In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system.
It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two fermions.
[1] Only a small subset of all possible many-body fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.
A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere.
The Slater determinant is named for John C. Slater, who introduced the determinant in 1929 as a means of ensuring the antisymmetry of a many-electron wave function,[2] although the wave function in the determinant form first appeared independently in Heisenberg's[3] and Dirac's[4][5] articles three years earlier.
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.
An antisymmetric wave function can be mathematically described as follows: This does not hold for the Hartree product, which therefore does not satisfy the Pauli principle.
This problem can be overcome by taking a linear combination of both Hartree products: where the coefficient is the normalization factor.
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[1][6] where the last two expressions use a shorthand for Slater determinants: The normalization constant is implied by noting the number N, and only the one-particle wavefunctions (first shorthand) or the indices for the fermion coordinates (second shorthand) are written down.
All skipped labels are implied to behave in ascending sequence.
The linear combination of Hartree products for the two-particle case is identical with the Slater determinant for N = 2.
The use of Slater determinants ensures an antisymmetrized function at the outset.
In the same way, the use of Slater determinants ensures conformity to the Pauli principle.
In chemistry one expresses this fact by stating that no two electrons with the same spin can occupy the same spatial orbital.
Many properties of the Slater determinant come to life with an example in a non-relativistic many electron problem.
are the nuclei and For simplicity we freeze the nuclei at equilibrium in one position and we remain with a simplified Hamiltonian where and where we will distinguish in the Hamiltonian between the first set of terms as
The two parts will behave differently when they have to interact with a Slater determinant wave function.
We start to compute the expectation values of one-particle terms In the above expression, we can just select the identical permutation in the determinant in the left part, since all the other N!
at the denominator Because of the orthonormality of spin-orbitals it is also evident that only the identical permutation survives in the determinant on the right part of the above matrix element This result shows that the anti-symmetrization of the product does not have any effect for the one particle terms and it behaves as it would do in the case of the simple Hartree product.
And finally we remain with the trace over the one-particle Hamiltonians Which tells us that to the extent of the one-particle terms the wave functions of the electrons are independent of each other and the expectation value of total system is given by the sum of expectation value of the single particles.
It is important to notice explicitly that the exchange term, which is always positive for local spin-orbitals,[8] is absent in simple Hartree product.
Since exchange bielectronic integrals are different from zero only for spin-orbitals with parallel spins, we link the decrease in energy with the physical fact that electrons with parallel spin are kept apart in real space in Slater determinant states.
The best Slater approximation to a given fermionic wave function can be defined to be the one that maximizes the overlap between the Slater determinant and the target wave function.
[9] The maximal overlap is a geometric measure of entanglement between the fermions.
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 refer to a Slater determinant of orthonormal orbitals,[10] but this term is rarely used.
Unlike fermions that are subject to the Pauli exclusion principle, two or more bosons can occupy the same single-particle quantum state.
Wavefunctions describing systems of identical bosons are symmetric under the exchange of particles and can be expanded in terms of permanents.