Slater determinant

In quantum mechanics, a Slater determinant is a simple approximate expression for a wave function of a multi-fermion system. The Slater determinant is constructed from a spin-orbital product. It is the simplest possible way to construct from the spin-orbital product (also known as an independent particle wave function) a wave function that satisfies the Pauli principle. That is, a Slater determinant changes sign upon transposition of the space and spin coordinates of any of its two fermions. It is common to refer also to a Slater determinant as an independent particle wave function, although the Slater determinant has built in some correlation between the motions of the particles. It is named for its inventor, John C. Slater, who published the construction as a simple answer to the complicated group theoretical constructions for antisymmetric wave functions of Hermann Weyl and Eugene Wigner that had been introduced in the 1920s.

Two-particle case

The simplest way to approximate the wavefunction of a many-particle system is to consider the product of properly chosen one-fermion wavefunctions (spin-orbitals) of the individual particles. For the two-particle case, we have

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

where x_i indicates a space and spin coordinate of particle i. This spin-orbital product can be used as an Ansatz for the molecular wavefunction and is known as an independent particle function. However, it is not satisfactory for fermions, such as electrons, because the wavefunction is not antisymmetric as it should according to the Pauli principle. This problem can be overcome by taking a linear combination of two orbital products

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

where the coefficient normalizes the left hand side (provided the spin-orbitals are orthonormal). This wavefunction is antisymmetric and no longer distinguishes between electrons. Moreover, it vanishes if the two spin-orbitals are proportional or if x₁ = x₂. This is equivalent to satisfying the Pauli exclusion principle.

The antisymmetric function can be written as a 2 x 2 determinant (dropping the normalization factor)

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

Generalization to N particles

The expression can be generalized to any number of fermions by writing it as a determinant. For an N-electron system, the Slater determinant is defined as

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

If this determinant is worked out, either by the Laplace expansion, or by the Leibniz rule, it becomes a sum of N! terms, which differ from each other by any of the N! permutations of the particle coordinates. The sign of the term is the parity (or signature) of the permutation. Since the permutation of rows i and j is equivalent to permutation of the coordinates x_i and x_j, and since a determinant changes sign upon permutation of two rows, it follows that a Slater determinants is antisymmetric under transposition of these two coordinates. Moreover determinant theory shows that the Slater determinant vanishes if the set {χ_i } 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 (Pauli exclusion principle). Mathematicians may recognize a Slater determinant as 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.

See also

• Antisymmetrizer