The javascript here finds the colors of all atoms and monatomic ions using the wave equation. You'll have to zoom in with the arrow keys to see the fine structure. At first, just try randomly pressing the three buttons 'new electron,' 'next config,' and 'clear.' Note that you have to re-set the number of electrons to 1 or it will keep getting larger. Red warnings at the end of the display advise which checkboxes to use to resolve multiple term symbols in an electron configuration, to resolve the fine structure or complete the numerical calculation.

For a second tutorial, reload the page to reset everything. Checkmark all the checkboxes on the upper right except 'other' (which is just experimental), and press 'new electron' until it says 'Magnesium.' Now press 'Next Config,' waiting a few seconds each time, until it is no longer flashing red. Now you can use the arrow keys to explore the spectrum! Also press 'compare external' to compare it numerically. With all the checkboxes checked it will be too slow for some ions of high atomic number.

Here is a video showing the javascript react while the perturbation coefficient, which multiplies the electrostatic multipole series, changes for Beryllium changing from 0 to 1; the green are actual which get permuted at the end fullscreen.html?file=quantumvideo.mp4. If you check 'use stored coef' it will use a stored number for this perturbation coefficient called pmult, usually 1 or 1.5.

The coarse correction allows an attempt to correct for the problems the perturbation causes for the sums of reciprocal squares, by artificially increasing the proton charge by a multiple called emult, but only in how it affects levels with differing reciprocal sums of squares (not affecting any spectral frequency which was resolved into a difference if the two electron configurations are the same or even have the same reciprocal sum of squares). Once you have pressed 'next config' until it stops flashing, the value of emult will be such that if the levels are ordered lexicographically giving the reciprocal sum of squares effect precedence, then the second successive differences within each subsequence where the set of l quantum numbers is constant, is minimized in the least squares sense.

The 'visible colors' button shows details about visible spectral lines, but the display will likely be misleading if care is not taken.

You can set pmult, emult and a fine multiplier fmult from the url, as in the examples below which also adjust the width, speed and panning of the spectrograph display to identify some spectral phenomenon from nature. You can also set any of the six checkboxes to be checked or un-checked, their names are docalc, domults, dofine, autocorrect, finecorrect, and usepmults; so you can set usepmults.checked=false in the url if you want it to come up unchecked by default. Also you can put for instance nextt2(6);nextt2(6);nextt2(6) for the first three Carbon electron configurations to be loaded at the outset.

To make functions in the webworker available in a browser's developer console, type "loadWorkerAsScript()" and this also loads things like a polynomial algebra script. Many of the functions are intuitively named, so casimirL("1s22p3") is the rotational casimir of the configuration 1s22p3, and times(A,B) is the product of matrices A,B.

Examples:

Here are the electromagnetic equations making the red and green Northern Lights index.html?allowForbidden=true&autocorrect.checked=false&panLeft=160&scaleWidth=4&panLeftSpeed=-.002&scaleWidthSpeed=.15&dofine.checked=true&nextt(8) They represent transitions between even functions.

Here is the green Magnesium star triplet, and press 'next config' a few times after the calculation is definitely done, to see the yellow one. index.html?panLeft=160&scaleWidth=4&panLeftSpeed=-.002&scaleWidthSpeed=.15&emult=4.5&autocorrect.checked=false&usepmults.checked=false&dofine.checked=true&finecorrect.checked=false&nextt2(12)&nextt2(12)&nextt2(12). Both are reversed because the p orbital is more than half full.

Here is the familiar bright yellow Sodium doublet index.html?autocorrect.checked=false&dofine.checked=true&nextt2(11)&nextt2(11)&panLeft=280&scaleWidth=20&panLeftSpeed=-3&scaleWidthSpeed=1.

And the Carbon spectrum, perhaps reminiscent of fire, index.html?panLeft=280&scaleWidth=20&panLeftSpeed=1&scaleWidthSpeed=.1&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6)&nextt2(6).

The arrow keys are for looking closer, and 'compare external' compares with what is actually found in nature

Details of the calculation are in 'view source' or in even more detail in chemistry.pdf
Also here is a link to other mathematical papers