I recently saw the Rayleigh Ritz variational approach used in spectral graph theory, so I was curious to look it up again in the quantum mechanics context. Anyway, there was a real sticking point quite quickly... When we pick our trial wave function, because we want our overlap integrals Sij=Sji, we pick real valued basis functions. Moreover, because f(z) = z conjugate is not holomorphic, we choose our constants to be real as well. So our trial wave functions is the sum of real constants and real functions. So my question is: is it true that we can approximate well complex valued wave functions with pure real quantities? It seems really sketchy, and my gut feeling is that this wouldn't work well at all.