References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 5.14.
The rare-earth element dysprosium has atomic number 66, and its ground state is listed as . This means that , (since after , the labels go in alphabetical order, so , and ), and . This isn’t enough information on its own to determine the electron configuration, since the outer shells don’t fill in strict order of and due to shielding effects. For these shells, the shell of level is filled before the shell of level , and the shell of before the of and so on. Given the maximum populations of the various shells ( has a maximum of 2, of 6, of 10 and of 14), a possible configuration for dysprosium is
We can check this against the given values. The last shell () contains 10 out of a possible 14 electrons. According to Hund’s first rule, these should be arranged to give the maximum possible spin, which would mean 3 pairs and 4 unpaired electrons with parallel spin. This gives which matches the listing above. The value of is difficult to check, since it depends on symmetry requirements which would be difficult (though possible, if you’re persistent) to calculate for the 4 unpaired electrons. However, the 4 unpaired electrons in the shell have a maximum possible of , so is certainly possible. Having and , we can apply Hund’s third rule which in this case says that since the last shell is more than half full.