Slater determinants

Multi-particle case

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 ${\displaystyle \{\chi _{i}\}}$ 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.