The condition $$ \Delta E = \frac{1}{2} \langle \chi | \hat{M}| \chi \rangle \ge 0 $$ for an arbitrary vector $$ \chi = \left[ \delta C\hspace{0.2cm} \delta C^*\right]^T $$ means that all eigenvalues of the matrix have to be larger than or equal zero. A necessary (but no sufficient) condition is that the matrix elements (for all \( ai \) ) $$ (\varepsilon_a-\varepsilon_i)\delta_{ab}\delta_{ij}+A_{ai,bj} \ge 0. $$ This equation can be used as a first test of the stability of the Hartree-Fock equation.