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 electrons (or other fermions). Only a small subset of all possible fermionic wave functions can be written as a single Slater determinant, but those form an important and useful subset because of their simplicity.
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 a single electron. 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, although the wave function in the determinant form first appeared independently in Heisenberg's and Dirac's articles three years earlier.
Definition
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 with coordinates and , we have
This expression is used in the Hartree 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 under exchange of any two of the fermions, as it must be according to the Pauli exclusion principle. 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. This wave function is now antisymmetric and no longer distinguishes between fermions (that is, one cannot indicate an ordinal number to a specific particle, and the indices given are interchangeable). Moreover, it also goes to zero if any two spin orbitals of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.
Multi-particle case
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
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. 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 with the same spin can occupy the same spatial orbital.
Example: Matrix elements in a many electron problem
Many properties of the Slater determinant come to life with an example in a non-relativistic many electron problem. The one particle terms of the Hamiltonian will contribute in the same manner as for the simple Hartree product, namely the energy is summed and the states are independent The multi-particle terms of the Hamiltonian, i.e. the exchange terms, will introduce a lowering of the energy of the eigenstates
Starting from an Hamiltonian:
where are the electrons and 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 "1" particle terms) and the last term which is the "2" particle term or exchange term
The two parts will behave differently when they have to interact with a Slater determinant wave function. We start to compute the expectation values
In the above expression, we can just select the identical permutation in the determinant in the left part, since all the other N! − 1 permutations would give the same result as the selected one. We can thus cancel N! at the denominator