Chemistry and The Periodic Table

Let us consider the consequences of the Pauli exclusion principle for the chemical properties of atoms: for each allowed wave funtion (or ``orbital'') of an electron in the Coulomb potential, fully specified by the quantum numbers , there are two (and only two) possible spin states for the electron: spin ``up'' () and spin ``down'' (). Since the Pauli principle only excludes occupation of exactly the same state by two electrons, this means that each orbital may be occupied by at most two electrons; and if both electrons are present in that orbital, their spins are necessarily ``paired'' - equal and opposite in direction, thus perfectly cancelling each other and contributing nothing to the net angular momentum or magnetic moment.

Determining the chemical properties of the elements thus becomes a simple matter of counting states: for each ``shell'' (n) we see how many different orbitals there are with different values of and and then multiply by two for the two possibilities to get the total number of electrons that will ``fit'' into that shell. Of course, this simple picture contains some crude approximations that we must expect to break down before things get very complicated, and they do. We assume, for example, that for a positive charge of +Ze on the nucleus, each time we ``add another electron'' it goes into the lowest available (unfilled) energy state and effectively reduces the effective charge of the nucleus by one. That is, the first electron ``sees'' a positive charge of +Ze but the second electron ``sees'' a positive charge of , and so on. This is not true, of course, for electrons in the same n shell; they all pretty much see the same apparent charge on the ``core'' of the atom (and see each others' charges as well). One important effect of such ``screening'' is that states of higher (whose wavefunctions do not penetrate as deeply into the core) ``see'' on average a lower effective charge on the core and therefore are slightly less bound (higher energy) than their low- neighbours. This provides the rule that (up to a point) lower states are filled first. A collection of states with the same n and is called a ``subshell.''

If we want to do this right for an arbitrary many-electron atom, we have several very difficult problems to solve: first, for large Z the innermost electrons have kinetic energies comparable to their rest mass energy; thus our nonrelativistic Schr´'odinger equation is inadequate and one must resort to more advanced descriptions. Second, the approximation that one shell simply ``screens'' part of the nucleus' charge from the next shell is not that great, especially for wavefunctions that have a significant probability density near the nucleus; moreover, electrons in the same shell definitely ``see'' each other and are affected by that interaction. One way of approaching this problem is to start with an initial guess based upon this approximation and then recalculate each electron wavefunction including the effects of the other electrons in their wavefunctions; then go back and correct again with the new guess as a starting point. Eventually this ``Hartree-Fock'' method should converge on the actual wavefunctions for all the electrons.... Finally, the coupling of orbital and spin angular momentum contributions can follow several possible scenarios, depending upon which couplings are the strongest; the final result must always be a single overall net angular momentum (and magnetic moment) for the atom as a whole; but for large atoms it is very difficult to predict even its magnitude; the only thing we can be sure of is that if there are an even number of electrons, protons and neutrons making up the atom, it will be a boson.

Having issued these caveats, we can go back to our ``state counting'':

• For each ``shell'' characterized by an energy (starting at n = 1, where n is the principle quantum number) we have n possible values of , ranging from to .
• For each orbital quantum number , we have possible values of the magnetic quantum number , ranging from to .
• For each orbital specified by , there are two possible values of : spin up () and spin down ().

Since (at least for small n) it is a long way (in energy) between shells, a ``closed shell'' is especially stable - it ``likes'' neither to give up an electron nor to acquire an extra one from another atom; it has ``zero valence.'' This means an atom with electrons [helium] is very unreactive chemically, and so is one with electrons [neon]. This goes on until the breakdown of our assumption that all the electrons for one shell are filled in before any electrons from the next shell. This breakdown comes at the subshell.

Section 9.1 of the text provides all the details you need to explore the periodic table on your own; it also offers at least a qualitative explanation for the fact that the subshell is actually lower in energy than the subshell and therefore fills first, despite having a higher n. This is the point at which our simple qualitative rules break down and we must rely on empirical information to further understand the chemical properties of the elements.

• Chemical Reactions