Think back to quantum 101 when you calculated the electron energy levels for an isolated hydrogen atom. They were exactly that, discrete levels, which your instructor summarily drew on the board, labelled n=0, n=1, etc etc. Everything was nice, the sun was shining, the birds were singing. Your professor, ever the harbinger of doom then said something like 'helium' and then 'interactions'. However you smiled goofily, assuming he referred to a mixer later that day with balloons; you planned to drop by because you were sweet on a girl called mary. A new set of lines appears on the board, labelled n=0, n=1, etc etc. He even labelled the orbits 4p4d etc and then he says it: 'spin'. This is when you notice out of the corner of your eye a rain cloud forming outside and he draws from the nice discrete levels a set of spidery lines. You hear 'spin splitting', he snaps the chalk, you laugh, although now nervously sensing your impending demise under a sea of symbols and myriad forthcoming interactions. You are now detached from time and space, the rain is falling, the mixer is off. With great alacrity the professor cries 'orbital splitting', more lines go up, another interaction that takes us farther from the simple coulombic electron-nucleus. you weep and then to end the lecture he throws up the spin-orbit interaction, a veritable dose of arsenic to your aspirations for an A. Through the sobs and tears, the diabolical laughter of the now triumphant professor, a sea of broken souls around him, you see that what was once a utopian series of single lines is approaching a set of bands.
The epilogue to our story is solid state physics 401 where you find that in reality the hero of our story, the electron has an energy which belongs to a band in a practical material composed of more than a few atoms. You learn about slater, kohn and sham, you look up some papers on PROLA. Despite the fact that you swore group theory was as useful as a paper umbrella in a hurricane, you can see how Mattheis and others used it to simplify a band structure calculation.
I hope my semi autobiographical rant has helped :)
