Two-level variational contamination estimate ($|c_b/c_a{|}^2\simeq (\tilde{E}-E_a)/(E_b-\tilde{E})$)

If a trial quantum state is dominated by two energy eigenstates of distinct energies $E_a<E_b$, then its Rayleigh quotient is approximately their probability-weighted mean. Rearranging that mean gives the displayed relative admixture. Neglected states must make small contributions to the energy numerator as well as to wavefunction normalization; small probabilities alone need not suffice if their energies are large.