Finding energy of an electron analytically

[foota]

How would you find the energy of an electron in a given orbital in an element? I'm pretty sure from what I understand that you could excite the electron and measure the wavelength of light given off and find the difference, but is there some formula that at least approximates this energy?

 

[nnnm4]

Formally, one may solve the Schrödinger equation. For complex elements (i.e. elements that have more than one electron present) numerical methods become necessary.

 

[alxm]

It can't be done analytically for anything other than hydrogen. (And some fake atoms like 'harmonium') There are approximations for helium that can be hand-calculated, but even those are either inaccurate or tedious.

There are several whole fields of science (e.g. quantum chemistry, solid-state physics) largely devoted to finding better and faster approximations of this. So obviously: no simple "formula" is known. It's a many-body problem, so there are some mathematical reasons to believe no such simple formula exists, either. (among other things, it'd likely prove P = NP) But people are developing new methods of calculating and approximating the solutions every day, as well as applying the existing methods to learning new stuff.