I tend to use a lot of Density Functional Theory (DFT) in my research. In short, it comes down to that if you have a molecule or solid, and the positions of the nuclei are given, there is a bijection between the N-electron antisymmetric groundstate wavefunction and the electron density. This means that any information you can get out of the groundstate wavefunction, you can recast in a functional over the electron density.

Now for the practical problem: What we're usually looking for is the energy of the electronic system. So in theory, it's possible to define a functional E[rho] = <psi|H|psi> where rho is the electron density defined for all of space, psi is the N-electron wavefunction and H is the hamiltonian. But E[rho] is not known. Worse, there is no known approximation scheme. There are many educated guesses which work rather well, but none of those are generally expandible to provide a better approximation.

Now, here's my question. How could you go about proving that:

1. There is no closed form expression for E[rho];
2. There exists no approximation scheme for E[rho]?