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I've been digging into emission spectra of different elements and found that such things as the Rydberg equation, Bohr's model, and quantum mechanics can only fully describe the single electron in the Hydrogen atom. How did we then make the leap to s,p,d,f shells of multi-electron atoms? How accurate is our analysis of these more complicated elements?

Rydberg Equation (side-note: Is this an empirical 'data-fitting' equation? What's the significance of that?)

$$\frac{1}{\lambda}=R_H\left( \frac{1}{n_1^2}-\frac{1}{n_2^2}\right)$$

Hydrogen: enter image description here

Helium: enter image description here

Iron: enter image description here

Potassium: enter image description here

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I guess in this case it would be important to establish the distinction between "analytic" analyses and numerical analyses. –  Justin L. Jun 24 '13 at 7:12

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The only atoms for which the Schrodinger equation has an analytic solution are the one electron atoms i.e. H, He$^+$, Li$^{2+}$ and so on. That's because with more than one electron the forces between electrons make the equation too hard to solve analytically. However, over the 90 or so years since Schrodinger proposed his equation a vast array of numerical methods for solving it have been developed, and of course modern computers are so powerful they can calculate the (electronic) structure of any atom with ease. This applies even to heavy atoms where relativistic effects need to be taken into account.

The Rydberg equation is an approximation because it does not take the electronic fine structure into account. However it's a pretty good approximation. It works because for a one electron atom the energy of the orbitals (ignoring fine structure) is proportional to 1/$n^2$, where $n = 1$ is the lowest energy orbital, $n = 2$ is the second lowest and so on.

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Just to elaborate this a little, I know people who have really honed the art of numerically solving the Schrodinger equation. They can more or less directly solve up to the 7 body problem with an arbitrary potential to machine precision - so for practical purposes there are no approximations necessary. It takes a lot of computer time though - a single run takes a month or so on their cluster. For larger atoms/molecules there are of course a gaggle of sophisticated approximation techniques which are probably what you are referring to. –  Michael Brown Jun 24 '13 at 7:58
    
The Rydberg equation and Schrodinger equation are identical approximations of the energy levels of H, that do not account for fine structure. The Sommerfeld equation and Dirac equation account for the fine structure. Sommerfeld in 1916 had the correct fine structure equation even before the Schrodinger equation (which doesn't account for fine structure) was developed. uw.physics.wisc.edu/~knutson/phy448/wilson-sommerfeld.pdf –  DavePhD May 3 at 15:21

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