Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2A / Section A / 17 / a / Solution

Writing $A(\mathbf{r})={\varphi }_{1sA}(\mathbf{r})$ and $B(\mathbf{r})={\varphi }_{1sB}(\mathbf{r})$, the normalized determinant isIts two spin orbitals are orthogonal because $⟨\alpha |\beta ⟩=0$, even if the spatial atomic functions have nonzero overlap. Exchanging both the spatial and spin coordinates of the two electrons reverses the sign, as required for fermionic antisymmetry.