A linear combination that describes an appropriately antisymmetrized multi-electron wavefunction for any desired orbital configuration is easy to construct for a two-electron system. However, interesting chemical systems usually contain more than two electrons. For these multi-electron systems, a relatively simple scheme for constructing an antisymmetric wavefunction from a product of one-electron functions is to write the wavefunction in the form of a determinant. John Slater introduced this idea so the determinant is called a Slater determinant. The Slater determinant for the two-electron wavefunction of helium is (3.9.27)|ψ⁡(r1,r2)⟩=12⁢|ϕ1⁢s⁡(1)⁢α⁡(1)ϕ1⁢s⁡(1)⁢β⁡(1)ϕ1⁢s⁡(2)⁢α⁡(2)ϕ1⁢s⁡(2)⁢β⁡(2)| We can introduce a shorthand notation for the arbitrary spin-orbital (3.9.28)ϕi⁢α⁡(r)=ϕi⁡α or (3.9.29)ϕi⁢β⁡(r)=ϕi⁡β as determined by the ms quantum number. A shorthand notation for the determinant in Equation 3.9.27 is then (3.9.30)|ψ⁡(r1,r2)⟩=2−12⁢D⁢e⁢t|⁢ϕ1⁢s⁢α⁡(r1)⁢ϕ1⁢s⁢β⁡(r2)| The determinant is written so the electron coordinate changes in going from one row to the next, and the spin orbital changes in going from one column to the next. The advantage of having this recipe is clear if you try to construct an antisymmetric wavefunction that describes the orbital configuration for uranium! Note that the normalization constant is (N!)−12 for N electrons. The generalized Slater determinant for a multi-electron atom with N electrons is then