Shortly after Schrödinger Equation had been put forward and spectacularly validated for small systems like the H atom, the scientific community, and among them P. Dirac, was so enthusiastic and optimistic to adfirm that Condensed Matter Physics, as well as Chemistry, had come to an end - its content being entirely contained in that powerful equation. Nothing more wrong, as probably Condensed Matter is the field where experimentally observed exceptional phenomena last sometime many decades before to receive a fully convincing theoretical explanation, and many of them are still waiting for at least a qualitative justification. This is the domain where the Theory is far beyond the Experiment.
The problem is that the solution of the Schrödinger Equation:

due to the presence of the electron-electron interaction term w (with w Coulomb, Yukawa or whatsoever interaction), is already not trivial in the case of a 2-electron system like the He atom or the H₂ molecule. Let's imagine the difficulty for a system containing a number N of Avogadro particles... We cannot factorize the hamiltonian in many single-particle hamiltonians and the system wavefunction in a product of N single-particle wavefunctions. So that in principle we have to calculate the full, many-body, system wavefunction, that is a function of N real space variables (the positions of all the particles of the system). This is the Many-Body problem, which occurs not only in Condensed Matter Physics or in Chemistry, but also in Nuclear Physics and wherever you have a system composed of many interacting particles.
Even if we cannot say that the problem has been solved (the Experiment is still going alone on its glorious path, and the Theory is still not offering predictions where to interestingly observe) many progresses have been done. The historical first theories elaborated to solve the problem were Thomas-Fermi (TF) and Hartree-Fock (HF). The Configuration Interaction (CI) is an extreme development of the Hartree-Fock theory. Today there are other, more refined theories who offer a more elegant and in some case also simpler, answer to the problem: The Many-Body Quantum Field Theory (also called Many-Body Perturbation Theory, even if perturbation theory cannot apply safely) and the Density Functional Theory (DFT), with its last development in Time-Dependent Density Functional Theory (TDDFT) to allow for the accounting of excitations.

The authorship and the responsability of the informations hereby provided belongs to Valerio Olevano.