Slater determinant

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, $\chi (\mathbf {x} )$, where $\mathbf {x}$ denotes the position and spin of the singular electron. Two electrons within the same spin orbital result 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 orthogonal 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–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, like that of a fermion. An antisymmetric wave function can be mathematically described as follows:

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})=-\Psi (\mathbf {x} _{2},\mathbf {x} _{1}).$

which does not hold for the Hartree product. 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

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2})={\frac {1}{\sqrt {2}}}\{\chi _{1}(\mathbf {x} _{1})\chi _{2}(\mathbf {x} _{2})-\chi _{1}(\mathbf {x} _{2})\chi _{2}(\mathbf {x} _{1})\}$

$={\frac {1}{\sqrt {2}}}{\begin{vmatrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})\end{vmatrix}}$

where the coefficient is the normalization factor. 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.

Generalizations

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

$\Psi (\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\chi _{1}(\mathbf {x} _{1})&\chi _{2}(\mathbf {x} _{1})&\cdots &\chi _{N}(\mathbf {x} _{1})\\\chi _{1}(\mathbf {x} _{2})&\chi _{2}(\mathbf {x} _{2})&\cdots &\chi _{N}(\mathbf {x} _{2})\\\vdots &\vdots &\ddots &\vdots \\\chi _{1}(\mathbf {x} _{N})&\chi _{2}(\mathbf {x} _{N})&\cdots &\chi _{N}(\mathbf {x} _{N})\end{matrix}}\right|\equiv \left|{\begin{matrix}\chi _{1}&\chi _{2}&\cdots &\chi _{N}\\\end{matrix}}\right|,$

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_i } 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.

See also

• Antisymmetrizer
• Electron orbital
• Fock space
• Quantum electrodynamics
• Quantum mechanics
• Physical chemistry
• Hund's rule
• Hartree–Fock method