Lecture Notes from CHM 1341
2 July 1996


Electron Configurations


If we try to photoionize an atom (as with the photoelectric effect) by raising the photon's energy (lowering its wavelength), we find that the least strongly bound (valence shell) electron will come off first. As the photon energy is raised further, that ejected electron's kinetic energy continues to increase linearly. At some point, we will have a photon of sufficient energy to extract the next most weakly bound electron, and we'll know when that happens because suddenly some really slow electrons will be added to that stream of screamers! As we record these onsets of electron release, we're mapping the values of the atomic energy levels in a technique known as Photoelectron Spectroscopy (PES).

This "drilling" down into the core of atoms reveals the nature and number of occupied energy states. That curious expression means that you can't take an electron out of an energy state it's not occupying, but you could put one in. In fact, the atomic jostling we mentioned before (in flames, for example) which causes electrons occupying low lying energy states to jump up to higher ones, absorbing the energy difference from the flame. Then, since Nature seems to seek energy minima, that excited atomic state is unstable, the electrons drop back down to lower energy states, releasing the energy difference into the line spectra light.

So the energy states are all there but only some of them (the lowest, usually) have their complement of electrons.

The energies discovered by PES confirm what one suspects from the nature of the Periodic Table, as shells fill, new shells arise (with one more node thus a greater reach from the nucleus) starting each period off with alkali metals each with a far out electron, easily ionized and easily donated. And the numerology of the Periodic Table starts to become rational with quantum mechanical explanations like radial vs angular tradeoffs in nodal surfaces giving an ever increasing number of orbitals to each new shell. But quantum mechanics alone doesn't answer all the questions...about the natural occupancy complement of these orbitals.

Pauli postulated and Dirac proved (requiring Einstein's relativity theory) that electrons act as if they are spinning. And since spinning charges generate magnetic fields, electrons and their atoms should be influenced by magnetics. Some are and some aren't; so it appears that some of these electron spins must cancel others by spinning in the opposite direction. It sounds natural that little north-south magnets would align themselves with south-north ones, and so they do unless (a) there's an odd number of them like silver, for instance, with 47, or (b) they're far enough from one another that they respond to other fields in the atom...like the impressive one generated by the nucleus as the electrons circulate it. Oops...that sounds backwards but trust me if A circles B, then it appears to A as if B is circling it too. (Doesn't it really look like the sun goes 'round the earth?) So these "orbital" fields are strong enough that the little electron magnets will align themselves with it if they're not tucked into the same suborbital with another electron.

What Dirac showed was that this "spin" was absolutely necessary by relativity theory.

As an aside, here's a quick and dirty version of the argument. The moon doesn't crash into the earth because it has angular momentum; any reduction in its distance causes its orbital velocity to speed up which throws it back on course. Comforting, no? Relevant too since any electron in an orbital with an angular node HAS angular momentum and cannot get to the nucleus, but the radial noded wonders haven't got that luxury. Indeed those suborbitals permit the electron to have non-zero likelihood of being as close to the nucleus as it cares. Problem! Because the closer it gets the faster it goes until it's ON the nucleus and going infinitely fast! But, says Einstein, nothing can go faster than light. So, says Dirac, the tiny angular momentum of each electron's spin prevents this catastrophe from occurring! Even electrons in orbitals without orbital angular momentum have their spin angular momentum which saves them by forcing them to swing 'round it in a finite radius arc. All this is a Mickey Mouse explanation because it is predicated on the particle view of an electron that is terribly wavelike in atoms. Dirac did it better by solving Schrödinger's equation relativistically.

What Pauli postulated was that no more than two electrons (one spin up and the other down) can occupy any orbital (atomic or molecular)! What Hund postulated was that for orbitals all at the same energy, the electrons will scatter themselves (spin up) to single occupancy as long as there are no more than those obitals number. If there are 5 such orbitals, Hund tells us that anything up to 5 electrons will occupy one each, but the sixth (and subsequent) electron is obliged to share one of the orbitals and (Pauli) have the opposite spin as its partner.

Hund's rule makes sense when one realizes that orbitals of the same energy have the same number of nodes and differ only in the regions of space they occupy. Electrons paired occupy the same region and thus maximize their mutual electrostatic repulsion. That repulsion can be minimized if there are enough orbitals for the electrons to scatter into singly.

So now with Schrödinger, Dirac, Pauli, and Hund, we can tackle the Periodic Table in earnest. The energy (and thus filling) order is determined by the steady march of the increasing nodes to be as follows in hydrogen itself:
1s << 2s=2p < 3s=3p=3d < 4s=4p=4d=4f < 5s=5p=5d=5f=5g < etc.
where the number are the shells and the letters the subshells. The letters' order isn't really meaningful; if follows from early spectroscopists' notation of Sharp, Principal, Diffuse, and Fundamental. These are kept out of tradition since no other notation has replaced it.

What should be immediately noticeable is that this isn't the ordering of the Period Table! The reason is that every atom beyond hydrogen incorporates electron-electron repulsion, which influences the order. However, a look at the Periodic Table confirms the order obtained by photoelectron spectroscopy, namely:
1s << 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d etc.
All that remains is to specify the number of s, p, d, and f orbitals available occupancy, a function of sharing nodes between radial and angular coordinates and Pauli's Principle which yield:
Orbitals                      s   p   d   f  etc.
Max. Suborbitals              1   3   5   7
Max. Electron Occupancy       2   6  10  14
and to note that n=1 (no nodes) can only have s. n=2 (1 node) has s and p. Each increase in n adds one more orbit type so n=4 has s, p, d, and f. There are more orbit types; they get enumerated alphabetically thereafter...so it's s, p, d, f, g, h, etc. But we run out of ground state electrons before we populate the first g orbital, and all our interesting 1st 3 row elements utilize only s and p orbitals.

If we display empty orbitals as boxes such as [ | | ] for the 3 suborbitals for p electrons and label spins as "u" and "d" for "up" and "down," then we can build atoms like:
            1s   2s      2p      3s      3p
Argon  Ar: [ud] [ud] [ud|ud|ud] [ud] [ud|ud|ud]   or [Ar] for short

                    4s         3d
Vanadium  V:  [Ar] [ud] [u |u |u |  |  ]     or  [Ar] 4s2 3d3


Chromium Cr:  [Ar] [u ] [u |u |u |u |u ]     or  [Ar] 4s1 3d5
where Cr (with only one more electron than V) finds is so seductive to half fill the 3d subshell that it steals an electron from the 4s orbital to accomplish this. Notice that it then leaves Cr with a whopping 6 unpaired electrons all aligned in the same direction. It's not surprising that chromium makes fine magnet material.


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Chris Parr University of Texas at Dallas Programs in Chemistry, Room BE3.506 P.O. Box 830688 M/S BE2.6 (for snailmail) Richardson, TX 75083-0688
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Last modified 2 July 1996.