Movement, yes, but does it explain why they just keep moving about in that cloud?

I don't think so, but I'm open to learning.

The Schoedinger equation is a partial diferential equation. When you solve it, you get the eigenvalues and eigenvectors.

The eigenvalues are the energies. The differences if energies are easy to measure experimentaly looking at the light of discharge lamps. Most of the science of early 20th century is triying to find a theory that explains the colors in that light.

The eigenvectors of the Schoedinger equations are the clouds distributions of the electrons. I'm not sure how difficult is to measure them directly, but they are important to get the size and shape of molecules and the positions of atoms inside crystals and a lot of other stuff.

The same theory that explains the the light of discharge lamps and the electronic clouds also explains the leds in the screen and the transistors in the procesor of yoyr phone.

And dark energy is not used in any of these calcuñations.

No, our current models (the basis for the calculations) do not incorporate dark energy.

Sure, the Schoedinger Equation describes the electron cloud, but I'm pretty sure it doesn't account for why it is moving, and not spiraling into the nucleus. It can't. Put a different way, describing something's movement does not explain why it's moving. Still the SE is as amazing achievement, but it is descriptive of effects, not causes.