One could rightfully state that the atomic-structure problem has been around for a very long time. Yes, this is true - in fact quantum mechanics has been invented to explain atomic properties. Then why do we still struggle to solve it?

Should we be embarrassed by our inability to solve this basic problem? Sure we can solve it approximately, but solving it accurately is another story.

So what is holding us back? It is the very same entanglement and complexity of Hilbert space (that is where wave functions live) that makes quantum computing so powerful. To illustrate this enormous complexity, I'll take my favorite atom, cesium. It has 55 electrons. With 3 degrees of freedom per electron (x, y, and z), the Cs wave function depends on coordinates. As a result of the calculation I would need to store the wave function. If I  were to take a very poor grid of 10 points per coordinate, the storage would require $latex 10^{155}$ memory units.

$latex 10^{155}$ is of course an astronomically large number - in fact it exceed an estimated number of atoms in the Universe, . So even if we were able to compute the Cs wave function, there is no plausible way to store it in usable form.