There are many terms for a given problem as well as a system of equations for all of the interactions. The answers for even 3 body gravitational problems are chaotic, and gravity is less complication than schrodinger.

However, using numerical methods, symmetry, and observation we can model how many-electron atoms behave.

Here is an example of how the mechanics of a helium atom is determined: http://www.physics.usyd.edu.au/~oliver/PHYS3060/Lectures/PHYS3060-QM11-Handout.pdf

Quantum numbers arise as discrete parameters for solutions to the schrodinger equation as well as being observable. The Pauli exclusion principle is empirically found. We combine concepts together to model a larger atom, and then further to model molecules (Molecular orbital theory).

The Schrodinger equation works well, but it isn't the whole story.

There are many terms for a given problem as well as a system of equations for all of the interactions. The answers for even 3 body gravitational problems are chaotic, and gravity is less complication than schrodinger.

However, using numerical methods, symmetry, and observation we can model how many-electron atoms behave.

Here is an example of how the mechanics of a helium atom is determined.

Quantum numbers arise as discrete parameters for solutions to the schrodinger equation as well as being observable. The Pauli exclusion principle is empirically found. We combine concepts together to model a larger atom, and then further to model molecules (Molecular orbital theory).

The Schrodinger equation works well, but it isn't the whole story.