I must fundamentally misunderstand what the variational method is. According to my textbook, it's used to find the minimum eigen energy of an operator (in particular, the time-independent schrodinger equation). This appears to be synonymous to finding the eigenvalues of the matrix representation of the operator after you operate on some finite basis and project the result back to the same basis. My book doesn't go into detail about how to choose or vary the guesses for the coefficients of the of the basis expansion of the wave function. In the back of my mind, I can't help but think why you wouldn't just do the inverse rayleigh method with shifting, which is a pretty standard way to finding specific eigenvalues. Hopefully what I just wrote isn't too vague. If you need clarification on what I'm talking about, don't hesitate to ask. Edit: I think I kind of see the difference when the expansion functions used aren't the actual eigenfunctions of the operator. I have to think carefully about what the cross terms mean though.. I'm guessing the minimum eigenvalue of the operator matrix when there are cross terms present is not the same as the minimum eigenvalue of the operator in general.