Slater determinant

A Slater determinant is an antisymmetric wavefunction built from one-particle orbitals to represent multiple identical fermions. Exchanging any two fermions reverses its sign, enforcing the Pauli exclusion principle.

Antisymmetrization: A procedure that projects a many-particle state onto components that change sign under particle exchange. The determinant is the standard compact construction that antisymmetrizes a product of orbitals.
Quantum state: A mathematical description containing the probabilities and other measurable properties of a quantum system. A Slater determinant is one particular representation of a many-fermion quantum state.
Hartree–Fock method: A self-consistent approximation that optimizes a single Slater determinant for an interacting fermion system. Its central approximation represents the many-electron state by one optimized determinant.
Bosonic permanent: A matrix function formed like a determinant but with all permutation signs positive, used for symmetric particle states. Bosons require exchange-symmetric states, so their orbital products use permanents rather than determinants.
Pauli exclusion principle: No two identical fermions can occupy the same quantum state. Repeated columns make the determinant vanish, forbidding duplicate occupied orbitals.
Hilbert space: A complete inner-product vector space used to represent quantum states. Orbitals and their antisymmetric products inhabit the system's quantum state space.
Configuration interaction: A method that expands a quantum state as a linear combination of Slater determinants. It adds correlation by mixing determinants with different orbital occupations.
Configuration state function: A spin-adapted linear combination of determinants with fixed spatial-orbital occupations. It preserves total-spin symmetry, which a lone determinant may not represent correctly.
Fermion: A particle with half-integer spin whose identical-particle states are antisymmetric under exchange. Slater determinants encode the exchange statistics of these particles.
Inner product: A mathematical operation that measures overlap and angle between vectors in a vector space. Orbital overlaps determine the normalization and overlaps of Slater determinants.

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