Slater determinants in the context of Closed shell

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. 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.

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 ${\displaystyle \chi (\mathbf {x} )}$, where ${\displaystyle \mathbf {x} }$ 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.