Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 60 / c / v / Solution

The real symmetric quadratic form is positive for every nonzero vector of occupied-virtual rotation parameters precisely when its matrix $X$ is positive definite. Then the Hartree-Fock determinant is a strict local minimum within the considered real orbital-rotation space. A negative eigenvalue gives an unstable lowering direction; a zero eigenvalue requires higher-order analysis. Stability in this space does not by itself prove global minimality or stability under more general spin/complex-orbital variations.