structure of atoms, molecules and solids. This work is of ongoing importance in chemistry, as well as in many areas of physics. He also made major contributions to microwave electronics....
, 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, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. 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,
, where denotes the position and spin of the singular electron. Two electrons within the same spin orbital result in no wave function.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 haveThis expression is used in the Hartree–Fock method as an ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
for the many-particle wave function and is known as a Hartree product
Hartree product
The Hartree product is a system wavefunction, given as a combination of wavefunctions of the individual particle wavefunctions. It is used as an ansatz in the Hartree-Fock method.For two particles,...
. 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; that is to say: on the one hand, the interchange of fermions must give rise to negation of the wave function because the fermions are different, yet on the other hand, they should still be indistinguishable. This problem can be overcome by taking a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
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 of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
.
Generalizations
The expression can be generalised to any number of fermions by writing it as a determinantDeterminant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
. 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 ensures an antisymmetrized function on 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 {χi } 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, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant |B| of...
. 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 more accurate theories (such as configuration interaction
Configuration interaction
Configuration interaction 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. Mathematically, configuration simply describes the linear combination...
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
- Electron orbitalElectron orbitalAn 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- See also :...
- Quantum electrodynamicsQuantum electrodynamicsQuantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
- Quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
- Physical chemistryPhysical chemistryPhysical chemistry is the study of macroscopic, atomic, subatomic, and particulate phenomena in chemical systems in terms of physical laws and concepts...
- Hund's rule
- Hartree–Fock method