SlaterDeterminanten

A Slater determinant is an antisymmetric N-particle wavefunction constructed as the determinant of a matrix of spin-orbitals evaluated at the coordinates of each particle. If φ_j(x) denotes the j-th spin-orbital and x_i denotes the spatial and spin coordinates of particle i, then the wavefunction Ψ(x1,...,xN) = (1/√N!) det [ φ_j(x_i) ] with i and j indexing particles and orbitals, respectively.

The determinant construction guarantees antisymmetry under exchange of any two particles: swapping two coordinates changes the sign of Ψ. If two spin-orbitals are identical, two rows or columns of the matrix become equal, yielding zero, which enforces the Pauli exclusion principle.

A common two-electron example is Ψ(r1,σ1; r2,σ2) = (1/√2) [ φ_a(r1,σ1) φ_b(r2,σ2) − φ_b(r1,σ1) φ_a(r2,σ2) ], where φ_a and φ_b are spin-orbitals. This explicit form shows how spatial and spin degrees of freedom are entangled in a single antisymmetric state.

Slater determinants are central in quantum chemistry and many-body physics. They provide a compact way to describe fermionic systems and form the basis of methods such as Hartree–Fock, where a single determinant is used, and configuration interaction or multi-configurational approaches, which combine multiple determinants to capture electronic correlation. They can be spin-adapted to obtain states with definite total spin.

In second quantization, a Slater determinant corresponds to a product state created by applying a set of fermionic creation operators to the vacuum, and its properties follow from the anti-commutation relations of these operators, often analyzed with Wick’s theorem.