What's a good measure of the Eigen-ness of a wavefunction?
$\hat{F}\psi_i=\varepsilon_i \psi_i \implies \langle\phi_k|\hat{F}|\psi_i \rangle = \varepsilon_i\langle\phi_k|\psi_i \rangle$.  So the Fock matrix $F_{ki}$ can be approximated by the product of orbital energy $\varepsilon_i$ and the overlap matrix $S_{ki}$. The closer these two values are, the higher the "Eigen-ness".

Of course $\phi_k$ and $\psi_i$ have to be non-orthogonal, so $\phi_k$ could be, for example, a basis function or an MO for another molecule.  I've used this to approximate intermolecular Fock matrix elements here.