Has anyone figured out how to predict atomic spectra?

Factual Questions
1

(I’ve searched GQ for this already, so I’m reasonably sure this question hasn’t already been posted.)

I have read what information I could find on this subject, and found that there’s already the Rydberg Equation that predicts the spectrum of hydrogen and hydrogen-like ions, which reads:



1    [sup] [/sup]     1     1
- = R Z[sup]2[/sup] (——— - ———)
l          n[sub]l[/sub]    n[sub]u[/sub]

or some variation thereof, where R is the Rydberg constant which is in the vicinity of 109,708 or so. This equation gives wavelengths of spectral lines, which is ultimately what I’m after. However it only works for atoms and ions that have one electron.

Digging deeper, I found that for more complex atoms, there is the Extended Ritz Formula, which calculates energy levels rather than lines themselves:



     [sup] [/sup]     Z[sup]2[/sup]
E[sub]nl[/sub] = R ————————  
       (n - d[sub]l[/sub])[sup]2[/sup]

             a          b
d[sub]l[/sub] = d[sub]0[/sub] + ———————— + ———————— + ...
         (n - d[sub]0[/sub])[sup]2[/sup]   (n - d[sub]0[/sub])[sup]4[/sup]

*Source: Adv. At. Mol. Opt. Phys. 32:93 (1994)*

where d[sub]0[/sub] is the “quantum defect” constant and a, b, … are constants that (I presume) must have something to do with electron structure.

Of course, the wavelength of a line is proportional to the reciprocal of the difference between the upper and lower energy levels.

What I was wondering is, has anyone yet figured out how to calculate the values of d[sub]0[/sub] and a, b, … ? Is there an equation I can use to just plug in an atom’s electron structure and calculate its energy levels and hence its spectrum?

This is something that I’d really like to know, and would probably figure it out eventually but wanted to ask if anybody else already has because that would save me a lot of trouble.

2

Well, someone can correct me if I’m wrong, but I think this is how it works.

There is in general an equation that describes the electron energy levels. It’s the Schwarzschild Equation, a second-order differential equation. You could in theory just plug in the electron structure and get the energy eigenvalues from this. Of course, being a differential equation, it’s not that simple. For neutral hydrogen, it’s solvable, only taking a page or two, and the energy eigenvalues are easy:
[ul]E = E[sub]0[/sub] n[sup]-2[/sup][/ul]For anything more complex than that, it cannot be solved so simply. (It may not even be able to be solved exactly at all.) So it’s easier just to write down what the energy levels are than using the theoretical equation.

3

Mmmm. Eigenvalues.

4

heh, eigenvalues.

“This isn’t rocket science, people.” – my quantum mechanics professor.

Now, this is very slightly off topic, but other molecular spectra have become quite predictable. Infrared spectra can be predicted somewhat accurately by a complex series of matrix equations – I don’t remember exactly, and I won’t bother searching for it, but the equations must be out there somewhere. Essentially, molecules resemble springs obeying Hooke’s law enough that you can predict their vibrational frequencies and intensities. This is most useful for predicting changes to an IR spectrum between slightly different molecules, but there are software programs (Hyperchem, for example) which can predict entire spectra. UV spectra can be predicted in roughly the same way, I suppose, as has already been discussed – the energy difference for possible transitions are predicted, and the results can be plotted.

Predicting NMR spectra is (at least in software) slightly less sophisticated. Chemical shifts are taken from a list of experimental values, and various adjustments are made to each value depending on the proton/carbon atom’s environment (e.g. + 0.02 for a methyl group gamma to a hydroxyl group). Peak height is determined by the number of protons involved (for proton NMR), but multiplicity is not predicted. This is useful for assigning spectra and having a rough idea of what to expect, but isn’t any better, really, than the old method of predicting spectra from paper tables, and isn’t useful experimentally.

5

Good info to go on. Thank you, Achernar and Roches.

Another clue could be the repeated use of (n - d)[sup]x[/sup] which suggests to me that the series might go on indefinitely, and that a, b, … have effects analogous to the charge of the nucleus. According to the article where I found the equations, a, b, … are usually positive for something called core-penetration, and negative for core-polarization. In lithium the [sup]2[/sup]S series can be closely approximated with values of d[sub]0[/sub] = .02984; a = 0.4 exactly, and b=0. So that would appear to be the effect of a pair of 1s electrons.

What I can do, if I have some spare time and Real Life doesn’t get in the way :D, is to play around with using multiples of 0.2 for a, b, … and seeing if there are values of d[sub]0[/sub] that will approximate energy levels for other alkalis.

6

Hmm… first off, Achernar meant the Schrodinger equation, not the Schwarzschild equation. He knew that, I’m sure, but what can I say? He’s an astrophysicist… :stuck_out_tongue:

Predicting atomic spectra with a pencil and paper is basically impossible, but predicting them by solving the Schrodinger equation numerically is actually not too difficult, at least for the first several lines. Things get rather dicey as we go to more and more highly excited states, though.

As far as predicting NMR spectra, if one is willing to do some rather tedious and expensive calculations, one can predict them without resorting to the kind of cheating that was described. The calculations, though, are pretty hideous, and one wouldn’t want to try to predict the NMR spectra of one’s favorite protein or anything.

7

Wow, thanks for pointing that out. I wonder how I made that mistake.

At least I didn’t call it the Schwarzenegger equation…

8

Oh, God, that would have been too much. “Jah, I have invended dhis equation to describe adomic spegdra. Id’s a second-order differential equation, and dhe eigenvalues are dhe energy lebvels. inserd generic bad one-liner here”

9

LOL :smiley: :eek: I wouldn’t dream of trying it out on pencil and paper. Computer is a much better tool for the job.

I had a couple of numbers mixed up; d[sub]0[/sub] should be 0.4 and a ends up being 0.0304578 for lithium. And it turns out the same value of a applies for boron, with b=a and d[sub]0[/sub] = 0.971153. Makes sense given their electron structures, no?

Okay, off to research the Schwarzenegger equation. :slight_smile: Perhaps I can use it to predict the spectrum of Californium. :smiley: