What's a good measure of the Eigen-ness of a wavefunction?

Just wondering if someone can recommend an easy-to-compute measure for the Eigen-ness of a wave function, such as the wave functions computed using ab initio method?

For a wave function solved using a small basis set, the Eigen-ness should be smaller than that of wave function computed using larger basis set.

Naturally for arbitrary wavefunction $|\psi>$ one can compute : $<\psi | (H/E-1)^2 | \psi>$, where E is the expectation value, which is the dispersion of the energy. This, however, is too difficult to implement.

• Can you clarify this a bit? I'm not sure I understand what you're asking. – Todd Minehardt Jun 13 '17 at 22:37
• As to a measure of Eigen-ness, it's meant to quantify how close a candidate wavefunction is to the actual eigen state of the Hamiltonian. For example, we know the exact solution of Hydrogen atom ground state, but if approximating it with a GTO, we measure how good the approximating function is by finding out the overlapping integral. But for general moleculare wavefunction, we don't know what the exact eignen state is, in this case, what's the best way of evaluating the candidate? – wang1908 Jun 13 '17 at 22:50
• For variational methods, you can know for sure that your wavefunction is more Eigen if it has a lower energy, but AFAIK there's no way to know how close or far you are from the maximally Eigen function. – hBy2Py Jun 14 '17 at 4:00
• The energy converges with basis set size. So you can compare how much it changes e.g. from double to triple to quadruple zeta. You can even extrapolate the complete basis set limit from this. – Feodoran Jun 14 '17 at 6:46

$\hat{F}\psi_i=\varepsilon_i \psi_i \implies \langle\phi_k|\hat{F}|\psi_i \rangle = \varepsilon_i\langle\phi_k|\psi_i \rangle$. 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.
• could you use an auxiliary basis for the set of $k$? – Fl.pf. Jun 14 '17 at 6:21
• I don't know of any right off hand. However, you don't need to use another set of basis functions. The basis functions that you use to describe $\psi_i$ are non-orthogonal to $\psi_i$ itself – Jan Jensen Jun 15 '17 at 6:31